A Hybridized Discontinuous Galerkin Method with Reduced Stabilization

Abstract

In this paper, we propose a hybridized discontinuous Galerkin (HDG) method with reduced stabilization for the Poisson equation. The reduce stabilization proposed here enables us to use piecewise polynomials of degree $k$ and $k-1$ for the approximations of element and inter-element unknowns, respectively, unlike the standard HDG methods. We provide the error estimates in the energy and $L^2$ norms under the chunkiness condition. In the case of $k=1$, it can be shown that the proposed method is closely related to the Crouzeix–Raviart nonconforming finite element method. Numerical results are presented to verify the validity of the proposed method.

1 Introduction

In this paper, we propose a new hybridized discontinuous Galerkin (HDG) method with reduced stabilization. We consider the Poisson equation with homogeneous Dirichlet boundary condition as a model problem:

$$\begin{aligned} -\Delta u&= f \quad \hbox { in } \varOmega , \end{aligned}$$

(1a)

$$\begin{aligned} u&= 0 \quad \hbox { on } \partial \varOmega . \end{aligned}$$

(1b)

Here $\varOmega \subset {\mathbb {R}}^2$ is a convex polygonal domain, and $f \in L^2(\varOmega )$ is a given function. For simplicity, we here deal with only the two-dimensional case, although the proposed method can be applied to the three-dimensional problems.

The HDG methods are already applied to various problems and are still being developed. For second order elliptic problems, the HDG methods were introduced and analyzed by Cockburn et al. [7, 8]. The embedded discontinuous Galerkin (EDG) methods, which is the continuous-approximation version of the HDG method, were analyzed in [9]. In these HDG methods, the numerical traces are hybridized in a mixed formulation. In [16], another approach of the HDG method was proposed. The hybridization presented in [16] is based on the hybrid displacement method proposed by Tong for linear elasticity problems [17]. The resulting scheme is almost equivalent to the IP-H [8].

In the formulations of HDG methods, element and hybrid unknowns are introduced. The element unknown can be eliminated by the hybrid unknown, which allows us to reduce the number of the globally coupled degrees of freedom. In all the standard HDG methods, we need to use two polynomials of equal degree for the approximations of the element and hybrid unknowns in order to achieve optimal convergences. The motivation of the reduced stabilization we propose is to use piecewise polynomials of degree $k$ and $k-1$ for approximations of the element and hybrid unknowns, respectively, which we call $P_k$–$P_{k-1}$ approximation. In [4, 5], reduced stabilization was introduced for the discontinuous Galerkin method (the RIP-method). In [14], Lehrenfeld first proposed a reduced HDG scheme by adjusting a numerical flux, which is equivalent to our proposed scheme. However, no error analysis was presented.

In [9], it is proved that the hybrid part of an EDG solution coincides with the trace of a finite element solution with the linear element. Analogously, the proposed method with $P_1$–$P_0$ approximation is closely related to the Crouzeix–Raviart nonconforming finite element method [12]. We prove that our approximate solution coincides with the Crouzeix–Raviart approximation at the midpoints of edges.

We provide a priori error estimates under the chunkiness condition. The optimal error estimates in the energy norm are proved. In terms of the $L^2$-errors, it is shown that the convergence rates are optimal when the scheme is symmetric. However, for the nonsymmetric schemes, we prove only the sub-optimal estimates in the $L^2$ norm due to the lack of adjoint consistency. We also provide the easy implementation by means of the Gaussian quadrature formula in the two-dimensional case. Unfortunately, such implementation is impossible in the three-dimensional case.

This paper is organized as follows. Sect. 2 is devoted to the preliminaries. In Sect. 3, we introduce reduced stabilization, and describe the proposed method. In Sect. 4, we provide the error estimates in the energy and $L^2$ norms under the chunkiness condition. In Sect. 5, numerical results are presented to verify the validity of our scheme. Finally, in Sect. 6, we end with a conclusion.

2 Preliminaries and Notation

2.1 Chunkiness Condition

Let $\{\mathcal {T}_h\}_h$ be a family of meshes of $\varOmega$. Each element $K \in {\mathcal {T}}_h$ is assumed to be a polygonal domain star-shaped with respect to a ball of which radius is $\rho _K$. Let $h_K = \mathrm{diam}K$ and $h = \max _{K \in \mathcal {T}_h} h_K$. We assume that the boundary $\partial K$ of $K \in {\mathcal {T}}_h$ is composed of $m$-faces and $m$ is bounded by $M$ from above independently of $h$. Let us denote $\mathcal {E}_h = \{ e \subset \partial K{:}K \in \mathcal {T}_h\}$. In this paper, we assume that the family $\{\mathcal {T}_h\}_h$ satisfies the chunkiness condition [3, 13]: there exists a positive constant $\gamma _C$ independent of $h$ such that

$$\begin{aligned} \frac{h_K}{\rho _K} \le \gamma _{C} \quad \forall K \in \mathcal {T}_h. \end{aligned}$$

(2)

From the chunkiness condition, a kind of cone condition follows [3, 13]. Let $\tilde{T}$ be a reference triangle with the height of $\gamma _T>0$. We assume that $\tilde{T}$ is an isosceles triangle. Let $\tilde{e}$ denote the base of $\tilde{T}$. For each $K \in \mathcal {T}_h$ and $e \subset \partial K$ (Fig. 1), let $F_{e,K}$ be an affine-linear mapping from $\tilde{T}$ onto $T \subset K$ such that $F_{e,K}(\tilde{e}) = e$ and the height of $T$ is equal to $\gamma _T h_e$. The constant $\gamma _T$ depends only on the chunkiness parameter $\gamma _C$. Note that there exists a constant $\gamma _E \ge 1$ such that

$$\begin{aligned} \frac{h_T}{h_e} \le \gamma _E. \end{aligned}$$

(3)

