Error Estimates of a Pressure-Stabilized Characteristics Finite Element Scheme for the Oseen Equations

Abstract

Error estimates with the optimal convergence order are proved for a pressure-stabilized characteristics finite element scheme for the Oseen equations. The scheme is a combination of Lagrange–Galerkin finite element method and Brezzi–Pitkäranta’s stabilization method. The scheme maintains the advantages of both methods; (i) It is robust for convection-dominated problems and the system of linear equations to be solved is symmetric. (ii) Since the P1 finite element is employed for both velocity and pressure, the number of degrees of freedom is much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the scheme is efficient especially for three-dimensional problems. The theoretical convergence order is recognized by two- and three-dimensional numerical results.

1 Introduction

The purpose of this paper is to prove the stability and convergence of a pressure-stabilized characteristics finite element scheme for the Oseen equations. The core part of this scheme consists of a characteristics (Lagrange–Galerkin) method and Brezzi–Pitkäranta’s stabilization method. A similar scheme has been proposed by us for the Navier–Stokes equations [21, 23]. To the best of our knowledge it is the earliest work which combines the two methods, characteristics and stabilization. The stability of this scheme has its own interesting structure (Theorem 1), which is effectively used in the convergence proof not only for the Oseen equations here but also for the Navier–Stokes equations in a forthcoming paper [25].

Historically, in order to deal with convection-dominated problems, a lot of ideas have been proposed, e.g., upwind methods [1, 4, 8, 15, 18, 20, 33, 35], characteristics(-based) methods [3, 11, 13, 14, 21–24, 26–29, 32] and so on. Our scheme belongs to the second group, and is the simplest one, i.e., the trajectory of fluid particle is discretized by the backward Euler method. It has common advantages of characteristics methods, robust stability with little numerical diffusion and symmetry of the resulting matrix of the system of linear equations.

For the purpose of reducing the number of degrees of freedom (DOF) we apply the stabilization method. We employ Brezzi–Pitkäranta’s pressure-stabilization method [7], where the cheapest P1 element is used for both velocity and pressure. The number of DOF of this element is much smaller than that of typical stable finite elements, P2/P1 (Hood-Taylor) and P1-bubble/P1 (mini) [17]. As for other stabilized methods, e.g., pressure-stabilizing/Petrov–Galerkin (PSPG) and Galerkin-least-square (GLS) methods, cf. [4, 15, 16, 35], the numbers of DOF become larger than that of P1/P1 element. Furthermore, the advantage of the symmetric matrix is lost when combined with the characteristics method, e.g., PSPG.

Thus, the scheme to be considered in this paper leads to a small-size symmetric resulting matrix, which can be solved by efficient iterative solvers for symmetric matrices, e.g., minimal residual method (MINRES), conjugate residual method (CR) and so on [2, 30]. It is, therefore, efficient especially in three-dimensional computation. The scheme is proved to be essentially unconditionally stable (the required condition is only (6)) and convergent with optimal error estimates. A stabilized characteristics finite element scheme with an $L^2$-type pressure-stabilization is proposed in [19] for the Navier–Stokes equations and the convergence is proved under a strong stability condition on the discretization parameters. Since the stability condition is caused from the estimate of the incompressibility, it will be still required in the convergence proof for the Oseen equations if the proof is applied directly.

The paper is organized as follows. A pressure-stabilized characteristics finite element scheme for the Oseen equations is shown in Sect. 2. The main results on stability and error estimates are stated in Sect. 3 and they are proved in Sect. 4. The theoretical convergence order is recognized in two- and three-dimensional computations in Sect. 5. Finally conclusions are given in Sect. 6.

2 A Pressure-Stabilized Characteristics Finite Element Scheme

In this section we set the Oseen problems and state our pressure-stabilized characteristics finite element scheme.

Let $\varOmega$ be a bounded domain in $\mathbb {R}^d (d=2,3), \varGamma \equiv \partial \varOmega$ be the boundary of $\varOmega$ and $T$ be a positive constant. Let $m$ be a non-negative integer. We use the Sobolev spaces $W^{1,\infty }(\varOmega ), W^{1,\infty }_0(\varOmega ), H^m(\varOmega )$ and $H^1_0(\varOmega )$. For any normed space $X$ with norm $\Vert \cdot \Vert _X$, we define function spaces $C^0([0,T]; X)$ and $H^m(0,T; X)$ consisting of $X$-valued functions in $C^0([0,T])$ and $H^m(0,T)$, respectively. We use the same notation $(\cdot , \cdot )$ to represent the $L^2(\varOmega )$ inner product for scalar-, vector- and matrix-valued functions. $L^2_0(\varOmega )$ is a function space defined by

$$\begin{aligned} L^2_0(\varOmega ) \equiv \bigl \{q\in L^2(\varOmega );\ (q, 1) = 0 \bigr \}. \end{aligned}$$

