Nonlinear discontinuous Petrov–Galerkin methods

Abstract

The discontinuous Petrov–Galerkin method is a minimal residual method with broken test spaces and is introduced for a nonlinear model problem in this paper. Its lowest-order version applies to a nonlinear uniformly convex model example and is equivalently characterized as a mixed formulation, a reduced formulation, and a weighted nonlinear least-squares method. Quasi-optimal a priori and reliable and efficient a posteriori estimates are obtained for the abstract nonlinear dPG framework for the approximation of a regular solution. The variational model example allows for a built-in guaranteed error control despite inexact solve. The subtle uniqueness of discrete minimizers is monitored in numerical examples.

1 Introduction

The discontinuous Petrov–Galerkin methodology (dPG) has recently been introduced with the intention to design the optimal test spaces in a Petrov–Galerkin scheme for maximal stability. On the continuous level, the weak form of a PDE may assume the general form $b(u,\cdot )=F$ with a unique solution u in some real Banach space X and some bilinear form $b:X\times Y\rightarrow {\mathbb {R}}$ for some real Hilbert space Y with scalar product $a: Y \times Y \rightarrow {\mathbb {R}}$ and a given right-hand side $F\in Y^*$, the dual to Y. Well-posedness is understood to lead to an inf-sup condition on the continuous level. Given some discrete trial space $X_h\subset X$, the restriction $b|_{X_h\times Y}$ clearly satisfies the inf-sup condition (even with a possibly slightly better inf-sup constant) but it is less clear how to choose the best trial space $M_h$, i.e. some subspace, $M_h\subset Y$ such that

$$\begin{aligned} 0<\beta (X_h,M_h):= \mathop {{{\mathrm{\phantom {p}inf}}}}\limits _{x_h\in X_h} \sup _{y_h\in M_h} \frac{b(x_h,y_h)}{\Vert x_h \Vert _X \Vert y_h \Vert _Y} \end{aligned}$$

(1.1)

is maximal under the condition that $\dim (X_h) = \dim (M_h)$ is fixed. The idealized dPG method computes the optimal test space utilizing some Riesz representations in the infinite-dimensional Hilbert space Y [18]. The practical realization utilizes, first, a test-search space $Y_h\subset Y$ with dimension $n = \dim (Y_h)$ much larger than the dimension $m = \dim (X_h)$ of the trial space $X_h$ and, second, a minimal residual method to compute the discrete solution as a minimizer

(1.2)

The method is in fact equivalent to a Petrov–Galerkin scheme with the bilinear form restricted to $X_h\times M_h$ for an appropriate subspace $M_h \subset Y_h$ of dimension m as pointed out in [8, Thm. 3.3]. Therefore, the large discrete space $Y_h$ (which is an input of the dPG scheme) is called test-search space [17] and the (implicit) test space $M_h$ is not visible in (1.2).

The computation of $x_h$ in (1.2) is equivalent to solving the normal equations and so possibly expensive. This guided Demkowicz and Gopalakrishnan [19] to break the norms in the test (and ansatz) spaces [6]. This allows a parallel computation of the dual norm separately for each individual element domain. As it stands today, the term dPG abbreviates “discontinuous Petrov–Galerkin” and stands for a minimal residual method with broken test or ansatz functions and solely outlines a paradigm. The dPG methodology allows various weak and ultra-weak formulations, where X and Y are completely different and b is not at all symmetric. The least-squares finite element methods can be seen as a (degenerated) subset of (an idealized) dPG with a degenerated test space in which the Lebesgue norm can be evaluated exactly.

To the best knowledge of the authors, not much is known about nonlinear versions of the methodology. One first choice is to linearize the problem and then apply the dPG schemes to the linear equations to generalize the Gauss-Newton method. There exist already suggestions for nonlinear applications, in which there are constraints plus a linear problem, e.g., for the contact problem in [21]. Concepts of nonlinear dPG in fluid mechanics have been discussed in [16]. Another usage of the term nonlinear is in nonlinear approximation theory and there is the contribution [22] on linear problems with an attempt to replace the Hilbert space Y by some uniformly convex Banach space.

This paper introduces a direct nonlinear dPG methodology and replaces the above bilinear form b by some nonlinear mapping $b:X\times Y\rightarrow {\mathbb {R}}$, which is linear and bounded in the second component to allow the computation of the dual norm in the minimal residual method. To stress the nonlinear dependence in the first component in X, the notation in this papers follows [24] and separates the linear components by a semi-colon so that the nonlinear dPG method replaces in (1.2) by .

The simplest case study for the nonlinear dPG methodology is an energy minimization problem with some Hilbert space setting and a nonlinearity with quadratic growth in the gradient. The scalar model example of this paper stands for a larger class of Hencky materials [28, Sect. 62.8] and is the first model problem in line towards real-life applications with a matrix-valued stress $\sigma (F)$ given as a nonlinear function of some deformation gradient F (such as the gradient $\nabla u$ of the displacement u) and the remaining equilibration equation

$$\begin{aligned} f+{{\mathrm{div}}}\sigma (\nabla u)=0 \quad \text {a.e. in }\varOmega \end{aligned}$$

(1.3)

for some prescribed source term f in the domain $\varOmega$. Although the existence of discrete solutions $x_h$ to (1.2) follows almost immediately, the closeness of $x_h$ to some continuous solution x is completely open (cf. Remark 2.11 below for a brief discussion).

One critical point is the role of the stability condition (1.1) in the nonlinear setting for a regular solution and its low-order discretizations (as the most natural first choice for nonlinear problems, partly because of limited known regularity properties). In the situation of the model scenario (1.3), the discrete stability follows from the stability of the continuous form for piecewise constant $\nabla u_h$ and so the local discrete stability simply follows from the linearization.

The overall structure of the nonlinear dPG of the type (1.2) but for a nonlinear map b with derivative $b'$ with respect to the first variable is also characterized as a nonlinear mixed formulation with solution $(x_h,y_h)\in X_h\times Y_h$ to

$$\begin{aligned} \begin{aligned} a(y_h,\eta _h) + b(x_h; \eta _h)&= F(\eta _h) \quad \text {for all } \eta _h \in Y_h, \\ b'(x_h; \xi _h, y_h)&= 0 \quad \text {for all } \xi _h \in X_h. \end{aligned} \end{aligned}$$

(M)

Another characterization in the lowest-order cases under consideration is that as a weighted least-squares functional on Courant finite element functions $S^1_0({\mathcal {T}})$ with homogeneous Dirichlet boundary values and the Raviart–Thomas finite element functions $RT_0({\mathcal {T}})$ with some mesh-dependent piecewise constant weight $S_0\in P_0({\mathcal {T}};{\mathbb {R}}^{n\times n})$

This is already a new result even for the linear cases in [9, 13] and opens the door to a convergence analysis of adaptive algorithms via a generalization of [11, 14].

This paper contributes the aforementioned equivalent characterizations and a first convergence analysis in the natural norms. The a priori result is local quasi-optimal convergence for the simple model problem in that any discrete solution $x_h\in X_h$, sufficiently close to the exact regular solution $x\in X$, satisfies

$$\begin{aligned} \Vert x-x_h \Vert _X \lesssim \mathop {{{\mathrm{\phantom {p}inf}}}}\limits _{\xi _h \in X_h} \Vert x-\xi _h \Vert _X. \end{aligned}$$

It has been discussed in [5, 9, 13] that the norm of the computed residual is almost a computable error estimator for linear problems and this paper extends it to the a posteriori error estimate

(1.4)

for the nonlinear model problem (1.3). Since is the computable residual, this leads to built-in error control despite inexact solve: The discrete quantities $(v_C,q_{RT })$ in (1.4) do not need to solve the nonlinear dPG discrete problem.

The analysis is given for the primal version of the nonlinear dPG for brevity but applies to the other formulations of Sect. 4.4 as well. The results of this paper can be generalized, e.g., to the Hencky material [28, Sect. 62.8], and then applied to more complicated real-life computational challenges where the advantages of the dPG methodology are more striking.

The remaining parts of of this paper are organised as follows. Section 2 discusses an abstract framework for different equivalent formulations of a dPG method for nonlinear problems and develops an abstract a priori estimate. Section 3 presents a model problem with a dPG discretization. Section 4 analyses this discretization and gives proofs of the existence of a solution and an a posteriori error estimate. Some numerical examples in Sect. 5 conclude the paper.

This paper employs standard notation of Sobolev and Lebesgue spaces $H^k({\varOmega })$, $H({{\mathrm{div}}},{\varOmega })$, $L^2({\varOmega })$, and $L^\infty (\varOmega )$ and the corresponding spaces of vector- or matrix-valued functions $H^k({\varOmega };{\mathbb {R}}^n)$, $L^2({\varOmega };{\mathbb {R}}^n)$, $L^\infty ({\varOmega };{\mathbb {R}}^n)$, $H^k({\varOmega };{\mathbb {R}}^{n \times n})$, $H({{\mathrm{div}}},{\varOmega };{\mathbb {R}}^{n \times n})$, $L^2({\varOmega };{\mathbb {R}}^{n \times n})$, and $L^\infty ({\varOmega };{\mathbb {R}}^{n \times n})$. For any regular triangulation ${\mathcal {T}}$ of $\varOmega$, let $H^k({\mathcal {T}}) :=\prod _{T \in {\mathcal {T}}} H^k(T) :=\{v \in L^2(\varOmega ) \,|\, \forall T \in {\mathcal {T}},\, v\vert _T \in H^k(T)\}$ denote the piecewise (or broken) Sobolev spaces and $(\nabla _{NC }v)\big \vert _{T} = \nabla (v\big \vert _{T})$ on $T\in {\mathcal {T}}$ the piecewise gradient for $v\in H^1({\mathcal {T}})$. Let abbreviate the energy norm. For every Hilbert space X, let denote the associated inner product and, for every normed space , the sphere in X. The measure is context-dependent and refers to the number of elements of some finite set or the length $\vert E \vert$ of an edge E or the area $\vert T \vert$ of some triangle T and not just the modulus of a real number or the Euclidean length of a vector.

Throughout the paper, $A\lesssim B$ abbreviates the relation $A \le CB$ with a generic constant $0<C$, which does not depend on the mesh-size of the underlying triangulation ${\mathcal {T}}$ but solely on the initial triangulation ${\mathcal {T}}_0$; $A\approx B$ abbreviates $A\lesssim B\lesssim A$, e.g., in (1.4).

2 Abstract framework

This section analyses an abstract nonlinear dPG method and presents an a priori error estimate.

2.1 Abstract nonlinear dPG

For an open set $D\ne \emptyset$ in a real Banach space X and a real Hilbert space Y with scalar product $a: Y \times Y \rightarrow {\mathbb {R}}$, let $B\in C^1(D;Y^*)$ be a differentiable nonlinear map with Fréchet derivative ${{\mathrm{D\!}}}B(x)\in L(X;Y^*)$ at $x \in D$. With the duality bracket in Y, associate the nonlinear map $b: X \times Y \rightarrow {\mathbb {R}}$, , which is linear and bounded in the second component. Let abbreviate the derivative ${{\mathrm{D\!}}}B(x) \in L(X;Y^*)$ with $b'(x;\xi ,\eta ):=\langle {{\mathrm{D\!}}}B(x;\xi ),\eta \rangle$ for $x\in D,\xi \in X,\eta \in Y$.

Given $F \in Y^*$, let $x \in D$ be a regular solution to the problem $B(x) = F$ in $Y^*$. That means that x solves $B(x) = F$ and the Fréchet derivative ${{\mathrm{D\!}}}B$ at x is a bijection from X to $Y^*$. The latter implies the inf-sup condition for the Fréchet derivative at the regular solution x, namely,

$$\begin{aligned} 0 < \beta (x) :=\mathop {{{\mathrm{\phantom {p}inf}}}}\limits _{\xi \in {{\mathcal {S}}(X)}} \sup _{\eta \in {{\mathcal {S}}(Y)}} b'(x; \xi , \eta ). \end{aligned}$$

(2.1)

The minimal residual formulation of the continuous problem seeks $x \in X$ with

$$\begin{aligned} x \in \mathop {\hbox {argmin}}\limits _{\xi \in D} \Vert F - B(\xi )\Vert _{Y^*}. \end{aligned}$$

(2.2)

The existence of a solution x to (2.2) is immediate from the assumption $B(x) = F$. In particular, the minimum is zero and any minimizer x in (2.2) solves $B(x) = F$. The situation is (in general) different on the discrete level with some discrete subspaces $X_h \subset X$ and $Y_h \subset Y$, the dPG scheme seeks a minimizer $x_h \in D_h:=X_h\cap D$ of the residual $F - B(\xi _h)$ in the norm of $Y_h^*$,

$$\begin{aligned} x_h \in \mathop {\hbox {argmin}}\limits _{\xi _h \in D_h} \Vert F - B(\xi _h) \Vert _{Y_h^*}. \end{aligned}$$

(dPG)

The existence of a solution to (dPG) requires further assumptions and follows in Proposition 4.3 for a model problem.