Fig. 1

Triangle condition

2.2 Function Spaces

We introduce the piecewise Sobolev spaces over $\mathcal {T}_h$, i.e., $H^k({\mathcal {T}}_h) = \{ v \in L^2(\varOmega ) : v|_K \in H^k(K) \ \forall K \in \mathcal {T}_h \}.$ The skeleton of ${\mathcal {T}}_h$ is defined by $\varGamma _h = \bigcup _{e\in {\mathcal {E}}_h} e.$ We use $L^2_D(\varGamma _h) = \{ \hat{v} \in L^2(\varGamma _h) : \hat{v} = 0 \mathrm on \partial \varOmega \}$ and $\varvec{V} = H^2(\mathcal {T}_h) \times L_D^2(\varGamma _h)$ for the hybridized formulation of the continuous problem. Let us define $\varvec{V}(h) = \{ (v, v|_{\varGamma _h}) : v \in H^2(\varOmega )\}$ $\subset H^2(\varOmega ) \times H^{3/2}(\varGamma _h)$, where $v|_{\varGamma _h}$ stands for the trace of $v_h$ on $\varGamma _h$. We use the following inner products:

$$\begin{aligned} (u,v)_{\mathcal {T}_h} = \sum _{K \in \mathcal {T}_h}\int _K uvdx,\quad \langle u, v \rangle _{\partial \mathcal {T}_h} = \sum _{K \in \mathcal {T}_h}\int _{\partial K} u v ds \end{aligned}$$

for $u, v \in L^2(\varOmega )$ or $L^2_D(\varGamma _h)$.

2.3 Finite Element Spaces and Projections

Let ${\mathcal {P}}^k({\mathcal {T}}_h)$ be the function space of element-wise polynomials of degree $k$ over ${\mathcal {T}}_h$, and ${\mathcal {P}}^l({\mathcal {E}}_h)$ be the space of edge-wise polynomials of degree $l$ over ${\mathcal {E}}_h$, where $k$ and $l$ are nonnegative integers. Then we define $V_h^k = {\mathcal {P}}^k({\mathcal {T}}_h)$ and $\hat{V}_h^l = {\mathcal {P}}^l({\mathcal {E}}_h) \cap L^2_D(\varGamma _h)$. We employ $\varvec{V}^{k,l}_h = V^k_h \times \hat{V}^l_h$ as finite element spaces of $\varvec{V}$. Let us denote by $\mathsf{P}_{k}$ the $L^2$-projection from $L^2(\varGamma _h)$ onto ${\mathcal {P}}^{k}(\mathcal {E}_h)$.

2.4 Mesh-Dependent Norms

Let $\Vert \cdot \Vert _m$ and $|\cdot |_m$ be the usual Sobolev norms and seminorms in the sense of [1], respectively. We introduce auxiliary mesh-dependent seminorms:

$$\begin{aligned} |v|_{1,h}^2&= \sum _{K\in {\mathcal {T}}_h} |v|_{1,K}^2 \quad \mathrm for \ v \in H^1({\mathcal {T}}_h), \end{aligned}$$

(4)

$$\begin{aligned} |v|_{2,h}^2&= \sum _{K\in {\mathcal {T}}_h} h_K^2|v|_{2,K}^2 \quad \mathrm for \ v \in H^2({\mathcal {T}}_h), \end{aligned}$$

(5)

$$\begin{aligned} |\varvec{v}|_\mathrm{j }^2&= \sum _{K\in {\mathcal {T}}_h} \sum _{e \subset \partial K} \frac{1}{h_e}\left\| \mathsf{P}_{k-1}(\hat{v} - v)\right\| ^2_{0,e} \quad \mathrm for \ \varvec{v}=\{v,\hat{v}\} \in \varvec{V}, \end{aligned}$$

(6)

where $h_e$ is the diameter of $e$. Note that $\mathsf{P}_{k-1} v$ in (6) is defined by $\mathsf{P}_{k-1}(\mathrm{trace}(v|_K))$, which is well-defined, whereas $v$ may be double-valued on $\partial K$. In our error analysis, we use the following energy norm:

$$\begin{aligned}&|||\varvec{v}|||\,^2 = |v|_{1,h}^2 +|v|_{2,h}^2 +|\varvec{v}|_\mathrm{j }^2 \quad \text { for } \varvec{v} = \{v,\hat{v}\} \in \varvec{V}. \end{aligned}$$

2.5 Trace and Inverse Inequalities

We here state the trace and inverse inequalities without proofs. The constants appearing in the inequalities are independent of $h$, $K \in \mathcal {T}_h$ and $e \subset \partial K$ under the chunkiness condition.

Lemma 1

(Trace inequality) Let $K \in {\mathcal {T}}_h$ and $e$ be an edge of $K$. There exists a constant $C$ independent of $K$, $e$ and $h$ such that

$$\begin{aligned} \Vert v\Vert _{0,e} \le Ch_e^{-1/2} \left( \Vert v\Vert _{0,K}^2 + h_K^2|v|_{1,K}^2 \right) ^{1/2} \qquad \forall v \in H^1(K). \end{aligned}$$

(7)

Proof

Refer to [13]. $\square$

Lemma 2

(Inverse inequality) Let $K \in {\mathcal {T}}_h$. There exists a constant $C$ independent of $K$ and $h$ such that

$$\begin{aligned} |v_h|_{1,K} \le Ch_K^{-1} \Vert v_h\Vert _{0,K} \qquad \forall v_h \in {\mathcal {P}}^k(K). \end{aligned}$$

(8)

Proof

Refer to [3]. $\square$

We will use the lemma below to bound the terms of the complementary projection, $\mathsf{I}-\mathsf{P}_{k-1}$.

Lemma 3

There exists a constant $C$ independent of $h$ such that, for all $v \in H^1(K)$,

$$\begin{aligned} h_e^{-1}\Vert (\mathsf{I}- \mathsf{P}_{k-1})v\Vert _{0,e}^2 \le C |v|_{1,K}^2. \end{aligned}$$

(9)

Proof

Let $\tilde{T}$ be a reference triangle and $\tilde{e}$ be the base of $\tilde{T}$ as illustrated in Fig. 1. Let $\tilde{v} \in {\mathcal {P}}^k(\tilde{T})$ be arbitrarily fixed. We define a linear functional on $H^1(\tilde{T})$ by

$$\begin{aligned} G(\tilde{w}) = \langle (\mathsf{I}-\mathsf{P}_{k-1})\tilde{v}, \tilde{w} \rangle _{\tilde{e}}. \end{aligned}$$