We often omit $[0,T]$ and/or $\varOmega$ if there is no confusion, e.g., $C^0(H^1)$ in place of $C^0([0,T]; H^1(\varOmega ))$. The integer $d$ is also often omitted from superscripts. For $t_0$ and $t_1\in \mathbb {R}$ we introduce a function space

$$\begin{aligned} Z^m(t_0, t_1) \equiv \left\{ v\in H^j\left( t_0, t_1; H^{m-j}(\varOmega )^d\right) ;\ j=0,\ldots ,m,\ \Vert v\Vert _{Z^m(t_0, t_1)} < \infty \right\} , \end{aligned}$$

where the norm $\Vert \cdot \Vert _{Z^m(t_0, t_1)}$ is defined by

$$\begin{aligned} \Vert v\Vert _{Z^m(t_0, t_1)} \equiv \biggl \{ \sum _{j=0}^m \Vert v\Vert _{H^j(t_0,t_1; H^{m-j}(\varOmega ))}^2 \biggr \}^{1/2}. \end{aligned}$$

We denote $Z^m(0, T)$ by $Z^m$.

We consider an initial boundary value problem; find $(u, p):\ \varOmega \times (0, T) \rightarrow \mathbb {R}^d\times \mathbb {R}$ such that

$$\begin{aligned} \frac{Du}{Dt} - \nabla \bigl ( 2\nu D(u) \bigr ) + \nabla p + \lambda u&= f \qquad \,\,\text{ in } \; \varOmega \times (0,T), \end{aligned}$$

(1a)

$$\begin{aligned} \nabla \cdot u&= 0 \qquad \,\,\text{ in }\; \varOmega \times (0,T), \end{aligned}$$

(1b)

$$\begin{aligned} u&= 0 \qquad \,\,\text{ on } \; \varGamma \times (0,T),\end{aligned}$$

(1c)

$$\begin{aligned} u&= u^0\qquad \text{ in }\; \varOmega ,\ \text{ at }\ t=0, \end{aligned}$$

(1d)

where $u$ is the velocity, $p$ is the pressure, $f\in C^0([0, T]; L^2(\varOmega )^d)$ is a given external force, $u^0\in H^1_0(\varOmega )^d$ is a given initial velocity, $\lambda :\ \varOmega \times (0, T)\rightarrow \mathbb {R}^{d\times d}$ is a given reaction rate, $\nu$ is a viscosity, $D(u)$ is the strain-rate tensor defined by

$$\begin{aligned} D_{ij}(u)\equiv \dfrac{1}{2}\left( \dfrac{\partial u_i}{\partial x_j}+\dfrac{\partial u_j}{\partial x_i}\right) \quad (i, j = 1,\ldots , d), \end{aligned}$$

and $D/Dt$ is a material derivative defined by

$$\begin{aligned} \frac{D}{Dt} \equiv \frac{\partial {}}{\partial {t}} + w\cdot \nabla \end{aligned}$$

for a given velocity $w:\ \varOmega \times (0, T)\rightarrow \mathbb {R}^d$.

Remark 1

If $w$ is replaced by $u$ and $\lambda = 0$, (1) becomes the Navier–Stokes problem. Here, we focus on the Oseen problem (1). The discussion on the Navier–Stokes problem will be presented in the forthcoming paper [25].

We impose assumptions for the given velocity $w$ and reaction rate $\lambda$.

Hypothesis 1

The function $w$ satisfies $w\in C^0([0,T]; W^{1,\infty }_0(\varOmega )^d)$.

Hypothesis 2

The function $\lambda$ satisfies $\lambda \in C^0([0,T]; L^{\infty }(\varOmega )^{d\times d})$.