2.2 Derivation of nonlinear dPG

A formal Lagrange ansatz leads to the minimization of the Lagrange functional ${\mathcal {L}}: D_h \times Y_h \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ defined for $(x_h, y_h, \lambda )\in X_h\times Y_h \times {\mathbb {R}}$ by

$$\begin{aligned} {\mathcal {L}}(x_h, y_h, \lambda ) :=F(y_h) - b(x_h; y_h) - \frac{\lambda }{2} \big ( a(y_h, y_h) - 1 \big ). \end{aligned}$$

The stationary points $x_h \in D_h$, $y_h \in Y_h$, and $\lambda \in {\mathbb {R}}$ of ${\mathcal {L}}$ are characterized by the first derivatives of ${\mathcal {L}}$ with respect to each argument in the sense that, for all $\eta _h \in Y_h$ and $\xi _h \in X_h$,

$$\begin{aligned} \lambda a(y_h, \eta _h) + b(x_h; \eta _h) = F(\eta _h),\quad b'(x_h; \xi _h, y_h)= 0,\quad a(y_h, y_h) = 1. \end{aligned}$$

For $\eta _h = y_h$, this implies $\lambda = F(y_h) - b(x_h; y_h)$. The substitution of $y_h$ by $\lambda y_h$ leads to a modified system of equations. The resulting mixed formulation of the nonlinear dPG method seeks $x_h \in X_h$ and $y_h \in Y_h$ with

$$\begin{aligned} \begin{aligned} a(y_h,\eta _h) + b(x_h; \eta _h)&= F(\eta _h) \quad \text {for all } \eta _h \in Y_h, \\ b'(x_h; \xi _h, y_h)&= 0 \quad \text {for all } \xi _h \in X_h. \end{aligned} \end{aligned}$$

Notice that this is known for linear problems (there, ) [17, Sect. 2.3].

2.3 Equivalent mixed formulation

It is known in linear problems that the dPG method is equivalent to the mixed problem (M) and this is generalized in this subsection to the nonlinear problem $B(x) = F$ at hand. Any local (or global) minimizer of $\varPhi (\xi _h):=\Vert F - B\xi _h \Vert _{Y_h^*}^2/2$ is a stationary point of $\varPhi$.

Definition 2.1

(stationary point) Any $x_h\in D_h:=D\cap X_h$ is a stationary point of the dPG discretization (dPG) if any directional derivative of $\varPhi (\xi _h):=\Vert F - B\xi _h \Vert _{Y_h^*}^2/2$ vanishes at $x_h$, i.e., $\lim _{\delta \rightarrow 0} (\varPhi (x_h+\delta \xi _h)-\varPhi (x_h))/\delta = 0$ for all $\xi _h\in X_h$.

Stationary points are exactly the solutions to (M).

Theorem 2.2

((dPG) $\Leftrightarrow$ (M))

• (a) Suppose $x_h$ is a stationary point of (dPG) and $y_h$ is the residual’s Riesz representation (i.e. ) in $Y_h$. Then $(x_h, y_h)$ solves (M).

• (b) Suppose that $(x_h, y_h)$ solves (M), then $x_h$ is a stationary point of (dPG).

Proof

(a) For any $\xi _h\in D_h$, the unique Riesz representation $\varrho _h(\xi _h) \in Y_h$ of the residual satisfies

$$\begin{aligned} \varPhi (\xi _h) = \frac{1}{2} \Vert \varrho _h(\xi _h) \Vert _Y^2. \end{aligned}$$

Given the stationary point $x_h\in D_h$ to (dPG) and $\xi _h \in X_h$, consider $\varPhi (x_h + t \xi _h)$ as a scalar function of the real parameter t with a derivative zero at $t = 0$. For $\vert t\vert$ small such that $x_h(t) :=x_h + t \xi _h\in D_h$ and $y_h(t) :=\varrho _h(x_h(t))$, it follows

A differentiation with respect to t shows for ${\dot{y}}_h :=\partial y_h(0) / \partial t$ and ${\dot{x}}_h :=\partial x_h(0) / \partial t = \xi _h$ that ${\dot{y}}_h$ exists and is the Riesz representation of in $Y_h$. Therefore, $\varPhi (x_h(t)) = a(y_h(t), y_h(t))/2$ is differentiable and the derivative vanishes at $t = 0$, which leads to

$$\begin{aligned} 0 = a({\dot{y}}_h, y_h) \quad \text {for } y_h :=y_h(0). \end{aligned}$$

It follows that

$$\begin{aligned} b'(x_h; \xi _h, y_h) = 0 \quad \text {for all } \xi _h \in X_h. \end{aligned}$$

Since $y_h = y_h(0) = \varrho _h(x_h)$, $(x_h, y_h)$ solves (M).

(b) Conversely, if $(x_h, y_h)$ solves (M) then, for any $\xi _h \in D_h$ and the above notation for the Riesz representation $y_h(t)$ of $F - B x_h(t)$ in $Y_h$,

$$\begin{aligned} \Vert F - B x_h(t) \Vert _{Y_h^*}^2 = a(y_h(t), y_h(t)) = F(y_h(t)) - b(x_h(t); y_h(t)) \end{aligned}$$

has a derivative with respect to t at $t = 0$, namely, for $y_h :=y_h(0)$

$$\begin{aligned} 2\,a({\dot{y}}_h, y_h) = F({\dot{y}}_h) - b'(x_h; \xi _h, y_h) - b(x_h; {\dot{y}}_h). \end{aligned}$$

Since $b'(x_h; \xi _h, y_h) = 0$ and $F({\dot{y}}_h) - b(x_h; {\dot{y}}_h) = a(y_h, {\dot{y}}_h)$, this implies $a({\dot{y}}_h, y_h) = 0$. Recall $\partial \varPhi (x(t))/\partial t\vert _{t = 0} = a({\dot{y}}_h, y_h) = 0$, and so $x_h$ is a stationary point of $\varPhi$. $\square$

Proposition 2.3

(necessary and sufficient second-order condition)

Assume that $\varPhi$ is twice differentiable. (a) If $x_h$ solves (dPG), then

(2.3)

(b) If, in addition,

(2.4)

then $x_h$ is locally unique.

Proof

The second derivative of $\varPhi (x_h(t))$ reads $a(\partial ^2y_h/\partial t^2, y_h) + \Vert \partial y_h/\partial t \Vert _Y^2$. Recall from the proof of Theorem 2.2 for $t=0$, that the Riesz representation ${\dot{y}}_h = \partial y_h(0)/\partial t$ satisfies

Another differentiation with respect to t shows that $\ddot{y}_h :=\partial ^2 y_h(0)/\partial t^2$ satisfies

Consequently, the second derivative of $\varPhi (x_h(t))$ at $t=0$ is

(2.5)

The assertion follows from this and standard arguments in the calculus of stationary and minimal points. $\square$

Remark 2.4

(linear problems) For a linear problem, vanishes and (2.4) holds. This implies local uniqueness in the linear situation (which is a global one).

The uniqueness of the discrete solution is observed in numerical examples; cf. Theorem 4.4 for a sufficient condition in the model example below.

2.4 Abstract a priori error analysis

This section presents a best-approximation result based on a discrete inf-sup condition and the existence of a Fortin operator.

Hypothesis 2.5

Throughout this paper, assume that there exists a linear bounded projection $\varPi _h: Y \rightarrow Y_h$ with $\varPi _h\vert _{Y_h} = {{\mathrm{id}}}\vert _{Y_h}$ and

$$\begin{aligned} b'(D_h; X_h, (1-\varPi _h)Y) = 0, \end{aligned}$$

(2.6)

i.e., for all $x_h \in D_h$ and all $y \in Y$, $\varPi _h y \in Y_h$ satisfies $b'(x_h; \xi _h, y - \varPi _h y) =0$ for all $\xi _h \in X_h$. Let $\Vert \varPi _h \Vert$ denote the bound of $\varPi _h$ in L(Y; Y).

The following theorem generalizes [1, Prop. 5.4.2] to the nonlinear problem at hand. A sufficiently fine initial triangulation guarantees that $B(x, \varepsilon )\cap X_h\subset D_h$ is nonempty.

Theorem 2.6

(discrete inf-sup condition) Given a regular solution x to $B(x) =F$, there exists an open ball $B(x, \varepsilon ):=\{{{\tilde{x}}}\in X\,|\, \Vert x-\tilde{x}\Vert _X<\varepsilon \}$ of radius $\varepsilon > 0$ around x such that, for all ${{\tilde{x}}}_h \in B(x, \varepsilon )\cap X_h\subset D_h$, the following discrete inf-sup condition holds

$$\begin{aligned} 0 < \frac{\beta (x;X_h,Y_h)}{2\Vert \varPi _h\Vert } \le \beta ({{\tilde{x}}}_h; X_h, Y_h) :=\mathop {{{\mathrm{\phantom {p}inf}}}}\limits _{\xi _h \in {{\mathcal {S}}(X_h)}} \sup _{\eta _h \in {{\mathcal {S}}(Y_h)}} b'({{\tilde{x}}}_h; \xi _h, \eta _h). \end{aligned}$$

Proof

The continuous inf-sup condition (2.1) and the continuity of ${{\mathrm{D\!}}}B$ in D lead to some $\varepsilon$ such that

$$\begin{aligned} B(x,\varepsilon )&\subset D, \end{aligned}$$

(2.7)

$$\begin{aligned} \beta (x) / 2&\le \mathop {{{\mathrm{\phantom {p}inf}}}}\limits _{\xi \in B(x,\varepsilon )} \beta (\xi ). \end{aligned}$$

(2.8)

Then ${{\tilde{x}}}_h \in B(x, \varepsilon )\cap X_h$ and (2.6) imply

$$\begin{aligned} \beta (x) / 2&\le \beta ({{\tilde{x}}}_h) \le \mathop {{{\mathrm{\phantom {p}inf}}}}\limits _{\xi _h \in {{\mathcal {S}}(X_h)}} \sup _{\eta \in {{\mathcal {S}}(Y)}} b'({\tilde{x}}_h; \xi _h, \eta )\\&= \mathop {{{\mathrm{\phantom {p}inf}}}}\limits _{\xi _h \in {{\mathcal {S}}(X_h)}} \sup _{\eta \in {{\mathcal {S}}(Y)}} b'({\tilde{x}}_h; \xi _h, \varPi _h\eta )\\&= \mathop {{{\mathrm{\phantom {p}inf}}}}\limits _{\xi _h \in {{\mathcal {S}}(X_h)}} \sup _{\eta \in {{\mathcal {S}}(Y)}} \Vert \varPi _h \eta \Vert _Y b'\big ( {\tilde{x}}_h; \xi _h, \varPi _h\eta \big / \Vert \varPi _h\eta \Vert _Y \big )\\&\le \Vert \varPi _h \Vert \mathop {{{\mathrm{\phantom {p}inf}}}}\limits _{\xi _h \in {{\mathcal {S}}(X_h)}} \sup _{\eta \in {{\mathcal {S}}(Y)}} b'\big ( {\tilde{x}}_h; \xi _h, \varPi _h\eta \big / \Vert \varPi _h\eta \Vert _Y \big )\\&= \Vert \varPi _h \Vert \mathop {{{\mathrm{\phantom {p}inf}}}}\limits _{\xi _h \in {{\mathcal {S}}(X_h)}} \sup _{\eta _h \in {{\mathcal {S}}(Y_h)}} b'({\tilde{x}}_h; \xi _h, \eta _h). \end{aligned}$$

Hence, any ${\tilde{x}}_h \in B(x, \varepsilon )$ satisfies $0 < \frac{\beta (x)}{2\Vert \varPi _h \Vert } \le \beta ({\tilde{x}}_h; X_h, Y_h)$. $\square$

Remark 2.7

(converse of Theorem 2.6) Given the discrete inf-sup condition

$$\begin{aligned} 0 < \mathop {{{\mathrm{\phantom {p}inf}}}}\limits _{\xi _h \in {{\mathcal {S}}(X_h)}} \sup _{\eta _h \in {{\mathcal {S}}(Y_h)}} b'({\tilde{x}}_h; \xi _h, \eta _h) \end{aligned}$$

(2.9)

at some point ${\widetilde{x}}_h \in D_h$, the techniques of [9, Lemma 10] guarantee the existence of a linear bounded projection $\varPi _h({\widetilde{x}}_h):Y\rightarrow Y_h$ with (2.6), which depends on ${\widetilde{x}}_h$. The above proof shows that the existence of $\varPi _h({\widetilde{x}}_h)$ is also sufficient for (2.9). The class of model examples allows for the simple more uniform Hypothesis 2.5 with $\varPi _h({\widetilde{x}}_h)=\varPi _h$ independent of ${\widetilde{x}}_h\in D_h$.