Note that the functional $G$ vanishes on ${\mathcal {P}}^{0}(\tilde{T})$. By the Schwarz and trace inequalities, we have

$$\begin{aligned} |G(\tilde{w})|&\le \Vert (\mathsf{I}-\mathsf{P}_{k-1})\tilde{v}\Vert _{0, \tilde{e}} \Vert \tilde{w}\Vert _{0,\tilde{e}} \\&\le \Vert (\mathsf{I}-\mathsf{P}_{k-1})\tilde{v}\Vert _{0, \tilde{e}} \cdot C h_{\tilde{e}}^{-1/2}(\Vert \tilde{w}\Vert _{0,\tilde{T}}^2+h_{\tilde{T}}^2|\tilde{w}|_{1,\tilde{T}}^2)^{1/2} \\&\le C \Vert (\mathsf{I}-\mathsf{P}_{k-1})\tilde{v}\Vert _{0, \tilde{e}} \Vert \tilde{w}\Vert _{1,\tilde{T}}. \end{aligned}$$

By the Bramble–Hilbert lemma, we have

$$\begin{aligned} |G(\tilde{w})| \le C\Vert (\mathsf{I}-\mathsf{P}_{k-1})\tilde{v}\Vert _{0, \tilde{e}} |\tilde{w}|_{1,\tilde{T}}. \end{aligned}$$

Taking $\tilde{w} = \tilde{v}$ gives us

$$\begin{aligned} \Vert (\mathsf{I}-\mathsf{P}_{k-1})\tilde{v}\Vert _{0, \tilde{e}} \le C |\tilde{v}|_{1,\tilde{T}}. \end{aligned}$$

(10)

Let $F(\tilde{\varvec{x}})=B \tilde{\varvec{x}} + \varvec{d}$ be an affine mapping from $\tilde{T}$ onto $T \subset K$ such that $F(\tilde{e}) = e$. Choosing $\tilde{v} =v \circ F$, we have

$$\begin{aligned} \Vert (\mathsf{I}-\mathsf{P}_{k-1})\tilde{v}\Vert _{0, \tilde{e}} = \frac{\mathrm {meas}(\tilde{e})^{1/2}}{\mathrm {meas}(e)^{1/2}} \Vert (\mathsf{I}-\mathsf{P}_{k-1})v\Vert _{0,e}, \end{aligned}$$

(11)

where $\mathrm {meas}(e)$ is the measure of $e$. From [6, Theorem 3.1.2.], it follows that

$$\begin{aligned} |\tilde{v}|_{1,\tilde{T}}&\le C \Vert B\Vert |\det B|^{-1/2} |v|_{1, T} \nonumber \\&\le C\frac{h_T}{2\rho _{\tilde{T}}} \frac{\mathrm {meas}(\tilde{T})^{1/2}}{\mathrm {meas}(T)^{1/2}} |v|_{1, T}, \end{aligned}$$

(12)

where $\rho _{\tilde{T}}$ is the radius of the inscribed ball of $\tilde{T}$. The measure of $T$ is given by

$$\begin{aligned} \mathrm {meas}(T) = \frac{\gamma _T h_e}{2} \mathrm {meas}(e). \end{aligned}$$

(13)

From (12), (13) and (3), we have

$$\begin{aligned} |\tilde{v}|_{1,\tilde{T}}&\le C\frac{h_T}{h_e^{1/2}\mathrm {meas}(e)^{1/2}} |v|_{1, T} \nonumber \\&\le C\gamma _E \frac{h_e^{1/2}}{\mathrm {meas}(e)^{1/2}}|v|_{1, T}. \end{aligned}$$

(14)

From (10), (11) and (14), it follows that

$$\begin{aligned} \Vert (\mathsf{I}-\mathsf{P}_{k-1})v\Vert _{0,e}&\le C h_e^{1/2} |v|_{1, T} \le C h_e^{1/2} |v|_{1, K}, \end{aligned}$$

which completes the proof. $\square$

2.6 Approximation Property

The approximation property in the energy norm follows from those of ${\mathcal {P}}^k(\mathcal {T}_h)$ and ${\mathcal {P}}^k(\mathcal {E}_h)$.

Lemma 4

(Approximation property) Let $v \in H^{k+1}(\varOmega )$ and $\varvec{v} = \{ v , v|_{\varGamma _h}\}$. We assume that the finite element space for the hybrid unknown is discontinuous. Then there exists a positive constant $C$ independent of $h$ such that

(15)

Proof

It is known that there exists $\varvec{w}_h =\{w_h, \hat{w}_h\} \in \varvec{V}_h^{k,k}$ such that

$$\begin{aligned}&\Vert v - w_h \Vert _{0} \le C h^{k+1} |v|_{k+1}, \end{aligned}$$

(16)

$$\begin{aligned}&|v - w_h |_{i,h} \le C h^{k+1-i} |v|_{k+1} \quad (i = 1,2), \end{aligned}$$

(17)

$$\begin{aligned}&\left( \sum _{K \in \mathcal {T}_h} \sum _{e\subset \partial K} h_e^{-1} \Vert v - \hat{w}_h\Vert ^2_{0,e}\right) ^{1/2} \le Ch^{k}|v|_{k+1}. \end{aligned}$$

(18)

Let us define $\varvec{v}_h = \{ w_h, \mathsf{P}_{k-1}\hat{w}_h\} \in \varvec{V}_h^{k,k-1}$. Note that $|\varvec{v} - \varvec{v}_h|_\mathrm{j} = |\varvec{v}_h|_\mathrm{j} = |\varvec{w}_h|_\mathrm{j}$. By the trace inequality for $K \in \mathcal {T}_h$ and $e \subset \partial K$, we have

$$\begin{aligned} \Vert \mathsf{P}_{k-1}(\hat{w}_h - w_h)\Vert _{0,e}&\le \Vert \hat{w}_h - w_h\Vert _{0,e}\\&\le \Vert \hat{w}_h- v \Vert _{0,e} + \Vert v- w_h\Vert _{0,e} \\&\le \Vert \hat{w}_h- v \Vert _{0,e} + Ch_e^{-1/2}(\Vert v- w_h\Vert _{0,K}^2+h_K^2|v- w_h|_{1,K}^2)^{1/2}. \end{aligned}$$

From the above and (16)–(18), it follows that $|||\varvec{v} - \varvec{v}_h|||\, \le Ch^k |v|_{k+1}$. $\square$

Remark

In the standard HDG methods, we can impose the continuity at nodes on the hybrid unknowns to reduce the number of degrees of freedom, which is so-called continuous approximation. However, the approximation property with respect to the energy norm does not hold for the continuous approximation since $\mathsf{P}_{k-1} v_h$ is not continuous at nodes in general even if $v_h$ is continuous. Indeed, it is found by numerical experiments that the convergence rates in the energy and $L^2$ norms are sub-optimal for the continuous approximation.