Let $V \equiv H^1_0(\varOmega )^d$ and $Q\equiv L^2_0(\varOmega )$ be function spaces. We define bilinear forms $a$ on $V\times V, b$ on $V \times Q, d(t)$ on $V \times V$ and $\fancyscript{A}$ on $(V \times Q) \times (V \times Q)$ by

$$\begin{aligned} a(u,v) \equiv 2\nu \bigl ( D(u), D(v) \bigr ),\quad b(v,q)&\equiv -( \nabla \cdot v, q ),\quad d(t)(u,v) \equiv ( \lambda (t)u, v ),\\ \fancyscript{A} ((u,p), (v,q))&\equiv a(u,v)+b(v,p)+b(u,q), \end{aligned}$$

respectively. Then, we can write the weak formulation of (1); find $(u, p):\ (0, T)\rightarrow V\times Q$ such that for $t\in (0,T)$

$$\begin{aligned} \Bigl ( \frac{Du}{Dt}(t), v \Bigr ) + \fancyscript{A}((u,p)(t), (v,q)) + d(t)(u(t), v)&= (f(t), v),\quad \forall (v, q)\in V\times Q, \end{aligned}$$

(2)

with $u(0) = u^0$.

Let $X: (0, T)\rightarrow \mathbb {R}^d$ be a solution of the system of ordinary differential equations,

$$\begin{aligned} \frac{dX}{dt} = w(X, t). \end{aligned}$$

(3)

Then, if $u$ is smooth, it holds that

$$\begin{aligned} \frac{Du}{Dt} (X(t), t) = \frac{d}{dt} u \bigl ( X(t), t \bigr ). \end{aligned}$$

Let $\varDelta t$ be a time increment, $N_T\equiv \lfloor T/\varDelta t \rfloor$ be a total number of time steps, $t^n \equiv n\varDelta t$ for $n=0,\ldots ,N_T$ and $X(\cdot ; x, t^n)$ be the solution of (3) satisfying an initial condition $X(t^n) = x$. Then, we can consider a first order approximation of the material derivative at $(x, t^n)~(n\ge 1)$ as follows.

$$\begin{aligned} \frac{Du}{Dt} (x, t^n)&= \frac{d}{dt} u \bigl ( X(t; x, t^n), t \bigr )\Bigr |_{t=t^n}\nonumber \\&= \frac{u ( X(t^n; x, t^n), t^n ) - u ( X(t^{n-1}; x, t^n), t^{n-1} )}{\varDelta t} + O(\varDelta t)\nonumber \\&= \frac{u^n - u^{n-1} \circ X_1^n}{\varDelta t} (x) + O(\varDelta t), \end{aligned}$$

(4)

where $X_1^n(x)$ is a function defined by

$$\begin{aligned} X_1^n(x) \equiv x - w^n(x)\varDelta t, \end{aligned}$$

and we have used notations, $u^n \equiv u(\cdot , t^n)$ and

$$\begin{aligned} v\circ X_1^n(x) \equiv v( X_1^n (x) ). \end{aligned}$$

The point $X_1^n(x)$ is called an upwind point of $x$. The approximation (4) of $Du/Dt$ is the basic idea to devise numerical schemes based on the method of characteristics. It has been combined with finite element and difference methods to lead to powerful numerical schemes for flow problems, cf. [11, 22, 24, 27, 29].

The next proposition gives a sufficient condition to guarantee all upwind points are in $\varOmega$.

Proposition 1

([29, Proposition 1]) Under Hypothesis 1 and the inequality

$$\begin{aligned} \varDelta t < \frac{1}{\Vert w\Vert _{C^0(W^{1,\infty }(\varOmega ))}}, \end{aligned}$$

(5)

it holds that for any $n=0,\ldots ,N_T$

$$\begin{aligned} X_1^n(\varOmega ) = \varOmega . \end{aligned}$$

Here we fix $\varDelta t_0$, which satisfies (5) and

$$\begin{aligned} \det \Bigl ( \frac{\partial {X_1^n}}{\partial {x}} \Bigr ) \ge \frac{1}{2},\ \ \forall n\in \{0,\ldots ,N_T\},\ \ \forall \varDelta t \in (0,\varDelta t_0]. \end{aligned}$$

In the following we suppose that

$$\begin{aligned} \varDelta t \in (0,\varDelta t_0]. \end{aligned}$$

(6)

For the sake of simplicity we assume $\varOmega$ is a polygonal $(d=2)$ or polyhedral $(d=3)$ domain. Let $\fancyscript{T}_h=\{K\}$ be a triangulation of $\bar{\varOmega }\ (= \bigcup _{K\in \fancyscript{T}_h} K ), h_K$ be a diameter of $K\in \fancyscript{T}_h$, and $h\equiv \max _{K\in \fancyscript{T}_h}h_K$ be the maximum element size. We consider a regular family of subdivisions $\{\fancyscript{T}_h\}_{h\downarrow 0}$ satisfying the inverse assumption [9], i.e., there exists a positive constant $\alpha _0$ independent of $h$ such that

$$\begin{aligned} \frac{h}{h_K} \le \alpha _0,\quad \forall K\in \fancyscript{T}_h,\quad \forall h. \end{aligned}$$