Theorem 2.8

(local best-approximation) Given a regular solution x to $B(x)=F$, there exist positive constants $\varepsilon > 0$ and $C(x,\varepsilon )>0$ such that any solution $(x_h, y_h)$ to (M) with $\Vert x - x_h \Vert _X < \varepsilon$ satisfies

$$\begin{aligned} \Vert x-x_h \Vert _X + \Vert y_h\Vert _Y \le C(x,\varepsilon ) \mathop {{{\mathrm{\phantom {p}inf}}}}\limits _{\xi _h \in X_h} \Vert x-\xi _h \Vert _X. \end{aligned}$$

The proof of the theorem requires the following lemma.

Lemma 2.9

Any $\varepsilon >0$ and $x_h\in B(x,\varepsilon )\subset D$ satisfy

(2.10)

$$\begin{aligned} \Vert B(x) - B(x_h)\Vert _{Y^*}&\le \sup _{\xi \in B(x,\varepsilon )} \Vert {{\mathrm{D\!}}}B(\xi )\Vert _{L(X;Y^*)} \Vert x - x_h \Vert _X. \end{aligned}$$

(2.11)

Proof

Given any $\eta \in {{\mathcal {S}}(Y)}$, the Taylor’s formula of b at $x_h$ with remainder reads

$$\begin{aligned}&b'(x_h; x - x_h, \eta ) - b(x; \eta )+ b(x_h; \eta ) \\&\quad = \int _0^1 \Big ( b'(x_h;x-x_h,\eta ) - b'(x_h+s(x-x_h);x-x_h,\eta ) \Big ) \, \mathrm {d} s. \end{aligned}$$

Since $\Vert x - x_h\Vert _X <\varepsilon$ implies $\Vert x - (x_h+s(x-x_h))\Vert _X <\varepsilon$ for $0\le s \le 1$, the triangle inequality proves

$$\begin{aligned}&b'(x_h; x - x_h, \eta ) + b(x_h; \eta ) - b(x; \eta )\\&\quad \le 2\sup _{\xi \in B(x,\varepsilon )} \vert b'( x;x-x_h,\eta ) - b'(\xi ;x-x_h,\eta ) \vert \\&\quad \le 2 \sup _{\xi \in B(x,\varepsilon )} \Vert {{\mathrm{D\!}}}B(x)- {{\mathrm{D\!}}}B(\xi )\Vert _{L(X;Y^*)} \Vert x - x_h \Vert _X. \end{aligned}$$

Since $\eta \in {{\mathcal {S}}(Y)}$ is arbitrary, this implies (2.10). The assertion (2.11) follows from the same arguments without the term $b'(x_h; x - x_h, \eta )$.$\square$

Proof (of Theorem 2.8)

Let ${\tilde{x}}_h$ be the best-approximation to x in $X_h$, i.e.,

$$\begin{aligned} \Vert x - {\tilde{x}}_h \Vert _X = \mathop {{{\mathrm{\phantom {p}inf}}}}\limits _{\xi _h \in D_h} \Vert x - \xi _h \Vert _X \le \Vert x - x_h \Vert _X < \varepsilon . \end{aligned}$$

Suppose $\varepsilon >0$ satisfies (2.7)–(2.8) and, with the continuity of ${{\mathrm{D\!}}}B$ at x,

$$\begin{aligned} \sup _{\xi \in B(x,\varepsilon )} \Vert {{\mathrm{D\!}}}B(\xi )\Vert _{L(X;Y^*)} \le 2 \Vert {{\mathrm{D\!}}}B(x)\Vert _{L(X;Y^*)}. \end{aligned}$$

(2.12)

The discrete inf-sup condition from Theorem 2.6 plus the Brezzi splitting lemma [3, Thm. 4.3 in Ch. III] with inf-sup constants, $\beta (x)/2$ and 1, and continuity constants, $2 \Vert {{\mathrm{D\!}}}B(x)\Vert _{L(X;Y^*)}$ and 1, for the bilinear form and scalar product a prove the global inf-sup condition $0<\gamma \le \beta (x_h;X_h,Y_h)$ for

$$\begin{aligned} \gamma := \,\,\mathop {{{\mathrm{\phantom {p}inf}}}}\limits _{(\tilde{\xi }_h, \tilde{\eta }_h) \in {{\mathcal {S}}(X_h \times Y_h)}} \sup _{(\xi _h, \eta _h) \in {{\mathcal {S}}(X_h \times Y_h)}} \Big (b'(x_h; \tilde{\xi }_h, \eta _h) + b'(x_h; \xi _h , \tilde{\eta }_h) + a(\tilde{\eta }_h,\eta _h)\Big ). \end{aligned}$$

independent of $\varepsilon$ with (2.7)–(2.8) and (2.12). Given $\gamma >0$ and $\beta (x)>0$ suppose, for some smaller $\varepsilon >0$ if necessary, that $\varepsilon >0$ satisfies (2.7)–(2.8), (2.12), and, from the continuity of ${{\mathrm{D\!}}}B$ at x,

$$\begin{aligned} \sup _{\xi \in B(x,\varepsilon )} \Vert {{\mathrm{D\!}}}B(x)- {{\mathrm{D\!}}}B(\xi )\Vert _{L(X;Y^*)} \le \min \{\gamma /4,\beta (x)/8\}. \end{aligned}$$

(2.13)

For the best-approximation ${\tilde{y}}_h=0$ to $y=0$ in $Y_h$ and $(\tilde{\xi }_h, \tilde{\eta }_h) = ({\tilde{x}}_h - x_h, {\tilde{y}}_h - y_h)$, this implies the existence of $(\xi _h, \eta _h) \in {{\mathcal {S}}(X_h \times Y_h)}$ with

$$\begin{aligned} \gamma \big ( \Vert {\tilde{x}}_h - x_h \Vert _X + \Vert y_h \Vert _Y \big ) \le b'(x_h; {\tilde{x}}_h - x_h, \eta _h) - b'(x_h; \xi _h, y_h) - a(y_h, \eta _h). \end{aligned}$$

Since $(x_h, y_h)$ solves (M) and ${\tilde{y}}_h = 0$, this leads to

$$\begin{aligned} \gamma \big ( \Vert {\tilde{x}}_h - x_h \Vert _X + \Vert y_h \Vert _Y \big )\le & {} b'(x_h; {\tilde{x}}_h - x, \eta _h)\\&+\,b'(x_h; x - x_h, \eta _h) + b(x_h; \eta _h) - b(x; \eta _h). \end{aligned}$$

Lemma 2.9 and (2.13) imply

$$\begin{aligned} b'(x_h; x - x_h, \eta _h)&+ b(x_h; \eta _h) - b(x; \eta _h)\le \gamma \Vert x - x_h \Vert _X/2. \end{aligned}$$

The combination of the preceding two displayed formulae reads

$$\begin{aligned} \frac{\gamma }{2} \Vert {\tilde{x}}_h - x_h \Vert _X + \gamma \Vert y_h \Vert _Y&\le b'(x_h;{\tilde{x}}_h - x_h,\eta _h). \end{aligned}$$

With (2.12), this is bounded from above by

$$\begin{aligned} \Vert {{\mathrm{D\!}}}B (x_h)\Vert _{L(X;Y^*)} \Vert x-{\tilde{x}}_h\Vert _X \le 2 \Vert {{\mathrm{D\!}}}B (x_h)\Vert _{L(X;Y^*)} \Vert x-{\tilde{x}}_h\Vert _X. \end{aligned}$$

The triangle inequality concludes the proof. $\square$

Remark 2.10

Under further smoothness conditions on the nonlinear mapping $b'$ the local existence and uniqueness of a discrete solution, e.g., follows from [25, Thm. 2].

Remark 2.11

The Newton–Kantorovich theorem [27, Section 5.2] is another tool for the proof of the existence of discrete solutions close to the regular solution. In the model problem of Sect. 3, the higher Fréchet derivatives for this argument do not exist, cf. Remark 3.3 for details.

2.5 Abstract a posteriori error analysis

This subsection is devoted to a brief abstract a posteriori error analysis of the nonlinear dPG. Given a discrete approximation $x_h$ close to the regular solution x to $B(x)=F$, the residual $F-B(x_h)\in Y^*$ has a norm that, in principle, is accessible in the sense that lower and upper bounds may be computable. The latter issue is a typical general task in the a  posteriori error analysis and will be adressed in Sect. 3 for a model example.

Theorem 2.12

(local a posteriori analysis) Let x be a regular solution to $B(x)=F$ with inf-sup constant $\beta (x)$ from (2.1). Then there exists some $\varepsilon >0$ such that any $x_h\in B(x,\varepsilon )\subset D$ satisfies

$$\begin{aligned} \frac{\beta (x)}{4} \Vert x - x_h\Vert _X \le \Vert F-B(x_h)\Vert _{Y^*} \le 2 \Vert {{\mathrm{D\!}}}B(x)\Vert _{L(X;Y^*)} \Vert x - x_h\Vert _X. \end{aligned}$$

Proof

With the choice of $\varepsilon >0$ from the proof of Theorem 2.8 it follows (2.7)–(2.8) and (2.12)–(2.13). The continuous inf-sup condition (2.1) implies the existence of $\eta \in {{\mathcal {S}}(Y)}$ with

$$\begin{aligned} \frac{\beta (x)}{2} \Vert x - x_h\Vert _X&\le b'(x_h;x-x_h,\eta )\\&\le b(x;\eta ) - b(x_h;\eta ) + \vert b'(x_h;x-x_h,\eta ) - b(x;\eta ) + b(x_h;\eta )\vert . \end{aligned}$$

Lemma 2.9 for the last term, $b(x;\eta )=F(\eta )$, and (2.13) show

$$\begin{aligned} \frac{\beta (x)}{2} \Vert x - x_h\Vert _X \le F(\eta ) - b(x_h;\eta ) + \frac{\beta (x)}{4} \Vert x - x_h\Vert _X. \end{aligned}$$

This proves the asserted reliability

$$\begin{aligned} \frac{\beta (x)}{4} \Vert x - x_h\Vert _X \le \Vert F(\eta ) - b(x_h;\eta )\Vert _{Y^*}. \end{aligned}$$

To prove the efficiency, utilize $F=B(x)$, Lemma 2.9, and (2.12) to verify

$$\begin{aligned} \Vert F-B(x_h)\Vert _{Y^*}&= \Vert B(x) - B(x_h)\Vert _{Y^*}\\&\le 2\Vert {{\mathrm{D\!}}}B(x)\Vert _{L(X;Y^*)} \Vert x-x_h\Vert _X. \quad \end{aligned}$$

$\square$

Remark 2.13

Since $y=0$ and $y_h$ is computed, the a posteriori error $\Vert y - y_h\Vert _Y = \Vert y_h\Vert _Y$ is already an error estimator and can be added on both sides of the reliability (resp. efficiency) a posteriori error estimate. This justifies the usage of the extended residual of the system (M).