3 Reduced HDG Method

3.1 The Standard HDG Scheme

To begin with, we present the standard HDG formulation: find $\varvec{u}_h = \{u_h, \hat{u}_h\} \in \varvec{V}_h^{k,k}$ such that

$$\begin{aligned} B_\mathrm{std}(\varvec{u}_h, \varvec{v}_h) = (f,v_h)_\varOmega \quad \forall \varvec{v}_h = \{v_h, \hat{v}_h\} \in \varvec{V}_h^{k,k}, \end{aligned}$$

(19)

where the bilinear form is defined by

$$\begin{aligned} B_\mathrm{std}(\varvec{u}_h, \varvec{v}_h)&= (\nabla u_h, \nabla v_h)_{\mathcal {T}_h} + \langle \varvec{n} \cdot \nabla u_h, \hat{v}_h - v_h \rangle _{\partial \mathcal {T}_h} \nonumber \\&+ s\langle \varvec{n} \cdot \nabla v_h, \hat{u}_h - u_h \rangle _{\partial \mathcal {T}_h} \nonumber \\&+ \langle \tau (\hat{u}_h - u_h), \hat{v}_h -v_h \rangle _{\partial \mathcal {T}_h}. \end{aligned}$$

(20)

Here $s$ is a real number and $\tau$ is a stabilization parameter. The parameter $\tau$ takes a constant value $\tau _e/h_e$ on each edge $e$ with $0 < \tau _0 \le \tau _e \le \tau _1$ for some $\tau _0, \tau _1$. We refer to [16] for the details of the derivation.

3.2 Reduced HDG Schemes

Let us sketch the main idea of our method. The second term in the convectional scheme (20) can be rewritten as

$$\begin{aligned} \langle \varvec{n} \cdot \nabla {u_h}, \hat{v}_h - v_h \rangle _{\partial \mathcal {T}_h} = \langle \varvec{n} \cdot \nabla {u_h}, \mathsf{P}_{k-1} (\hat{v}_h - v_h) \rangle _{\partial \mathcal {T}_h} \end{aligned}$$

since $\varvec{n} \cdot \nabla u_h \in {\mathcal {P}}^{k-1}(\mathcal {T}_h)$. The stabilization term is correspondingly decomposed into

$$\begin{aligned} \langle \tau (\hat{u}_h - u_h), (\hat{v}_h - v_h) \rangle _{\partial \mathcal {T}_h}&= \langle \tau \mathsf{P}_{k-1} (\hat{u}_h - u_h), \mathsf{P}_{k-1}(\hat{v}_h - v_h) \rangle _{\partial \mathcal {T}_h} \\&+ \langle \tau (\mathsf{I}- \mathsf{P}_{k-1} )(\hat{u}_h - u_h), (\mathsf{I}- \mathsf{P}_{k-1})(\hat{v}_h - v_h) \rangle _{\partial \mathcal {T}_h} \end{aligned}$$

Our reduced stabilization is obtained by dropping the second term in the right-hand side. The proposed scheme reads: find $\varvec{u}_h = \{u_h, \hat{u}_h\} \in \varvec{V}_h^{k,k-1}$ such that

$$\begin{aligned} B_{h}(\varvec{u}_h, \varvec{v}_h) = (f,v_h)_\varOmega \quad \forall \varvec{v}_h = \{v_h, \hat{v}_h\} \in \varvec{V}_h^{k,k-1}, \end{aligned}$$

(21)

where the bilinear form is defined by

$$\begin{aligned} B_h(\varvec{u}_h, \varvec{v}_h)&= (\nabla u_h, \nabla v_h)_{\mathcal {T}_h} + \langle \varvec{n} \cdot \nabla u_h, \hat{v}_h - v_h \rangle _{\partial \mathcal {T}_h} \nonumber \\&\quad + s\langle \varvec{n} \cdot \nabla v_h, \hat{u}_h - u_h \rangle _{\partial \mathcal {T}_h} \nonumber \\&\quad + \langle \tau \mathsf{P}_{k-1} (\hat{u}_h - u_h), \mathsf{P}_{k-1}(\hat{v}_h - v_h) \rangle _{\partial \mathcal {T}_h}. \end{aligned}$$

(22)

3.3 Local Conservativity

Let $K$ be an element of $\mathcal {T}_h$, and let $\chi _K$ denote a characteristic function on $K$. Taking $\varvec{v}_h = \{\chi _K,0\}$ in (21), we find that our method as well as the other HDG methods satisfies the followingt local conservation property

$$\begin{aligned} -\int _{\partial K} \hat{\varvec{\sigma }}(\varvec{u}_h) \cdot \varvec{n} ds = \int _K f dx, \end{aligned}$$

(23)

where $\hat{\varvec{\sigma }}$ is a numerical flux defined by

$$\begin{aligned} \hat{\varvec{\sigma }}(\varvec{u}_h) = \nabla u_h + \tau (\hat{u}_h- u_h)\varvec{n}. \end{aligned}$$

3.4 Implementation Using the Gaussian Quadrature Formula

In this section, we will show that the reduced stabilization term can be easily calculated by means of the Gaussian quadrature formula in the two-dimensional case. We can also avoid the calculation of the $L^2$ projections in the reduced stabilization term by using it.

For simplicity, we consider the case of the interval $I = [-1,1]$. Let $\varphi _m$ be the Legendre polynomial of order $m \ge 0$ on $I$. Let $f$ be a smooth function on $I$. The $k$-point Gauss-Legendre quadrature rule on $I$ is given by

$$\begin{aligned} \mathcal {G}_{k}[f] = \sum _{i=1}^k w_i f(a_i), \end{aligned}$$

where $\{a_i, w_i\}_{i=1}^k$ are the quadrature points and weights. The standard stabilization term for $P_k$–$P_k$ approximation can be exactly computed by using the $(k+1)$-point Gauss-Legendre quadrature rule. If we use the $k$-point quadrature rule instead of $(k+1)$-point one, then the reduced stabilization term is obtained.