(7)

We define function spaces $X_h, M_h, V_h$ and $Q_h$ by

$$\begin{aligned} X_{h}&\equiv \{ v_h\in C^0(\bar{\varOmega })^d;\ v_h|_K \in P_1(K)^d,\ \forall K \in \fancyscript{T}_h \},\\ M_{h}&\equiv \{ q_h\in C^0(\bar{\varOmega });\ q_h|_K \in P_1(K),\ \forall K \in \fancyscript{T}_h \}, \end{aligned}$$

$V_h \equiv X_h\cap V$ and $Q_h \equiv \ M_h \cap Q$, respectively, where $P_1(K)$ is a polynomial space of linear functions on $K\in \fancyscript{T}_h$. Let $\delta _0$ be a positive constant and $(\cdot ,\cdot )_K$ be the $L^2(K)^d$ inner product. We define bilinear forms $\fancyscript{C}_h$ on $H^1(\varOmega ) \times H^1(\varOmega )$ and $\fancyscript{A}_h$ on $(V \times H^1(\varOmega )) \times (V \times H^1(\varOmega ))$ by

$$\begin{aligned} \fancyscript{C}_h (p, q)&\equiv \delta _0 \sum _{K\in \fancyscript{T}_h} h_K^2 ( \nabla p, \nabla q )_K, \\ \fancyscript{A}_h ((u,p), (v,q))&\equiv a(u,v)+b(v,p)+b(u,q) -\fancyscript{C}_h (p, q),\nonumber \end{aligned}$$

(8)

respectively. Let $f_h = \{f_h^n\}_{n=1}^{N_T} \subset L^2(\varOmega )^d$ and $u^0_h \in V_h$ be given. Our pressure-stabilized characteristics finite element scheme is to find $\{(u_h^n,\, p_h^n)\}_{n=1}^{N_T}\subset V_h \times Q_h$ such that for $n=1,\ldots , N_T$

$$\begin{aligned} \Bigl (\frac{u^n_h-u^{n-1}_h\circ X_1^n}{\varDelta t}, v_h\Bigr ) + \fancyscript{A}_h((u^n_h, p^n_h), (v_h, q_h)) + d^n(u_h^{n-1}, v_h)&= (f_h^n, v_h),\nonumber \\&\forall (v_h, q_h) \in V_h\times Q_h,\nonumber \\ \end{aligned}$$

(9)

where we have simply denoted $d(t^n)$ by $d^n$.

Remark 2

(i) We can replace the third term by $d^n(u_h^{n}, v_h)$ and prove the stability and convergence of the scheme. The scheme, however, loses such an advantage of the Galerkin characteristics method that the resulting matrix is symmetric unless $\lambda$ is symmetric. That is the reason why we consider scheme (9). (ii) Scheme (9) is equivalent to

$$\begin{aligned} \Bigl (\frac{u^n_h}{\varDelta t}, v_h\Bigr ) + a(u_h^n,v_h)+b(v_h,p_h^n)&= \Bigl ( f_h^n+\frac{u^{n-1}_h\circ X_1^n}{\varDelta t}, v_h\Bigr ) -d^n(u_h^{n-1}, v_h),\quad \forall v_h \in V_h,\\ b(u_h^n,q_h) -\fancyscript{C}_h(p_h^n, q_h)&= 0,\quad \forall q_h\in Q_h. \end{aligned}$$

Selecting specific bases of $V_h$ and $Q_h$ and expanding $u_h^n$ and $p_h^n$ in terms of the associated basis functions, we get an algebraic system involving a symmetric matrix of the form

$$\begin{aligned} \left( \begin{array}{ll} A &{} \ B^T \\ B &{} -C \end{array}\right) . \end{aligned}$$

Here, $A, B$ and $C$ are sub-matrices derived from $\tfrac{1}{\varDelta t}(u^n_h, v_h) + a(u^n_h, v_h), b(u_h^n, q_h)$ and $\fancyscript{C}_h(p_h^n, q_h)$, respectively, and the matrix is independent of the velocity $w$ and current time $t^n$.

3 Main Results

In this section we present the main results of stability and error estimates, which are proved in Sect. 4.