Remark 2.14

The constants $\beta (x)/4$ (resp. $2\Vert {{\mathrm{D\!}}}B(x)\Vert _{L(X;Y^*)}$) in Theorem 2.12 follow from the choice of $\varepsilon$ in the a priori error analysis in the proof of Theorem 2.8. For smaller and smaller values of $\varepsilon$, those constants could be replaced by any number $<\beta (x)$ (resp. $>\Vert {{\mathrm{D\!}}}B(x)\Vert _{L(X;Y^*)}$) in the following sense. For any $0<\lambda <1$ there exists some $\varepsilon >0$ such that any $x_h\in B(x,\varepsilon )$ satisfies $\lambda \beta (x)\le \Vert F-B(x_h)\Vert _{Y^*}\le (1+\lambda ) \Vert {{\mathrm{D\!}}}B(x)\Vert _{L(X;Y^*)}$.

3 Model problem

This section introduces a nonlinear model problem and a low-order dPG discretization and establishes two further equivalent characterizations of the nonlinear dPG method: reduced discretization and weighted least-squares.

3.1 Convex energy minimization

The nonlinear model problem involves a nonlinear function $\phi \in C^2(0, \infty )$ with $0 < \gamma _1 \le \phi (t) \le \gamma _2$ and $0 < \gamma _1 \le \phi (t) + t \phi '(t) \le \gamma _2$ for all $t \ge 0$ and universal positive constants $\gamma _1, \gamma _2$. Given $f \in L^2(\varOmega )$ and the convex function $\varphi$, $\varphi (t) :=\int _0^t s \, \phi (s) \, \mathrm {d} s$ for $t\ge 0$, the model problem minimizes the energy functional

$$\begin{aligned} E(v) :=\int _\varOmega \varphi (\vert \nabla v(x)\vert ) \, \mathrm {d} x- \int _\varOmega fv \, \mathrm {d} x\quad \text {among all } v \in H^1_0(\varOmega ). \end{aligned}$$

The convexity of $\varphi$ and the above assumptions on $\phi$ lead to growth-conditions and sequential weak lower semicontinuity of E and guarantee the unique existence of a minimizer u of E in $H^1_0(\varOmega )$ [29, Thm. 25.D]. The equivalent Euler-Lagrange equation reads

$$\begin{aligned} \int _\varOmega \phi (\vert \nabla u \vert ) \nabla u \cdot \nabla v \, \mathrm {d} x= \int _\varOmega fv \, \mathrm {d} x\quad \text {for all } v \in H^1_0(\varOmega ) \end{aligned}$$

(3.1)

and has the unique solution u in $H^1_0(\varOmega )$. The stress variable $\sigma (A) :=\phi (\vert A \vert ) A$ defines a function $\sigma \in C^1({\mathbb {R}}^n; {\mathbb {R}}^n)$ with Fréchet derivative

$$\begin{aligned} {{\mathrm{D\!}}}\sigma (A) = \phi (\vert A \vert ) I_{n\times n} + \phi '(\vert A \vert ) \vert A \vert {{\mathrm{sign}}}(A) \otimes {{\mathrm{sign}}}(A) \end{aligned}$$

(3.2)

with the sign function ${{\mathrm{sign}}}(A) :=A\,/\,\vert A \vert$ for $A \in {\mathbb {R}}^n \setminus \lbrace 0 \rbrace$ and the closed unit ball ${{\mathrm{sign}}}(0) :=\overline{B(0, 1)}$ in ${\mathbb {R}}^n$. The prefactor $\phi '(\vert A \vert ) \vert A \vert$ makes ${{\mathrm{D\!}}}\sigma$ a continuous function in ${\mathbb {R}}^n$. In fact ${{\mathrm{D\!}}}\sigma \in C^0({\mathbb {R}}^{n\times n}_{\hbox {sym}})$ is bounded with eigenvalues in the compact interval $[\gamma _1,\gamma _2]\subset (0,\infty )$.

Remark 3.1

($\hbox {Lip}(\sigma ) \le \gamma _2$) For $A,B\in {\mathbb {R}}^n$, the argument $\sigma (A) - \sigma (B) = \int _0^1 {{\mathrm{D\!}}}\sigma (sA+(1-s)B) (A-B) \, \mathrm {d} s$ and (3.2) imply the global Lipschitz continuity of $\sigma$,

$$\begin{aligned} \vert \sigma (A) - \sigma (B) \vert \le \int _0^1 \vert {{\mathrm{D\!}}}\sigma (sA+(1-s)B) (A-B) \vert \, \mathrm {d} s\le \gamma _2 \vert A - B \vert . \end{aligned}$$

Example 3.2

In the following examples, $0 \le \phi '' \le 2$ is bounded as well as $\phi '$ and ${{\mathrm{D\!}}}\sigma$ from (3.2) is globally Lipschitz continuous. (a) $\phi (t) :=2 + (1 + t)^{-2}$ with $\gamma _1 = 1 < \gamma _2 = 3$ [15] and $\hbox {Lip}({{\mathrm{D\!}}}\sigma ) \le 4$ and (b) $\phi (t) :=2 - (1 + t^2)^{-1}$ with $\gamma _1 = 1 < \gamma _2 = 4$ and $\hbox {Lip}({{\mathrm{D\!}}}\sigma ) \le 2$.

Remark 3.3

(second derivative) A formal calculation with $s(j) :=({{\mathrm{sign}}}A)_j$, $s(j,k) :=({{\mathrm{sign}}}A)_j ({{\mathrm{sign}}}A)_k$ etc. and the Kronecker symbol $\delta _{jk}$ for $j, k, \ell = 1, \dots , n$ leads at any $A \in {\mathbb {R}}^n$ to

$$\begin{aligned} {{\mathrm{{\text {D}}^2\!}}}\sigma (A)_{j,k,\ell } = \phi '(\vert A \vert )(\delta _{jk} s(\ell ) + \delta _{j\ell } s(k) + \delta _{k\ell }s(j)) + (\phi ''(\vert A \vert ) \vert A \vert - \phi '(\vert A \vert )) s(j,k,\ell ). \end{aligned}$$

Although ${{\mathrm{{\text {D}}^2\!}}}\sigma (A)$ may be bounded (at least in the Example 3.2.a and b), it may be discontinuous for $A \rightarrow 0$. In Example 3.2.b, $\phi '(0) = 0$ and ${{\mathrm{{\text {D}}^2\!}}}\sigma$ is continuous with ${{\mathrm{{\text {D}}^2\!}}}\sigma (0) = 0$. The associated trilinear form , however, is not well-defined on $X\times Y\times Y$ because the product of three Lebesgue functions in $L^2(\varOmega )$ is, in general, not in $L^1(\varOmega )$.

3.2 Breaking the test spaces

Let ${\varOmega }\subseteq {\mathbb {R}}^n$ be a bounded Lipschitz domain with polyhedral boundary $\partial {\varOmega }$. Let ${\mathcal {T}}$ denote a regular triangulation of the domain ${\varOmega }$ into n-simplices and let ${\mathcal {E}}$ (resp. ${\mathcal {E}}(T)$) denote the set of all sides in the triangulation (resp. of an n-simplex $T\in {\mathcal {T}}$).

The unit normal vector $\nu _T$ along the boundary $\partial T$ of an n-simplex $T\in {\mathcal {T}}$ (is constant along each side of T and) points outwards. For any side $E=\partial T_+\cap \partial T_-\in {\mathcal {E}}$ shared by two simplices, the enumeration of the neighbouring simplices $T_\pm$ is globally fixed and so defines a unique orientation of the unit normal $\nu _E = \nu _{T_+}\vert _E$. Let $h_T$ denote the diameter of $T\in {\mathcal {T}}$, $h_{max }:=\max _{T\in {\mathcal {T}}} h_T\le \hbox {diam}({\varOmega })$ and $h_{\mathcal {T}}|_K=h_K$ for any $K\in {\mathcal {T}}$. The barycenter ${{\mathrm{mid}}}(T)$ of $T\in {\mathcal {T}}$ defines the piecewise constant function ${{\mathrm{mid}}}({\mathcal {T}})\in P_0({\mathcal {T}};{\mathbb {R}}^n)$ by ${{\mathrm{mid}}}({\mathcal {T}})|_K :={{\mathrm{mid}}}(K)$ for any $K\in {\mathcal {T}}$ and ${{\mathrm{mid}}}(E)$ is the barycenter of $E\in {\mathcal {E}}$. The piecewise affine function equals $x - {{\mathrm{mid}}}(T)$ at $x \in T \in {\mathcal {T}}$.

Recall that $H^k({\mathcal {T}}) :=\prod _{T \in {\mathcal {T}}} H^k(T) :=\{v \in L^2(\varOmega ) \,|\, \forall T \in {\mathcal {T}},\, v\vert _T \in H^k(T)\}$ denotes the piecewise Sobolev space. Define the discrete spaces

Definition 3.4

For a triangulation ${\mathcal {T}}$ with skeleton $\partial {\mathcal {T}}:=\bigcup _{T\in {\mathcal {T}}} \bigcup _{E\in {\mathcal {E}}(T)} E$ and $T\in {\mathcal {T}}$, recall the local trace spaces $H^{1/2}(\partial T)$ and $H^{-1/2}(\partial T) = (H^{1/2}(\partial T))^\star$ and

$$\begin{aligned} H^{-1/2}(\partial {\mathcal {T}})&:=\{ t = (t_T)_{T\in {\mathcal {T}}}\in {\textstyle \prod _{T\in {\mathcal {T}}} } H^{-1/2}(\partial T) \,|\,\\&\exists q\in H({{\mathrm{div}}},{\varOmega }),\forall T \in {\mathcal {T}},t_T=(q|_T)|_{\partial T}\cdot \nu _T\} \end{aligned}$$

endowed with the minimal extension norm, for $t\in H^{-1/2}(\partial {\mathcal {T}})$,

$$\begin{aligned} \Vert t \Vert _{H^{-1/2}(\partial {\mathcal {T}})}&:=\min \{\Vert q\Vert _{H({{\mathrm{div}}},\varOmega )}\,|\, q\in H({{\mathrm{div}}},{\varOmega }),\forall T \in {\mathcal {T}}, t_T=(q|_T)|_{\partial T}\cdot \nu _T\}. \end{aligned}$$

The duality brackets extend the $L^2$ scalar product in $L^2(\partial T)$ and lead to the duality bracket on the skeleton for any $t=(t_T)_{T\in {\mathcal {T}}}\in \prod _{T\in {\mathcal {T}}} H^{-1/2}(\partial T)$ and $s=(s_T)_{T\in {\mathcal {T}}}\in \prod _{T\in {\mathcal {T}}}H^{1/2}(\partial T)$ defined by

$$\begin{aligned} \langle t, s\rangle _{\partial {\mathcal {T}}} :=\sum _{T\in {\mathcal {T}}} \langle {t_T},{s_T}\rangle _{\partial T}. \end{aligned}$$

Remark 3.5

($RT_0({\mathcal {T}})\equiv P_0({\mathcal {E}})$)

The spaces $RT_0({\mathcal {T}})$ and $P_0({\mathcal {E}})$ are isomorphic [8, Lemma 3.2] in the sense that any $q_{RT }\in RT_0({\mathcal {T}})$ and $E\in {\mathcal {E}}$ with fixed unit normal vector $\nu _E$ satisfies $q_{RT }|_E\cdot \nu _E \in P_0(E)$. Conversely, for any $t_0\in P_0({\mathcal {E}})$, there exists a unique $q_{RT }\in RT_0({\mathcal {T}})$ with $q_{RT }|_E\cdot \nu _E =t_0|_E$ for any $E\in {\mathcal {E}}$, in short notation $q_{RT }\cdot \nu =t_0$ in $\partial {\mathcal {T}}$. Since $\Vert t_0 \Vert _{H^{-1/2}(\partial {\mathcal {T}})}\approx \Vert q_{RT }\Vert _{H({{\mathrm{div}}},\varOmega )}$, this identification justifies the embedding $P_0({\mathcal {E}})\subseteq H^{-1/2}(\partial {\mathcal {T}})$, where any $T\in {\mathcal {T}}$ and $E\in {\mathcal {E}}(T)$ satisfy $(q_{RT }\cdot \nu _T)|_E= \pm t_0|_E$ with the sign $\pm = \nu _T \cdot \nu _E$ depending on the (globally fixed) choice of the orientation of the unit normal $\nu _E \in \{\nu _{T_\pm }|_E\}$.