Lemma 5

Let $\mathsf{P}_{k-1}$ denote the $L^2$-projection from $L^2(I)$ onto ${\mathcal {P}}^{k-1}(I)$. Then we have, for all $\hat{u}_h, \hat{v}_h \in {\mathcal {P}}^{k}(I)$,

$$\begin{aligned} \mathcal {G}_{k}[ \hat{u}_h \hat{v}_h] = \int _I \mathsf{P}_{k-1}\hat{u}_h \mathsf{P}_{k-1}\hat{v}_h ds. \end{aligned}$$

(24)

Proof

We can write $\hat{u}_h = \sum _{j=1}^k u_j \varphi _j$ and $\hat{v}_h = \sum _{j=1}^k v_j \varphi _j$. Note that the Legendre polynomial $\varphi _k$ vanishes at the quadrature points, i.e., $\varphi _k(a_i) = 0$ for $1 \le i \le k$, and that $\mathcal {G}_{k}$ is exact for polynomials of degree $\le 2k-1$. Then we have

$$\begin{aligned} \mathcal {G}_{k}[ \hat{u}_h \hat{v}_h]&= \sum _{i=1}^k \left[ w_i \sum _{j=1}^k u_j\varphi _j(a_i) \cdot \sum _{j=1}^k v_j\varphi _j(a_i)\right] \\&= \sum _{i=1}^{k}\left[ w_i \sum _{j=1}^{k-1} u_j\varphi _j(a_i) \cdot \sum _{j=1}^{k-1} v_j\varphi _j(a_i)\right] \\&= \sum _{i=1}^{k}\left[ w_i\mathsf{P}_{k-1} u_h(a_i) \mathsf{P}_{k-1} v_h(a_i)\right] \\&= \mathcal {G}_{k}[\mathsf{P}_{k-1} \hat{u}_h \mathsf{P}_{k-1} \hat{v}_h] \\&= \int _I \mathsf{P}_{k-1} \hat{u}_h \mathsf{P}_{k-1} \hat{v}_h ds, \end{aligned}$$

which completes the proof. $\square$

Remark

In the three-dimensional case, the efficient implementaion using the Gaussian cubature formula is impossible since there exists almost no cubature formula such that the nodes are the common zeros of orthogonal polynomials (see, for example, [10, 15]). Even for a triangle, it is known that there does not exist such a cubature formula of degree $\ge 3$, see [11]. Only in the case of $k=1$, our method can be easily implemented by the barycentric rule.

3.5 Relation with the Crouzeix–Raviart Nonconforming Finite Element Method

In [9], it is proved that the numerical trace of the EDG method coincides with the approximate solution given by the conforming finite element method on a skeleton $\varGamma _h$. In this section, we reveal the relation between the Crouzeix–Raviart nonconforming finite element and our symmetric scheme($s=1$) with $P_1$–$P_0$ triangular elements. The meshes considered here are assumed to be triangular. Let $\varPi _h$ denote the Crouzeix–Raviart interpolation operator with respect to a mesh $\mathcal {T}_h$. For $\hat{u} \in L^2(\varGamma _h)$, the interpolation $\varPi _h \hat{u} \in {\mathcal {P}}^1(\mathcal {T}_h)$ is given by

$$\begin{aligned} \int _{\varGamma _h} (\varPi _h \hat{u}) \hat{v}_h ds = \int _{\varGamma _h} \hat{u} \hat{v}_h ds \quad \forall \hat{v}_h \in {\mathcal {P}}^0(\mathcal {T}_h). \end{aligned}$$

Theorem 1

Let $\varvec{u}_h = \{u_h, \hat{u}_h\} \in \varvec{V}_h^{1,0}$ be the approximate solution provided by (21) with $s=1$ and $u_\mathrm{CR}$ be the Crouzeix–Raviart approximation. Then we have

$$\begin{aligned} \varPi _h \hat{u}_h = u_\mathrm{CR}. \end{aligned}$$

(25)

In particular, we have for all $e \in \mathcal {E}_h$,

$$\begin{aligned} \int _e \hat{u}_h ds = \int _e u_\mathrm{CR} ds. \end{aligned}$$

(26)

Proof

By the definition of the Crouzeix–Raviart interpolation, we have

$$\begin{aligned} \langle \varvec{n} \cdot \nabla {u_h},\varPi _h\hat{v}_h - \hat{v}_h \rangle _{\partial \mathcal {T}_h} = 0. \end{aligned}$$

and

$$\begin{aligned} \langle \tau \mathsf{P}_{k-1}(\hat{u}_h - u_h), \mathsf{P}_{k-1}(\varPi _h\hat{v}_h - \hat{v}_h) \rangle _{\partial \mathcal {T}_h} = 0. \end{aligned}$$

(27)

Taking $\varvec{v}_h = \{\varPi _h \hat{v}_h, \hat{v}_h\} \in \varvec{V}_h^{1,0}$ in (21) yields

$$\begin{aligned} B_h(\varvec{u}_h, \varvec{v}_h)&= (\nabla u_h, \nabla (\varPi _h \hat{v}_h))_{\mathcal {T}_h} + s\langle \varvec{n} \cdot \nabla (\varPi _h \hat{v}_h), \hat{u}_h - u_h \rangle _{\partial \mathcal {T}_h} \nonumber \\&= (1-s)(\nabla u_h, \nabla (\varPi _h v_h))_{\mathcal {T}_h}+ s( \nabla (\varPi _h \hat{u}_h), \nabla (\varPi _h \hat{v}_h))_{\mathcal {T}_h} \end{aligned}$$

(28)

When $s=1$, the resulting equation for $\hat{u}_h$ reads

$$\begin{aligned} ( \nabla (\varPi _h \hat{u}_h), \nabla (\varPi _h \hat{v}_h))_{\mathcal {T}_h} = (f, \varPi _h \hat{v}_h)_\varOmega \quad \forall \hat{v}_h \in {\mathcal {P}}^0(\mathcal {E}_h). \end{aligned}$$

(29)

The solution of the equation above is uniquely determined to be $u_\mathrm{CR}$. Hence we have $\varPi _h \hat{u}_h = u_\mathrm{CR}$. $\square$

Remark

For higher-order polynomials, (28) does not hold since the Laplacian of $u_h$ does not vanish. In the case of a polygonal element, the reduced stabilization term does not vanish, that is, (27) does not hold. Therefore we might not find a discrete equation in terms of only the hybrid unknown, like (29), in general cases.