Definition 3.6

Define $S_0\in P_0({\mathcal {T}};{\mathbb {R}}^{n\times n})$ and $H_0:L^2(\varOmega )\rightarrow P_0({\mathcal {T}};{\mathbb {R}}^n)$ for $T\in {\mathcal {T}}$ and $f\in {L^2({\varOmega })}$ by

(3.3)

Remark 3.7

An analysis of the eigenvalues of the piecewise symmetric positive semi-definite matrix $S_0$ shows that any $T\in {\mathcal {T}}$ and $v\in {\mathbb {R}}^n$ satisfies

$$\begin{aligned} \vert v \vert \le \vert (I_{n\times n} + S_0|_T) v \vert \le (1+h_T^2) \vert v \vert \text { and } \vert v \vert \le \vert (I_{n\times n} + S_0|_T)^{1/2} v \vert \le (1+h_T) \vert v \vert . \end{aligned}$$

Furthermore, for the maximal mesh-size $h_{max } = \max h_{\mathcal {T}}$ in ${\mathcal {T}}$.

3.3 Lowest-order dPG discretization

The nonlinear model problem of this paper concerns the nonlinear map $\sigma : {\mathbb {R}}^n \rightarrow {\mathbb {R}}^n$ of Sect. 3.1. A piecewise integration by parts in (3.1) and the introduction of the new variable $t:=\sigma (\nabla u)\cdot \nu$ on $\partial {\mathcal {T}}$ leads to the nonlinear primal dPG method with $F(v):=\int _\varOmega f v\, \mathrm {d} x$ and $b: X \times Y \rightarrow {\mathbb {R}}$ for $X :=H^1_0(\varOmega ) \times H^{-1/2}(\partial {\mathcal {T}})$ and $Y :=H^1({\mathcal {T}})$ defined by

$$\begin{aligned} b(u, t; v) :=\int _\varOmega \sigma (\nabla u) \cdot \nabla _{NC }v \, \mathrm {d} x- \langle t, v \rangle _{\partial {\mathcal {T}}} =:\langle B(u,t), y \rangle _Y. \end{aligned}$$

(3.4)

for all $x = (u, t) \in X :=H^1_0(\varOmega ) \times H^{-1/2}(\partial {\mathcal {T}})$ and $y = v \in Y = H^1({\mathcal {T}})$ with associated norms and the scalar product a in Y. Given the subspaces $X_h :=S^1_0({\mathcal {T}}) \times P_0({\mathcal {E}})$ and $Y_h :=P_1({\mathcal {T}})$, the discrete problem minimizes the residual norm and seeks $(u_h, t_h) = x_h \in X_h$ with

$$\begin{aligned} \Vert F -B(x_h)\Vert _{Y_h^*} = \min _{\xi _h \in X_h} \Vert F- B(\xi _h)\Vert _{Y_h^*}. \end{aligned}$$

(3.5)

The derivative ${{\mathrm{D\!}}}\sigma : {\mathbb {R}}^n \rightarrow {\mathbb {R}}^{n \times n}$ gives rise to the map

$$\begin{aligned} b'(u,t; w, s, v) :=\int _\varOmega \nabla w \cdot ({{\mathrm{D\!}}}\sigma (\nabla u) \nabla _{NC }v) \, \mathrm {d} x- \langle s, v \rangle _{\partial {\mathcal {T}}}. \end{aligned}$$

(3.6)

This defines a bounded bilinear form for any $x = (u,t) \in X$ and the operator B associated with b belongs to $C^1(X;Y^*)$. Recall the equivalent mixed formulation from (M) for the model problem at hand, which seeks $(u_h, t_h) \in X_h$ and $v_h \in Y_h$ with

$$\begin{aligned} \begin{aligned} a(v_h, \eta _h) + b(u_h, t_h; \eta _h)&= F(\eta _h) \quad \text {for all } \eta _h \in Y_h,\\ b'(u_h, t_h; w_h, s_h, v_h)&=0 \quad \text {for all } (w_h, s_h) \in X_h. \end{aligned} \end{aligned}$$

(3.7)

Remark 3.8

(regular solution) Since ${{\mathrm{D\!}}}\sigma (\nabla u)\in L^\infty ({\varOmega };{\mathbb {R}}^{n\times n}_{\hbox {sym}})$ uniformly positive definite, the splitting lemma from the linear theory [6, Thm. 3.3] implies the inf-sup condition (2.1) for the nondegenerate bilinear form . Hence, the solution $x\in X$ to $B(x)=F$ is regular.

3.4 Reduced discretization

The dPG discretization (3.5) can be simplified to a modified problem that seeks $(u_h, v_h) \in S^1_0({\mathcal {T}}) \times CR^1_0({\mathcal {T}})$ with

$$\begin{aligned} a(v_h, w_{CR }) + \int _\varOmega \sigma (\nabla u_h) \cdot \nabla _{NC }w_{CR }\, \mathrm {d} x&= \int _\varOmega f w_{CR }\, \mathrm {d} x\text { for all } w_{CR }\in CR^1_0({\mathcal {T}}),\nonumber \\ \int _\varOmega \nabla w_{C }\cdot \big ( {{\mathrm{D\!}}}\sigma (\nabla u_h) \nabla _{NC }v_h \big ) \, \mathrm {d} x&= 0\quad \text { for all } w_{C }\in S^1_0({\mathcal {T}}). \end{aligned}$$

(R)

Theorem 3.9

((3.7)$\;\Leftrightarrow \;$(R))

(a) If $(u_h, t_h; v_h) \in X_h \times Y_h$ solves (3.7), then $v_h \in CR^1_0({\mathcal {T}})$ and $(u_h, v_h) \in S^1_0({\mathcal {T}}) \times CR^1_0({\mathcal {T}})$ solves (R).

(b) For any solution $(u_h, v_h) \in S^1_0({\mathcal {T}}) \times CR^1_0({\mathcal {T}})$ to (R), there exists a unique $t_h\in P_0({\mathcal {E}})$ such that $(u_h,t_h; v_h)$ solves (3.7).

The proof utilizes the following discrete inf-sup condition of a linear primal dPG method [19]. Let the bilinear forms $a_{NC }:H^1({\mathcal {T}})\times H^1({\mathcal {T}})\rightarrow {\mathbb {R}}$ and ${\widetilde{b}}: X \times Y \rightarrow {\mathbb {R}}$ be defined by

$$\begin{aligned} a_{NC }(v_1, v_2)&:=\int _\varOmega \nabla _{NC }v_1 \cdot \nabla _{NC }v_2 \, \mathrm {d} x\quad \text { for }v_1,v_2\in H^1({\mathcal {T}}),\\ {\widetilde{b}}(x, y)&:=a_{NC }(u, w) - \langle t, w \rangle _{\partial {\mathcal {T}}}\quad \text { for }x = (u, t) \in X, y = w \in Y. \end{aligned}$$

Lemma 3.10

The bilinear form ${\widetilde{b}}: X_h \times Y_h \rightarrow {\mathbb {R}}$ satisfies the discrete inf-sup condition

$$\begin{aligned} 0 < {\widetilde{\beta }}_h :=\mathop {{{\mathrm{\phantom {p}inf}}}}\limits _{\xi _h \in {{\mathcal {S}}(X_h)}} \sup _{\eta _h \in {{\mathcal {S}}(Y_h)}} {\widetilde{b}}(\xi _h, \eta _h). \end{aligned}$$

(3.8)

Proof

The proof follows the arguments from [8, Thm. 3.5] for the bilinear form ${\widetilde{b}}$ in the lowest-order scheme at hand. $\square$

Proof (of Theorem 3.9)

(a) Since $b'(x_h; 0, s_h, v_h) = -\langle s_h, v_h \rangle _{\partial {\mathcal {T}}} = 0$ for all $s_h \in P_0({\mathcal {E}})$, $v_h \in CR^1_0({\mathcal {T}})$. Then, (3.7) reduces to (R).

(b) Conversely, suppose $(u_h, v_h)$ solves (R), then the second equation in (3.7) follows from the second equation in (R) and $v_h \in CR^1_0({\mathcal {T}})$. The first equation in (R) leads to the first equation in (3.7) for any $t_h \in P_0({\mathcal {E}})$ and test functions in $CR^1_0({\mathcal {T}})$. In other words, the linear functional

vanishes on $CR^1_0({\mathcal {T}}) \subset \ker \varLambda _h$. It remains to show that there exists $t_h \in P_0({\mathcal {E}})$ with , because then $(u_h, t_h, v_h)$ solves (3.7). To prove the existence of such a $t_h$ for $\varLambda _h \in Y_h^*$ with $CR^1_0({\mathcal {T}}) \subset \ker \varLambda _h$, recall the bilinear form ${\widetilde{b}}$ from Lemma 3.10 with discrete inf-sup condition (3.8) and consider the linear problem that seeks $(u_h,t_h,v_h)\in X_h\times Y_h$ with

$$\begin{aligned} \begin{aligned} a_{NC }(v_h, w_1) + {\widetilde{b}}(u_h,t_h,w_1)&= -\varLambda _h(w_1) \quad \text {for all } w_1 \in Y_h,\\ {\widetilde{b}}(w_C,s_0,v_h)&=0 \quad \text {for all } (w_{C }, s_0) \in X_h. \end{aligned} \end{aligned}$$

(L)

Since $\langle r_0, v_1 \rangle _{\partial {\mathcal {T}}} = 0$ for $v_1\in P_1({\mathcal {T}})$ and for all $r_0 \in P_0({\mathcal {E}})$ implies $v_1 \in CR^1_0({\mathcal {T}}) \subset P_1({\mathcal {T}})$, the kernel

$$\begin{aligned} Z_h :=\lbrace v_1 \in P_1({\mathcal {T}}) \,|\, {\widetilde{b}}(x_h, v_1) = 0 \text { for all } x_h \in X_h \rbrace \end{aligned}$$

of ${\widetilde{b}}$ consists of particular Crouzeix–Raviart functions, $Z_h \subset CR^1_0({\mathcal {T}})$, and the discrete Friedrichs inequality [4, p. 301] shows that $a_{NC }$ is $Z_h$-elliptic.

Hence, the Brezzi splitting lemma [3, Thm. 4.3 in Ch. III] applies to the linear system (L) and (L) has a unique solution $(u_h, t_h, v_h) \in X_h \times Y_h$. The test of the first equation in (L) with $w_1 \in CR^1_0({\mathcal {T}}) \subset \ker \varLambda _h$ shows in $CR^1_0({\mathcal {T}})$. The second equation in (L) implies $v_h \in CR^1_0({\mathcal {T}})$ and this proves $v_h = - u_h$. This leads to in $P_1({\mathcal {T}})$. The uniqueness of $t_h$ follows from the fact that in $P_1({\mathcal {T}})$ implies $t_h=0$. $\square$

3.5 Least-squares formulation

Recall $S_0\in P_0({\mathcal {T}};{\mathbb {R}}^{n\times n})$ and $H_0:L^2(\varOmega )\rightarrow P_0({\mathcal {T}};{\mathbb {R}}^n)$ from (3.3) to define an equivalent least-squares formulation.

Theorem 3.11

(dPG is LS) Any $x_h=(u_C,t_0)\in X_h$ and $p_{RT }\in RT_0({\mathcal {T}})$ with $p_{RT }\cdot \nu = t_0$ in $\partial {\mathcal {T}}$ satisfy

(3.9)

Consequently, any solution $x_h=(u_C,t_0)\in X_h$ to (3.7) and $p_{RT }\cdot \nu = t_0$ in $\partial {\mathcal {T}}$ from Remark 3.5 minimizes the weighted least-squares functional (3.9).

Proof

Let $v_1 \in P_1({\mathcal {T}}) \equiv Y_h$ be the Riesz representation of satisfies

$$\begin{aligned} a( v_1 , w_1 ) = b(x_h; w_1 )-F( w_1 ). \end{aligned}$$

The substitution of $t_0 =p_{RT }\cdot \nu$ based on the isometry in Remark 3.5 and an integration by parts lead to

$$\begin{aligned} b(x_h; w_1 )-F( w_1 )&= \int _\varOmega (\sigma (\nabla u_C) -p_{RT }) \cdot \nabla _{NC }w_1 \, \mathrm {d} x- \int _\varOmega (f+{{\mathrm{div}}}p_{RT }) w_1 \, \mathrm {d} x. \end{aligned}$$

With , this results in

$$\begin{aligned}&\int _\varOmega \varPi _0 v_1 \varPi _0 w_1 \, \mathrm {d} x+ \int _\varOmega ( I_{n\times n}+S_0)\nabla _{NC }v_1 \cdot \nabla _{NC }w_1 \, \mathrm {d} x\\&\qquad = \int _\varOmega (\sigma (\nabla u_C) -\varPi _0 p_{RT })\cdot \nabla _{NC }w_1 \, \mathrm {d} x- \int _\varOmega (\varPi _0 f+{{\mathrm{div}}}p_{RT }) \varPi _0 w_1 \, \mathrm {d} x\\&\qquad \phantom {{}={}}\qquad - \int _\varOmega H_0 f\cdot \nabla _{NC }w_1 \, \mathrm {d} x. \end{aligned}$$

For any $T\in {\mathcal {T}}$, the choices $w_1 =\chi _T$ and , $k=1,\dots ,n$, show

$$\begin{aligned} \varPi _0 v_1&= - ({{\mathrm{div}}}p_{RT }+\varPi _0 f),\\ ( I_{n\times n}+S_0)\nabla _{NC }v_1&= \sigma (\nabla u_C) -\varPi _0 p_{RT }- H_0 f. \end{aligned}$$

The Riesz isometry and for any $v_1\in P_1({\mathcal {T}})$ conclude the proof. $\square$

4 Mathematical analysis of dPG for the model problem

This section analyses the low-order dPG method presented in Sect. 3 and proves an a posteriori result next to the existence of a solution and applies the abstract framework from Sect. 2. Recall the discrete spaces $X_h :=S^1_0({\mathcal {T}}) \times P_0({\mathcal {E}})$, $Y_h :=P_1({\mathcal {T}})$, and the nonlinear map from (3.4).

4.1 Well-posedness

This subsection is devoted to the equivalence of the dPG residuals and the errors. For $q_{RT }\in RT_0({\mathcal {T}})$ and $v_C\in S^1_0({\mathcal {T}})$, the isomorphism between $RT_0({\mathcal {T}})$ and $P_0({\mathcal {E}})$ from Remark 3.5 leads to the abbreviation . Recall the energy norm in $H^1_0(\varOmega )$.

Theorem 4.1

The exact solution $u\in H^1_0(\varOmega )$ to the model problem (3.1) with stress $p:=\sigma (\nabla u)\in H({{\mathrm{div}}},\varOmega )$ and any discrete $(v_C, q_{RT })\in S^1_0({\mathcal {T}})\times RT_0({\mathcal {T}})$ satisfy the equivalence

The proof is based on a lemma on the nonlinear least-squares formulation. The related least-squares formulation is associated with the nonlinear residual for the first-order system of (3.1) and defined, for $(p,u)\in H({{\mathrm{div}}},{\varOmega })\times H^1_0({\varOmega })$, by

$$\begin{aligned} {{\mathrm{{\mathcal {R}}}}}(f;p,u) :=(f+{{\mathrm{div}}}p, p - \sigma (\nabla u)). \end{aligned}$$

Lemma 4.2

Any $( p,u),( q,v)\in H({{\mathrm{div}}},{\varOmega }) \times H^1_0({\varOmega })$ satisfy

Proof

Following [23, Thm. 4.4], the fundamental theorem of calculus shows

$$\begin{aligned} {{\mathrm{{\mathcal {R}}}}}(f; q,v) - {{\mathrm{{\mathcal {R}}}}}(f; p,u)&= \int _0^1 \frac{\mathrm {d}}{\mathrm {d} s} {{\mathrm{{\mathcal {R}}}}}(f; p + s( q - p), u + s(v -u)) \, \mathrm {d} s\\&= \int _0^1 {{\mathrm{{\mathcal {R}}}}}'( p + s( q - p), u + s(v -u); q - p, v-u) \, \mathrm {d} s. \end{aligned}$$

For $x\in {\varOmega }$ and $0\le s\le 1$, define $F(s):=\nabla u(x) + s\nabla (v -u)(x)$ and

$$\begin{aligned} M(x) :=\int _0^1 \Big (\phi (\vert F(s) \vert ) I_{n\times n} + \phi '(\vert F(s) \vert ) \frac{F(s) \otimes F(s)}{\vert F(s) \vert }\Big ) \, \mathrm {d} s. \end{aligned}$$

Then

Since the assumptions on $\phi$ show that $M\in L^2({\varOmega };{\mathbb {R}}^{n\times n})$ is pointwise symmetric and positive definite with eigenvalues in the real compact interval $[\gamma _1,\gamma _2]\subset (0,\infty )$, the triangle inequality shows

For the reverse estimate, the positive definiteness of M provides the unique existence of a solution $\alpha \in H^1_0({\varOmega })$ to the weighted problem

$$\begin{aligned} \int _\varOmega M\nabla \alpha \cdot \nabla \gamma \, \mathrm {d} x= \int _\varOmega (q - p) \cdot \nabla \gamma \, \mathrm {d} x\text { for any }\gamma \in H^1_0(\varOmega ). \end{aligned}$$

An integration by parts shows $r:=q-p-M\nabla \alpha \in H({{\mathrm{div}}},{\varOmega })$ with ${{\mathrm{div}}}r = 0$. The Friedrichs inequality with constant $C_{F }$ (i.e. ) implies

The orthogonality of $\nabla H^1_0({\varOmega })$ and $H({{\mathrm{div}}},{\varOmega })\cap \{{{\mathrm{div}}}=0\}$ in $L^2({\varOmega })$ shows

The two previous displayed inequalities, the triangle inequality, and the abbreviation $e:=v-u$ yield

$\square$

Proof (of Theorem 4.1)

Since ${{\mathrm{{\mathcal {R}}}}}(f;p,u)=0$, Lemma 4.2 with $(q,v):=(q_{RT },v_C)$ shows

The $L^2$-orthogonality of $(1-\varPi _0) q_{RT }$ and $(1-\varPi _0) f$ onto piecewise constants implies

$$\begin{aligned}&\Vert f+{{\mathrm{div}}}q_{RT }\Vert _{L^2(\varOmega )}^2 + \Vert q_{RT }- \sigma (\nabla v_C) \Vert _{L^2(\varOmega )}^2\\&\qquad = \Vert \varPi _0 f+{{\mathrm{div}}}q_{RT }\Vert _{L^2(\varOmega )}^2 + \Vert (1-\varPi _0) q_{RT }\Vert _{L^2(\varOmega )}^2\\&\qquad \phantom {{}={}} + \Vert (1-\varPi _0) f \Vert _{L^2(\varOmega )}^2 + \Vert \varPi _0 q_{RT }-\sigma (\nabla v_C) \Vert _{L^2(\varOmega )}^2. \end{aligned}$$

The triangle inequality and the estimates of Remark 3.7 result in

Recall that $S_0$ is pointwise positive semi-definite, hence $I_{n\times n}+S_0$ is positive definite and Remark 3.7 also proves

The proof of the converse estimate utilizes the last estimate and the triangle inequality to show

This and the aforementioned orthogonalities imply

$\square$

4.2 Existence and uniqueness of discrete solutions

The existence of discrete solutions follows from variational arguments, while their uniqueness is fairly open.

Proposition 4.3

The discrete problem (3.5) has a solution.

Proof

The proof follows with the direct method in the calculus of variations and, in the present case of finite dimensions, from the global minimum of a continuous functional on a compact set from the growth condition

$$\begin{aligned} \lim _{\Vert \xi _h \Vert _X \rightarrow \infty } \Vert F-B\xi _h \Vert _{Y_h^*} = \infty . \end{aligned}$$

(4.1)

The latter property follows from Theorem 4.1 up to some perturbation terms. Theorem 3.11 shows

The combination with Theorem 4.1 shows that the right-hand side of Theorem 4.1 is bounded from above by

Hence the left-hand side in Theorem 4.1 is controlled by this and so

Since f and x are fixed, this implies (4.1) and concludes the proof. $\square$

The uniqueness of the exact solution (u, t) on the continuous level does not imply the uniqueness of discrete solutions. There is, however, a sufficient condition for a global unique discrete solution. Notice that $v_h=v=0$ on the continuous level $h=0$ satisfies (4.2).

Theorem 4.4

(a posteriori uniqueness) Suppose that $(u_h,v_h)\in S^1_0({\mathcal {T}})\times CR^1_0({\mathcal {T}})$ solves (R) with ${{\mathrm{D\!}}}\sigma \in C({\mathbb {R}}^n;{\mathbb {R}}^{n\times n}_{\hbox {sym}})$ globally Lipschitz continuous and

$$\begin{aligned} \hbox {Lip}({{\mathrm{D\!}}}\sigma )(1+C_{F }^2)/\gamma _1^2\Vert \nabla _{NC }v_h\Vert _{L^{\infty }({{\varOmega }})} <1 \end{aligned}$$

(4.2)

with the Friedrichs constant $C_{F }$ from in $H^1_0({\varOmega })$. Then (R) has exactly one solution $(u_h,v_h)\in S^1_0({\mathcal {T}})\times CR^1_0({\mathcal {T}})$.

Proof

Suppose that $({\widetilde{u}}_h,{\widetilde{v}}_h)\in S^1_0({\mathcal {T}})\times CR^1_0({\mathcal {T}})$ solves (R) as well and so

$$\begin{aligned} a(v_h,v_h)&= F(v_h) - \int _{\varOmega }\sigma (\nabla u_h)\cdot \nabla _{NC }v_h \, \mathrm {d} x\\&= a({\widetilde{v}}_h, v_h) + \int _{\varOmega }(\sigma (\nabla {\widetilde{u}}_h) - \sigma (\nabla u_h))\cdot \nabla _{NC }v_h \, \mathrm {d} x. \end{aligned}$$

This and the second equation of (R) imply

$$\begin{aligned} a(v_h-{\widetilde{v}}_h,v_h) = \int _{\varOmega }\nabla _{NC }v_h \cdot (\sigma (\nabla {\widetilde{u}}_h) - \sigma (\nabla u_h) - {{\mathrm{D\!}}}\sigma (\nabla u_h)\nabla ({\widetilde{u}}_h - u_h) )\, \mathrm {d} x. \end{aligned}$$

Since $\sigma \in C^1({\mathbb {R}}^n)$ is bounded and ${{\mathrm{D\!}}}\sigma$ Lipschitz continuous, any $A,B\in {\mathbb {R}}^n$ with $F(s):=(1-s)A+sB$ for $0\le s\le 1$ satisfy

$$\begin{aligned}&\vert \sigma (B)-\sigma (A) - {{\mathrm{D\!}}}\sigma (A)(B-A)\vert = \vert \int _0^1 ({{\mathrm{D\!}}}\sigma (F(s))- {{\mathrm{D\!}}}\sigma (A))(B-A)\, \mathrm {d} s\vert \\&\quad \le \hbox {Lip}({{\mathrm{D\!}}}\sigma ) \vert B-A\vert \int _0^1 \vert F(s) - A\vert \, \mathrm {d} s= \frac{1}{2} \hbox {Lip}({{\mathrm{D\!}}}\sigma )\vert B-A\vert ^2. \end{aligned}$$

With $A=\nabla u_h(x)$ and $B=\nabla {\widetilde{u}}_h(x)$ for a.e. x and an integration over ${\varOmega }$, this leads in the preceding identity to

(4.3)

The discrete solutions of (R) lead to the same minimal discrete residual norm and hence

This shows $a(v_h - {\widetilde{v}}_h, v_h + {\widetilde{v}}_h)=0$ and the combination with (4.3) is

(4.4)

On the other hand, ${{\mathrm{D\!}}}\sigma (A) \in {\mathbb {R}}^{n\times n}_{sym }$ has eigenvalues in the compact interval $[\gamma _1, \gamma _2] \subset (0, \infty )$ and so, for all $A, B \in {\mathbb {R}}^n$,

$$\begin{aligned} \begin{aligned} \gamma _1 \vert A-B \vert ^2&\le \int _0^1 (A-B) \cdot {{\mathrm{D\!}}}\sigma (B + s(A-B)) (A-B) \, \mathrm {d} s\\&= (\sigma (A) - \sigma (B))\cdot (A-B) \le \gamma _2 \vert A-B \vert ^2. \end{aligned} \end{aligned}$$

(4.5)

With $A=\nabla u_h(x)$ and $B=\nabla {\widetilde{u}}_h(x)$, and an integration over a.e. $x\in {\varOmega }$, this shows

The first identity in (R) for $(u_h,v_h)$ and $({\widetilde{u}}_h,{\widetilde{v}}_h)$, respectively, results in

The combination with the previous inequality shows

(4.6)

The substitution in (4.4) results in

$$\begin{aligned} \Vert v_h - {\widetilde{v}}_h\Vert _{Y_h}^2 \le \hbox {Lip}({{\mathrm{D\!}}}\sigma )(1+C_{F }^2)/\gamma _1^2\Vert \nabla _{NC }v_h\Vert _{L^{\infty }({{\varOmega }})} \Vert v_h - {\widetilde{v}}_h\Vert _{Y_h}^2. \end{aligned}$$

This and (4.2) show $v_h = \widetilde{v}_h$. Then (4.6) implies $u_h = {\widetilde{u}}_h$. $\square$

4.3 Best-approximation

For any $v\in H^1({\mathcal {T}})$, the nonconforming interpolation $I_{NC }^{loc }v \in P_1({\mathcal {T}})$ is defined, on each triangle $T\in {\mathcal {T}}$, by piecewise linear interpolation of the values

(4.7)

at the midpoints of the sides $E\in {\mathcal {E}}(T)$.