4 Error Analysis

First, we prove the consistency, boundedness and coercivity of the bilinear form of our method.

Lemma 6

(Consistency) Let $u$ be the exact solution of (1a)(1b), and $\varvec{u} =\{u, u|_{\varGamma _h}\}$. Then we have

$$\begin{aligned} B_h(\varvec{u}, \varvec{v}_h) = (f,v_h)_\varOmega \quad \forall \varvec{v}_h \in \varvec{V}_h^{k,k}. \end{aligned}$$

(30)

Proof

Since $\hat{u} - u = 0$ on $\varGamma _h$ and the normal derivative of $u$ is single-valued, we have

$$\begin{aligned} B_h(\varvec{u}, \varvec{v}_h)&= (\nabla u, \nabla v_h)_{\mathcal {T}_h} - \langle \varvec{n} \cdot \nabla {u}, v_h \rangle _{\partial \mathcal {T}_h}\nonumber \\&= (-\Delta u, v_h)_{\mathcal {T}_h} \nonumber \\&= (f,v_h)_\varOmega . \end{aligned}$$

(31)

$\square$

Lemma 7

(Boundedness) There exists a constant $C_b$ independent of $h$ such that

(32)

Proof

Let ${\varvec{w}} = \{w, \hat{w}\} = \{\bar{w}+w_h, \bar{w}|_{\varGamma _h}+\hat{w}_h \}$ and $\bar{\varvec{v}} = \{v, \hat{v}\} = \{\bar{v}+v_h, \bar{v}|_{\varGamma _h} + \hat{v}_h\}$, where $\bar{w}, \bar{v} \in H^2(\varOmega )$ and $\{v_h, \hat{v}_h\}, \{w_h, \hat{w}_h\} \in \varvec{V}_h^{k,k-1}$. We estimate each term in the bilinear form separately. By the Schwarz inequality, we have

$$\begin{aligned} |(\nabla w, \nabla v)_K| \le \Vert \nabla w\Vert _{0,K} \Vert \nabla v\Vert _{0,K}. \end{aligned}$$

(33)

To bound the second term in the bilinear form, we decompose it as

$$\begin{aligned} \langle \varvec{n} \cdot \nabla {w}, \hat{v} - v \rangle _{\partial \mathcal {T}_h}&= \langle \varvec{n} \cdot \nabla {(\bar{w} + w_h)}, (\bar{v} + \hat{v}_h) - ( \bar{v}+v_h) \rangle _{\partial \mathcal {T}_h} \nonumber \\&= \langle \varvec{n} \cdot \nabla {\bar{w}}, \hat{v}_h - v_h \rangle _{\partial \mathcal {T}_h} +\langle \varvec{n} \cdot \nabla {w_h}, \hat{v}_h - v_h \rangle _{\partial \mathcal {T}_h}. \end{aligned}$$

(34)

Since $\langle \varvec{n} \cdot \nabla {\bar{w}}, z \rangle _{\partial \mathcal {T}_h} = 0$ for any single-valued function $z$, we have

$$\begin{aligned} \langle \varvec{n} \cdot \nabla {\bar{w}}, \hat{v}_h - v_h \rangle _{\partial \mathcal {T}_h}&=\langle \varvec{n} \cdot \nabla {\bar{w}}, \mathsf{P}_{k-1}(\hat{v}_h - v_h) \rangle _{\partial \mathcal {T}_h} +\langle \varvec{n} \cdot \nabla {\bar{w}}, (\mathsf{I}-\mathsf{P}_{k-1})(\hat{v}_h - v_h) \rangle _{\partial \mathcal {T}_h} \nonumber \\&=\langle \varvec{n} \cdot \nabla {\bar{w}}, \mathsf{P}_{k-1}(\hat{v}_h - v_h) \rangle _{\partial \mathcal {T}_h} +\langle \varvec{n} \cdot \nabla {\bar{w}}, (\mathsf{I}-\mathsf{P}_{k-1})(\bar{v} - v_h) \rangle _{\partial \mathcal {T}_h}. \end{aligned}$$

(35)

Similarly, noting that $\langle \varvec{n} \cdot \nabla {w_h}, (\mathsf{I}-\mathsf{P}_{k-1})z \rangle _{\partial \mathcal {T}_h} = 0$ for any single-valued function $z$, we have

$$\begin{aligned}&\langle \varvec{n} \cdot \nabla {w_h}, \hat{v}_h - v_h \rangle _{\partial \mathcal {T}_h} \nonumber \\&\quad =\langle \varvec{n} \cdot \nabla {w_h}, \mathsf{P}_{k-1}(\hat{v}_h - v_h) \rangle _{\partial \mathcal {T}_h} +\langle \varvec{n} \cdot \nabla {w_h}, (\mathsf{I}-\mathsf{P}_{k-1})(\hat{v}_h - v_h) \rangle _{\partial \mathcal {T}_h} \nonumber \\&\quad =\langle \varvec{n} \cdot \nabla {w_h}, \mathsf{P}_{k-1}(\hat{v}_h - v_h) \rangle _{\partial \mathcal {T}_h} +\langle \varvec{n} \cdot \nabla {w_h}, (\mathsf{I}-\mathsf{P}_{k-1})(\bar{v} - v_h) \rangle _{\partial \mathcal {T}_h}. \end{aligned}$$

(36)

From (34), (35) and (36), it follows that

$$\begin{aligned} \langle \varvec{n} \cdot \nabla {w}, \hat{v} - v \rangle _{\partial \mathcal {T}_h} = \langle \varvec{n} \cdot \nabla {w}, \mathsf{P}_{k-1}(\hat{v}_h - v_h) + (\mathsf{I}-\mathsf{P}_{k-1})(\bar{v} - v_h) \rangle _{\partial \mathcal {T}_h}. \end{aligned}$$