Proposition 4.5

The operator $\varPi :=I_{NC }^{loc }$ satisfies Hypothesis 2.5.

Proof

Given $v \in H^1({\mathcal {T}})$, set $v_h :=I_{NC }^{loc }v \in P_1({\mathcal {T}})$. For every $K\in {\mathcal {T}}$, an integration by parts leads to

$$\begin{aligned} \nabla \big (v_h\big \vert _{K}\big ) = \int _{\partial K} v_h \cdot \nu _K\, \mathrm {d} s= \int _{\partial K} v\cdot \nu _K \, \mathrm {d} s= \int _K \nabla v \, \mathrm {d} x\Big / \vert K \vert . \end{aligned}$$

Since $\nabla w_C\in P_0({\mathcal {T}};{\mathbb {R}}^n)$ and (3.2) shows ${{\mathrm{D\!}}}\sigma (\nabla u_C) \in P_0({\mathcal {T}};{\mathbb {R}}^{n\times n})$ for all $w_C,u_C \in S^1_0({\mathcal {T}})$, this implies

$$\begin{aligned} \int _\varOmega \nabla w_C \cdot ({{\mathrm{D\!}}}\sigma (\nabla u_C) \nabla _{NC }(v-v_h)) \, \mathrm {d} x=0. \end{aligned}$$

Moreover, (4.7) guarantees that any $s_0\in P_0({\mathcal {E}})$ satisfies

$$\begin{aligned} \langle s_0, v-v_h \rangle _{\partial {\mathcal {T}}} = 0. \end{aligned}$$

Consequently, any $x_h=(u_C,t_0)\in X_h$ and $\xi _h=(w_C,s_0)\in X_h$ satisfy

$$\begin{aligned} b'(x_h; \xi _h,v-v_h)&= \int _\varOmega \nabla w_C \cdot ({{\mathrm{D\!}}}\sigma (\nabla u_C) \nabla _{NC }(v-v_h)) \, \mathrm {d} x- \langle s_0, v-v_h \rangle _{\partial {\mathcal {T}}}= 0. \quad \end{aligned}$$

$\square$

The estimates for the function ${{\mathrm{D\!}}}\sigma$ from Sect. 3.1 lead to an explicit generic constant for the best-approximation estimate from Theorem 2.8 without any local hypothesis.

Theorem 4.6

(best-approximation)

Let $x = (u,t) \in X$ be the unique solution to $B(x) = F$ for the nonlinear map B from (3.4) in Sect. 3.3. Any discrete solution $(u_h,t_h;v_h) \in X_h\times Y_h$ to (3.5) satisfies

Proof

Given the best-approximation $x_h^*=(u_h^*,t_h^*)\in X_h$ to (u, t) in $X_h$ and let $A=\nabla u_h^*(x)$ and $B=\nabla u_h(x)$ in (4.5) and integrate over a.e. $x\in {\varOmega }$. Then

Since , the last term is equal to

$$\begin{aligned} \begin{aligned} F(u_h^*&- u_h) - b(x_h; u_h^* - u_h) + b(x_h^*; u_h^* - u_h)- b(u; u_h^* - u_h)\\&= a(v_h, u_h^* - u_h) + \int _\varOmega \big ( \sigma (\nabla u_h^*) - \sigma (\nabla u) \big ) \cdot \nabla (u_h^* - u_h) \, \mathrm {d} x. \end{aligned} \end{aligned}$$

(4.8)

The Lipschitz continuity from Remark 3.1 leads to

(4.9)

and the last term in (4.8) is controlled by

(4.10)

Since $x_h$ is a global discrete minimizer,

The Lipschitz continuity (4.9) of $\sigma$ and the structure of the map b from (3.4) show that the last term is . The combination of with (4.8) and (4.10) shows

A triangle inequality concludes the proof with explicit constants

$\square$

The following a posteriori error estimate holds for any discrete approximation, and even for inexact solve, and generalizes the built-in error control despite inexact solve of [5, Thm. 2.1] to the nonlinear model problem at hand.

Theorem 4.7

(a posteriori) There exist universal constants $\kappa \approx 1\approx C_{d {F }}$ such that the exact solution $(u,t)\in X$ of $B(x)=F$ and any discrete $(v_C,s_0)\in S^1_0({\mathcal {T}})\times P_1({\mathcal {T}})$ satisfy

Remark 4.8

The proof reveals that $C_{d {F }}$ is the constant in the discrete Friedrichs inequality [4, p. 301] in $CR^1_0({\mathcal {T}})$. The explicit bounds of $C_{d {F }}$ in [10] allow quantitative estimates in 2D and show in particular $C_{d {F }}\le 6.24$ for a convex domain with $\hbox {diam}({\varOmega })\le 1$ and a triangulation with right isosceles triangles.

Remark 4.9

The proof reveals that $\kappa$ is the constant in interpolation error estimate for the nonconforming interpolation operator $\Vert h_{\mathcal {T}}^{-1} (1-I_{NC })v\Vert \le \kappa \Vert \nabla _{NC }(1-I_{NC })v \Vert$ for $v\in H^1(\varOmega )$. An estimate with the first positive root $j_{1,1}$ of the Bessel function of the first kind in [7, Thm. 4] in 2D reads $\kappa =(1/48+1/j_{1,1}^2)^{1/2}= 0.29823$.

Proof (of Theorem 4.7)

The estimate (4.5) with $A = \nabla u(x)$, $B = \nabla u_h(x)$, $e:=u-u_h$, and an integration over a.e. $x\in \varOmega$ leads to

(4.11)

Since $(u,t) \in X$ solves in $Y^*$ and with the nonconforming interpolation operator (4.7), this is equal to

The interpolation error estimate for the nonconforming interpolation operator with constant $\kappa$ [7, Thm. 4] yields

The discrete Friedrichs inequality [4, p. 301] and $I_{NC }e\in CR^1_0({\mathcal {T}})$ prove

The Cauchy inequality in ${\mathbb {R}}^2$ and the theorem of Pythagoras imply

$\square$

4.4 Other nonlinear dPG methods

This section illustrates the plethora of dPG methodology by introducing the primal mixed, the dual, and the ultraweak dPG method for the nonlinear model problem. All three methods concern the first-order system of (3.1) with the convex function $\varphi$ and and its dual $\varphi ^*$ so that the relation $p=\sigma (\nabla u)$ is equivalent to $\nabla u ={{\mathrm{D\!}}}\varphi ^*(\vert p \vert ){{\mathrm{sign}}}{p}$ on the continuous level. Recall the space of functions with piecewise divergence $H({{\mathrm{div}}};{\mathcal {T}}) :=\prod _{T \in {\mathcal {T}}} H({{\mathrm{div}}};T)$ from [8] as well as the piecewise version $RT^{NC }_k({\mathcal {T}})\subset H({{\mathrm{div}}};{\mathcal {T}})$ of $RT_k({\mathcal {T}})$, and the subspace $S^k_0({\mathcal {E}})\equiv S^k_0({\mathcal {T}})\big \vert _{\partial {\mathcal {T}}}$ of

$$\begin{aligned} H^{1/2}_0(\partial {\mathcal {T}})&:=\{ s = (s_T)_{T\in {\mathcal {T}}}\in {\textstyle \prod _{T\in {\mathcal {T}}} } H^{1/2}(\partial T) \,|\, \exists v\in H^1_0({\varOmega }),\\&\qquad \forall T \in {\mathcal {T}},s_T=(v|_T)|_{\partial T}\}. \end{aligned}$$

Recall the primal nonlinear dPG method (dPG) in Sect. 3.3 with b from (3.4) and general polynomial degree $k\ge 0$ and $m\ge k$ in the discrete spaces

$$\begin{aligned} X_h := S^{k+1}_0({\mathcal {T}}) \times P_k({\mathcal {E}}) \text { and }Y_h := P_{m+1}({\mathcal {T}}). \end{aligned}$$

The primal mixed nonlinear dPG method departs from a piecewise integration by parts and employs the spaces and discrete subspaces

$$\begin{aligned} X&:= L^2(\varOmega ;{\mathbb {R}}^n) \times H^1_0({\mathcal {T}}) \times H^{1/2}(\partial {\mathcal {T}})\text { and }Y:= L^2(\varOmega ;{\mathbb {R}}^n) \times H^1({\mathcal {T}}),\\ X_h&:= P_k({\mathcal {T}};{\mathbb {R}}^n) \times S^{k+1}_0({\mathcal {T}}) \times P_k({\mathcal {E}}) \text { and }Y_h := P_m({\mathcal {T}};{\mathbb {R}}^n) \times P_{m+1}({\mathcal {T}}). \end{aligned}$$

For $(p,u,t)\in X$ and $(q,v)\in Y$, define (dPG) with $F(q,v) := (f,v)_{L^2(\varOmega )}$ and

$$\begin{aligned} b(p,u,t;q,v)&:= (p-\sigma (\nabla u),q)_{L^2(\varOmega )}+(p,\nabla _{NC }v)_{L^2(\varOmega )} -\langle t,v\rangle _{\partial {\mathcal {T}}}. \end{aligned}$$

The dual nonlinear dPG method utilizes the spaces and discrete subspaces

$$\begin{aligned} X&:= H({{\mathrm{div}}};\varOmega ) \times L^2(\varOmega ) \times H^{1/2}_0(\partial {\mathcal {T}})\text { and }Y:= H({{\mathrm{div}}};{\mathcal {T}}) \times L^2(\varOmega ),\\ X_h&:= RT_k({\mathcal {T}})\times P_k({\mathcal {T}}) \times S^{k+1}_0({\mathcal {E}})\text { and } Y_h := RT_m^{NC }({\mathcal {T}}) \times P_m({\mathcal {T}}) . \end{aligned}$$

For F as before and $(p,u,s)\in X$ and $(q,v)\in Y$, define (dPG) with $b(p,u,s;q,v):=$

$$\begin{aligned} ({{\mathrm{D\!}}}\varphi ^*(\vert p \vert ){{\mathrm{sign}}}{p},q)_{L^2(\varOmega )} + (u,{{\mathrm{div}}}_{NC }q)_{L^2(\varOmega )}-({{\mathrm{div}}}p,v)_{L^2(\varOmega )} -\langle q\cdot \nu , s\rangle _{\partial {\mathcal {T}}}. \end{aligned}$$

The ultraweak nonlinear dPG method utilizes a piecewise integration by parts in both equations of the first-order system and the spaces

$$\begin{aligned} X&:= L^2(\varOmega ;{\mathbb {R}}^n) \times L^2(\varOmega ) \times H^{1/2}(\partial {\mathcal {T}}) \times H^{1/2}_0(\partial {\mathcal {T}})\text { and } Y:= H({{\mathrm{div}}};{\mathcal {T}}) \times H^1({\mathcal {T}}),\\ X_h&:= P_k({\mathcal {T}};{\mathbb {R}}^n) \times P_k({\mathcal {T}}) \times P_k({\mathcal {E}}) \times S^{k+1}_0({\mathcal {E}}) \text { and } Y_h:= RT_m^{NC }({\mathcal {T}}) \times P_{m+1}({\mathcal {T}}). \end{aligned}$$

For F from the above primal mixed method and $(p,u,t,s)\in X$ and $(q,v)\in Y$, define (dPG) with

$$\begin{aligned} b(p,u,t,s;q,v)&:= ({{\mathrm{D\!}}}\varphi ^*(\vert p \vert ){{\mathrm{sign}}}{p},q)_{L^2(\varOmega )} + (p,\nabla _{NC }v)_{L^2(\varOmega )}\\&\quad \quad + (u, {{\mathrm{div}}}_{NC }q)_{L^2(\varOmega )}-\langle q\cdot \nu , s\rangle _{\partial {\mathcal {T}}}-\langle t, v\rangle _{\partial {\mathcal {T}}}. \end{aligned}$$

The linear version is analysed in [2, 6, 8, 18]. The four nonlinear dPG methods may be further analysed in the spirit of this section.