(37)

By the trace inequality and Lemma 3, we get

$$\begin{aligned} |\langle \varvec{n} \cdot \nabla {w}, \hat{v} - v \rangle _{\partial \mathcal {T}_h}|&\le C(|w|_{1,h}^2 + h^2|w|_{2,h}^2)^{1/2} ( |{\varvec{v}}|_\mathrm{j}^2 + |\bar{v} - v_h|_{1,h}^2)^{1/2} \nonumber \\&\le C|||{\varvec{w}}|||\,|||{\varvec{v}}|||\,. \end{aligned}$$

(38)

In tha same manner, the third term in the bilinear form can be bounded. The stabilization term is bounded as

$$\begin{aligned} \left| \langle \tau \mathsf{P}_{k-1} (\hat{w}- w), \mathsf{P}_{k-1} (\hat{v}-v) \rangle _{\partial \mathcal {T}_h}\right|&= \left| \langle \tau \mathsf{P}_{k-1} ( \hat{w}_h- w_h), \mathsf{P}_{k-1} (\hat{v}_h - v_h) \rangle _{\partial \mathcal {T}_h}\right| \nonumber \\&\le \tau _1|\varvec{w}|_\mathrm{j } |\varvec{v}|_\mathrm{j }. \end{aligned}$$

(39)

Combining (33), (38) and (39), we obtain

$$\begin{aligned} |B_h(\varvec{w}, \varvec{v})| \le C_b |||\varvec{w}|||\, |||\varvec{v}|||\,, \end{aligned}$$

(40)

where the constant $C_b$ depends on the constants of the trace inequality and $\tau _1$, but is independent of $h$. The proof is completed. $\square$

Lemma 8

(Coercivity) Assume that $\tau _0$ is sufficiently large. Then there exists a constant $C_c > 0$ independent of $h$ such that

$$\begin{aligned} B_h(\varvec{v}_h, \varvec{v}_h) \ge C_c |||\varvec{v}_h|||\,^2 \qquad \forall \varvec{v}_h \in \varvec{V}_h^{k,k-1}. \end{aligned}$$

(41)

When $s=-1$, it holds for any $\tau >0$.

Proof

Letting $\varvec{u}_h = \varvec{v}_h$ in (21), we have

$$\begin{aligned} B_h(\varvec{v}_h, \varvec{v}_h)&\ge |v_h|^2_{1,h} -|1-s|\left| \langle \varvec{n} \cdot \nabla {v_h}, \hat{v}_h-v_h \rangle _{\partial \mathcal {T}_h}\right| +\tau _0|\varvec{v}_h|_\mathrm{j }^2. \end{aligned}$$

(42)

Note that

$$\begin{aligned} \langle \varvec{n} \cdot \nabla {v_h}, \hat{v}_h-v_h \rangle _{\partial \mathcal {T}_h}&= \langle \varvec{n} \cdot \nabla {v_h}, \mathsf{P}_{k-1}(\hat{v}_h-v_h) \rangle _{\partial \mathcal {T}_h} . \end{aligned}$$

(43)

By the trace inequality and Young’s inequality, it follows that, for any $\varepsilon >0$,

$$\begin{aligned} B_h(\varvec{v}_h, \varvec{v}_h)&\ge (1-C\varepsilon ) |v_h|^2_{1,h} +(\tau _0-\varepsilon ^{-1})|\varvec{v}_h|_\mathrm{j }^2. \end{aligned}$$

(44)

If $\tau _0 > C+1$, then we can take $\varepsilon =(\tau _0^{-1}+C^{-1})/2$. Therefore we obtain

$$\begin{aligned} B_h(\varvec{v}_h, \varvec{v}_h)&\ge \frac{1}{2} (|v_h|^2_{1,h} + |\varvec{v}_h|_\mathrm{j }^2) \nonumber \\&\ge C |||\varvec{v}_h|||\,^2, \end{aligned}$$

(45)

where we have used the inverse inequality. If $s=-1$, then the second term in the right-hand side in (42) vanishes. From this, we see that (45) holds for any $\tau > 0$ when $s=-1$. $\square$

Next, we prove the error estimates with respect to the energy norm.

Theorem 2

(Quasi-best approximation) Let $u$ be the exact solution of (1a)(1b) and $\varvec{u} := \{u, u|_{\varGamma _h}\} \in \varvec{V}$. Let $\varvec{u}_h \in \varvec{V}_h^{k, k-1}$ be an approximate solution provided by our method (21). Then we have

$$\begin{aligned} |||\varvec{u} - \varvec{u}_h|||\, \le C \inf _{\varvec{v}_h \in \varvec{V}_h^{k,k-1}} |||\varvec{u} - \varvec{v}_h|||\,, \end{aligned}$$

(46)

where $C$ is a positive constant independent of $h$.

Proof

Let $\varvec{v}_h \in \varvec{V}_h^{k,k-1}$ be arbitrary. By the coercivity, consistency and boundedness, we have

$$\begin{aligned} C_c |||\varvec{u}_h - \varvec{v}_h|||\,^2&\le B_h(\varvec{u}_h - \varvec{v}_h, \varvec{u}_h - \varvec{v}_h) \nonumber \\&= B_h(\varvec{u} - \varvec{v}_h, \varvec{u}_h - \varvec{v}_h) \nonumber \\&\le C_b |||\varvec{u} - \varvec{v}_h|||\, |||\varvec{u}_h - \varvec{v}_h|||\,, \end{aligned}$$

(47)

from which it follows that

$$\begin{aligned} |||\varvec{u}_h - \varvec{v}_h|||\, \le \frac{C_b}{C_c}|||\varvec{u} - \varvec{v}_h|||\,. \end{aligned}$$

(48)

By the triangle inequality, we have

$$\begin{aligned} |||\varvec{u} - \varvec{u}_h|||\,&\le |||\varvec{u} - \varvec{v}_h|||\, + |||\varvec{v}_h - \varvec{u}_h|||\, \\&\le \left( 1+ \frac{C_b}{C_c}\right) |||\varvec{u} - \varvec{v}_h|||\,, \end{aligned}$$

which implies (46). $\square$

By the approximation property, the optimal-order error estimate in the energy norm follows immediately.

Theorem 3

Let the notation be the same in Theorem 2. If $u \in H^{k+1}(\varOmega )$, then we have

$$\begin{aligned} |||\varvec{u} - \varvec{u}_h|||\, \le C h^k|u|_{k+1}. \end{aligned}$$

Finally, we prove the $L^2$-error estimates.

Theorem 4

($L^2$-error estimates) Let the notation be the same as in Theorem 2. If $u \in H^{k+1}(\varOmega )$, then we have

$$\begin{aligned} \Vert u - u_h \Vert _0&\le C h^{k+1}|u|_{k+1} \quad { for}\,\,s=1, \end{aligned}$$

(49)

$$\begin{aligned} \Vert u - u_h \Vert _0&\le C h^{k}|u|_{k+1} \quad { for}\,\,s \ne 1, \end{aligned}$$

(50)

where $C$ is a positive constant independent of $h$.

Proof

First, we prove (49). We can use Aubin-Nitsche’s trick for $s=1$. Let $\psi \in H^2(\varOmega ) \cap H_0^1(\varOmega )$ be the exact solution of the equation $-\Delta \psi = u - u_h$, and define $\varvec{\psi }= \{\psi , \psi |_{\varGamma _h}\}$. For any $\varvec{\psi }_h \in \varvec{V}^{k,k-1}_h$, by the consistency and boundedness, we have

$$\begin{aligned} \Vert u-u_h\Vert _0^2&= B_h(\varvec{u} - \varvec{u}_h, \varvec{\psi }) \nonumber \\&= B_h(\varvec{u} - \varvec{u}_h, \varvec{\psi }- \varvec{\psi }_h) \nonumber \\&\le C_b|||\varvec{u} - \varvec{u}_h|||\,|||\varvec{\psi }- \varvec{\psi }_h|||\,. \end{aligned}$$

(51)

Note that there exists $\varvec{\psi }_h \in \varvec{V}^{k,k-1}_h$ such that

$$\begin{aligned} |||\varvec{\psi }- \varvec{\psi }_h|||\, \le Ch|\psi |_2 \le Ch\Vert u-u_h\Vert _0. \end{aligned}$$

By Lemma 3, we obatin (49). For the proof of the case $s \ne 1$, we show the following inequality in a similar manner presented in [2].

$$\begin{aligned} \Vert w - w_h \Vert _0 \le C |||\varvec{w} - \varvec{w}_h|||\, \quad \forall \varvec{w} = \{w, w|_{\varGamma _h}\} \in \varvec{V}(h), \varvec{w}_h \in \varvec{V}_h^{k,k-1}. \end{aligned}$$

(52)

Let $\varphi \in H^2(\varOmega ) \cap H_0^1(\varOmega )$ be the solution of $-\Delta \varphi = w-w_h$ and define $\varvec{\varphi }=\{\varphi , \varphi |_{\varGamma _h}\}$. Then we have

$$\begin{aligned} \Vert w - w_h\Vert _0^2&= (\nabla \varphi , \nabla (w-w_h))_{\mathcal {T}_h} - \langle \varvec{n} \cdot \nabla \varphi , \hat{w}_h-w_h \rangle _{\partial \mathcal {T}_h} \\&\le C\Vert \varphi \Vert _{2,\varOmega } |||\varvec{w} - \varvec{w}_h|||\,. \end{aligned}$$

Since $\Vert \varphi \Vert _{2,\varOmega } \le C\Vert w-w_h\Vert _0$, we have (52). From Theorem 3, (50) follows immediately. $\square$

5 Numerical Results

5.1 Discontinuous Approximation

We consider the following test problem:

$$\begin{aligned} -\Delta u&= 2\pi ^2 \sin (\pi x) \sin (\pi y) \quad \mathrm{in}\,\,\varOmega , \end{aligned}$$

(53)

$$\begin{aligned} u&= 0 \quad \mathrm{on}\,\,\partial \varOmega , \end{aligned}$$

(54)

where the domain $\varOmega$ is the unit square and the source function is chosen so that the exact solution is $u(x,y) = \sin (\pi x)\sin (\pi y)$. We employed unstructured triangular meshes and $P_k$–$P_{k-1}$ discontinuous approximation for $1\le k \le 3$. The schemes are all symmetric. Tables 1 and 2 display the convergence histories of the reduced and standard HDG schemes, respectively. The mesh size is given by $h \approx 0.1 \times 2^{-(l-1)}$. It can be observed that the convergence rates of the piecewise $H^1$-error and $L^2$-error are optimal in all the cases, which agrees with our theoretical results. We also see that the absolute errors of the reduced HDG method is approximately as same as those of the standard HDG method. It suggests that the complementary projection part of the hybrid quantity, namely $(\mathsf{I}- \mathsf{P}_{k-1}) \hat{u}_h$, actually does not contribute to accuracy.

Table 1 Convergence history of the reduced HDG methods with discontinuous approximations

Table 2 Convergence history of the standard HDG methods with discontinuous approximations

5.2 Continuous Approximation

As mentioned in Sect 2.6, the approximation property does not hold for the continuous approximations. As a result, the convergence order in the energy and $L^2$ norms may not be optimal. We carried out numerical experiments to observe the convergence rate. The same test problem as in the previous is considered. We computed the approximate solutions by the reduced HDG method with the $P_2$–$P_1$ and $P_3$–$P_2$ continuous approximations. The same meshes as in the previous were used. The results are shown at Table 3. We observe that the convergence rates in the piecewise $H^1$ and $L^2$ norms are sub-optimal, which indicates the reduced stabilization is not suitable for the continuous approximations.

Table 3 Convergence history of the reduced HDG methods with continuous approximations

6 Conclusion

We proposed a new hybridized discontinuous Galerkin method with reduced stabilization. We devised an efficient implementation of our method by means of the Gaussian quadrature formula in the two-dimensional case. The error estimates in the energy and $L^2$ norms were proved under the chunkiness condition. It was also shown that our method with $P_1$–$P_0$ approximation is closely related to the Crouzeix–Raviart nonconforming finite element method. Numerical results confirmed the validity of the proposed schemes.