5 Numerical experiments

This section presents numerical experiments with the LS-FEM of Sect. 3.5.

5.1 Computational realization

Given $f \in L^2(\varOmega )$, the discrete solution of (3.7) is computed by a Newton scheme with an initial iterate from the solution of the scaled linear Poisson model problem. Let $S^1_0({\mathcal {T}})$ be endowed with the energy norm and $RT_0({\mathcal {T}})$ with and let denote the norm of the dual space of $S^1_0({\mathcal {T}}) \times RT_0({\mathcal {T}})$. The first Fréchet derivative of belongs to the dual space of $S^1_0({\mathcal {T}}) \times RT_0({\mathcal {T}})$. After at most 5 Newton iterations, every displayed discrete solution $(u_h, p_h)$ in the following subsections satisfy up to machine precision. In the case of successive mesh-refinement, the iteration starts with the prolongated solution from the coarser triangulation and terminates in at most 3 or 4 iterations.

Table 1 presents the errors of the Newton iterate $(u_h^{(j)}, p_h^{(j)})$ for $j=0,1,\dots ,5$ on fixed triangulations of the square domain from Sect. 5.2 and the L-shaped domain from Sect. 5.4 with the convex function $\phi$ from Example 3.2.a. The iterations (A) and (B) utilize a uniform triangulation of the square domain with $4\,096$ triangles ($\texttt {ndof}= 8\,193$) and an initial iterate from a Poisson model problem for (A) and a weighted Poisson problem with constant weight 2.5 in (B). The adaptive mesh of the L-shaped domain with $3\,450$ triangles ($\texttt {ndof}= 6.901$) in (C) and (D) has been generated by the algorithm from Sect. 5.3 below at level $\ell =12$. Iteration (C) starts with a weighted Poisson solution with constant factor 2.5 and (D) with the prolongated solution from the previous mesh.

From the very beginning of the Newton iteration, all values in Table 1 provide numerical evidence for Q-quadratic convergence.

Table 1 Convergence history of Newton iteration for 4 representative examples

In order to investigate the uniqueness of discrete solutions, the minimal and the maximal eigenvalue $\lambda _{min }$ and $\lambda _{max }$ of the Hessian matrix of the least-squares functional is computed, where $(u_h, q_h) \in X_h$ and $\lambda \in {\mathbb {R}}$ satisfy, for all $({\widetilde{v}}_h, {\widetilde{q}}_h) \in X_h$,

$$\begin{aligned} {{\mathrm{{\text {D}}^2\!}}}{LS}(f; u_h, p_h; v_h, q_h, {\widetilde{v}}_h, {\widetilde{q}}_h) = \lambda \big ( a_{NC }(v_h, {\widetilde{v}}_h) + (q_h, {\widetilde{q}}_h)_{H({{\mathrm{div}}},\varOmega }\big ). \end{aligned}$$

(5.1)

The value $\lambda _{min }$ is uniformly bounded from zero for the examples in the following subsections, so that every computed discrete solution $(u_h, p_h)$ is a local minimizer.

For any discrete approximation $(u_h, p_h)$, Theorems 3.11 and 4.7 verify the a posteriori error estimator $\eta ^2({\mathcal {T}}) :={LS }(f; u_h, p_h) + \Vert h_{\mathcal {T}}f \Vert _{L^2(\varOmega )}^2$ even for inexact solve in its computation. In view of a lacking proof in Sect. 5.4 below that the computed discrete solution is in fact a global discrete minimizer (at least up to machine precision), it is only by this universal a posteriori error control that we know that the computed approximations converge to the exact solution.

5.2 Numerical example on square domain

This subsection considers the nonlinear model problem for the exact solution

$$\begin{aligned} u(x) :=\cos (\pi x_1/2) \cos (\pi x_2/2) \quad \text {for } x \in \varOmega :=(-1,1)^2 \end{aligned}$$

with homogeneous Dirichlet boundary condition, $f :=-{{\mathrm{div}}}(\sigma (\nabla u))$, and $\phi$ from Example 3.2.a. This defines the exact stress function $p:=\sigma (\nabla u) \in H({{\mathrm{div}}},\varOmega )$.

Figure 1 displays the error estimator $\eta _\ell :=\eta ({\mathcal {T}}_\ell )$ at the discrete solutions $(u_\ell , p_\ell )$ on each level $\ell$ of a sequence of uniform triangulations as well as the error to the exact solution (u, p). The reference energy $E(u) = -5.774337908509$ in the energy difference has been approximated by the energies of $P_1$-conforming finite element solution with an Aitken $\Delta ^2$ extrapolation. The eigenvalues of (5.1) in all experiments of Fig. 1 satisfy $1.597\le \lambda _{min }\le 1.722$ and $9.943\le \lambda _{max }\le 16.128$ and so prove that the discrete solutions are local minimizers. The parallel graphs confirm the equivalence of the built-in error estimator $\Vert y_\ell \Vert _{Y} = ({LS}(f; u_\ell , p_\ell ))^{1/2}$ with the exact error from Theorem 4.1.

Fig. 1

Convergence history for a sequence of uniform triangulations of the square domain with exact solution u from Sect. 5.2

With the Friedrichs constant $C_{F }= \sqrt{2}/\pi$ of the square domain, the criterion (4.2) is equivalent to $\Vert \nabla _{NC }v_h \Vert _{L^\infty (\varOmega )} < \gamma _1^2 \hbox {Lip}({{\mathrm{D\!}}}\sigma )^{-1} (1+C_{F }^2)^{-1} = 0.17239892$ and so Fig. 1 shows that the criterion (4.2) holds for each level $\ell \ge 6$ and Theorem 4.4 implies global uniqueness of the computed $(u_\ell , p_\ell )$. This proves that there exists only one local minimizer in the discrete problem (dPG).

5.3 Adaptive mesh-refinement

The natural adaptive algorithm with collective Dörfler marking [20] utilizes the local error estimator $\eta ^2({\mathcal {T}}, T):=\Vert ( I_{n\times n}+S_0) ^{-1/2}\big (\varPi _0 p_{RT }- \sigma ( \nabla u_C) + H_0 f\big )\Vert _{L^{2}({T})}^2+ \Vert \varPi _0 f + {{\mathrm{div}}}p_{RT }\Vert _{L^{2}({T})}^2 + \Vert h_T f \Vert _{L^2(T)}^2$ for any $(u_C,p_{RT })\in S^1_0({\mathcal {T}})\times RT_0({\mathcal {T}})$ and $T\in {\mathcal {T}}$ as follows.

• Input: Regular triangulation ${\mathcal {T}}_0$ of the polygonal domain $\varOmega$into simplices.

• for any level $\ell =0,1,2,\dots$ do

  • Solve generalized LS-FEM with respect to triangulation ${\mathcal {T}}_\ell$ and solution $(u_\ell , p_\ell )$.

  • Compute error estimator $\eta _\ell :=\eta ({\mathcal {T}}_\ell )$.

  • Mark a subset ${\mathcal {M}}_\ell \subseteq {\mathcal {T}}_\ell$ of (almost) minimal cardinality $\vert {\mathcal {M}}_\ell \vert$ with

    $$\begin{aligned} 0.3\,\eta _\ell ^2 \le \eta _\ell ^2({\mathcal {M}}_\ell ) :=\sum _{T \in {\mathcal {M}}_\ell } \eta ^2({\mathcal {T}}_\ell , T) \end{aligned}$$

  • Compute smallest regular refinement ${\mathcal {T}}_{\ell +1}$ of ${\mathcal {T}}_\ell$ with ${\mathcal {M}}_\ell \subseteq {\mathcal {T}}_\ell \setminus {\mathcal {T}}_{\ell +1}$ by newest-vertex bisection (NVB). od

• Output: Sequence of discrete solutions $(u_\ell , p_\ell )_{\ell \in {\mathbb {N}}_0}$ and triangulations $({\mathcal {T}}_\ell )_{\ell \in {\mathbb {N}}_0}$.

See [26] for details on adaptive mesh-refinement and NVB.

5.4 Numerical example on L-shaped domain

This subsection considers $f \equiv 1$ on the L-shaped domain $\varOmega :=(-1,1)^2\setminus [0,1]^2 \subset {\mathbb {R}}^2$ with homogeneous Dirichlet boundary condition $u\vert _{\partial \varOmega } \equiv 0$ and unknown exact solution u. Figure 2 displays the corresponding discrete solutions $u_h$ on a uniform triangulation of $\varOmega$ for the different functions $\phi$ from Example 3.2.a and b.

Fig. 2

Solution $u_h$ for different functions $\phi$ on a uniform triangulation of the L-shaped domain into 768 ($\texttt {ndof}= 1\,537$). a $\phi$ from Example 3.2.a. b $\phi$ from Example 3.2.b

Figure 3 shows two typical adaptively generated triangulations with considerable refinement at the re-entrant corner for different functions $\phi$. At first glance, the meshes appear similar and resemble the undisplayed adaptive triangulation from the Poisson model problem.

Fig. 3

Adaptively refined triangulation ${\mathcal {T}}_\ell$ for different functions $\phi$. a $\phi$ from Example 3.2.a, mesh with $1\,783$ triangles ($\texttt {ndof}=3\,567$). b $\phi$ from Example 3.2.b, mesh with $1\,838$ triangles ($\texttt {ndof}=3\,677$)

For $\phi$ from Example 3.2.a, Fig. 4 shows the convergence history plot of the natural least-squares error estimator $\eta _\ell =\eta ({\mathcal {T}}_\ell )$ and the difference of the energy $E(u_\ell )$ of the solution $u_\ell$ and a reference energy $E(u) = -3.657423002939 \times 10^{-2}$ (computed by the energies of $P_1$-conforming finite element solutions with an Aitken $\Delta ^2$ extrapolation).

The eigenvalues of (5.1) in all experiments satisfy $1.787\le \lambda _{min }\le 1.914$ and $16.682\le \lambda _{max }\le 17.932$ and so prove that all the discrete solutions are local minimizers. The function $\phi$ from Example 3.2.b leads to (undisplayed) similar results.

For the L-shaped domain, the smallest eigenvalue $\lambda _1 = 9.6397238$ of the Laplacian with homogeneous Dirichlet boundary conditions yields the Friedrichs constant $C_{F }= 1/\sqrt{\lambda _1} = 0.32208293$. Since $\Vert \nabla _{NC }v_\ell \Vert _{L^\infty (\varOmega )} \ge \gamma _1^2 \hbox {Lip}({{\mathrm{D\!}}}\sigma )^{-1} (1+C_{F }^2)^{-1} = 0.22650326$ for all level $\ell \in {\mathbb {N}}_0$ in Fig. 4, Theorem 4.4 is not applicable to any triangulation ${\mathcal {T}}_\ell$ of the computation at hand.

To guarantee optimal convergence rates for least-squares FEMs with an alternative a posteriori error estimator, the choice of a sufficiently small bulk parameter is crucial [11, 14]. However, for the natural error estimator with the values of the least-squares functional, the plain convergence proof of [12] requires the bulk parameter sufficiently close to 1. For the nonlinear model problem at hand, the convergence history plot in Fig. 4 provides numerical evidence for optimal convergence rates for adaptive mesh-refinement of Sect. 5.3 and suboptimal convergence for uniform refinement.

Fig. 4

Convergence history for adaptive mesh-refinement (solid lines) and uniform mesh-refinement (dotted lines) with $\phi$ from Example 3.2.a