We use $c$ with or without subscript to represent the generic positive constant independent of the discretization parameters $h$ and $\varDelta t$, and it can take different values at different places. For $k\in \mathbb {N}, c(A_1,\ldots , A_k)$ means a positive constant depending on $A_1,\ldots , A_k$, which monotonically increases as each $A_i, i\in \{1,\ldots ,k\}$, increases. Constants $c_0, c_1, c_2$ and $c_*$ have particular meanings in this paper,

$$\begin{aligned} c_0&=c(\Vert w\Vert _{C^0(L^{\infty })}), \quad c_1 = c(\Vert \lambda \Vert _{C^0(L^{\infty })}), \nonumber \\ c_2&=c( \Vert w\Vert _{C^0(W^{1,\infty })}, \Vert \lambda \Vert _{C^0(L^{\infty })}, 1/\nu , \nu ), \nonumber \\ c_*&= c\bigl ( \Vert w\Vert _{C^0(W^{1,\infty })}, \Vert \lambda \Vert _{C^0(L^\infty )}, 1/\nu , \nu , T \bigr ). \end{aligned}$$

(10)

Constants $c^\prime , c_0^\prime$ and $c_1^\prime$, having the same meaning as $c, c_0$ and $c_1$, are used when they are distinguished from $c, c_0$ and $c_1$ near by, respectively. We use norms and seminorms, $\Vert \cdot \Vert _k \equiv \Vert \cdot \Vert _{H^k(\varOmega )}\ (k=0, 1, 2), \Vert \cdot \Vert _{V_h}\equiv \Vert \cdot \Vert _V \equiv \Vert \cdot \Vert _1, \Vert \cdot \Vert _{Q_h} \equiv \Vert \cdot \Vert _Q \equiv \Vert \cdot \Vert _0$,

$$\begin{aligned} \Vert u\Vert _{l^{\infty }(X)}&\equiv \max _{n=0,\ldots , N_T} \Vert u^n\Vert _X,&\Vert u\Vert _{l^{2}(X)}&\equiv \Bigl \{ \varDelta t\sum _{n=1}^{N_T} \Vert u^n\Vert _X^2 \Bigr \}^{1/2},\\ |q|_h \equiv \Bigl \{&\sum _{K\in \fancyscript{T}_h} h_K^2 (\nabla q, \nabla q)_K \Bigr \}^{1/2},&|p|_{l^{\infty }(|\cdot |_h)}&\equiv \max _{n=0,\ldots , N_T} |p^n|_h,\\ |p|_{l^{2}(|\cdot |_h)}&\equiv \Bigl \{ \varDelta t\sum _{n=1}^{N_T} |p^n|_h^2 \Bigr \}^{1/2}, \end{aligned}$$

for $X=H^k(\varOmega )$ and $H^k(\varOmega )^d~(k=0, 1)$. $\overline{D}_{\varDelta t}$ is the backward difference operator defined by

$$\begin{aligned} \overline{D}_{\varDelta t} a^n \equiv \frac{a^n - a^{n-1}}{\varDelta t}. \end{aligned}$$

Firstly we show the stability result.

Theorem 1

(Stability) Suppose that Hypotheses 1 and 2 hold. Assume condition (6). Let $f_h=\{f_h^n\}_{n=1}^{N_T}\subset L^2(\varOmega )^d$ and $u^0_h\in V_h$ be given and $(u_h,p_h)$ be the solution of (9). Suppose that there exists $p_h^0\in Q_h$ such that

$$\begin{aligned} b(u^0_h, q_h) -\fancyscript{C}_h(p_h^0, q_h) = 0, \quad \forall q_h\in Q_h. \end{aligned}$$

(11)

Then, there exists a constant $c_*$ of (10) such that

$$\begin{aligned} \Vert u_h\Vert _{l^\infty (H^1)},\ \ \Vert \overline{D}_{\varDelta t}u_h\Vert _{l^2(L^2)},\ \ |p_h|_{l^\infty (|\cdot |_h)}, \ \ \Vert p_h\Vert _{l^2(L^2)} \le c_*( \Vert u^0_h\Vert _1 + |p_h^0|_h + \Vert f_h\Vert _{l^2(L^2)} ). \end{aligned}$$

(12)

Remark 3

The relation (11) is satisfied if $(u_h^0, p_h^0)\in V_h\times Q_h$ is chosen as a Stokes projection of $(u^0, 0)$ (cf. Definition 1 below).

Secondary we give error estimates after preparing a (pressure-stabilized) Stokes projection using P1/P1-element and a hypothesis.

Definition 1

(Stokes projection) For $(u,p)\in V \times Q$ we define the Stokes projection $(\hat{u}_h, \hat{p}_h)\in V_h\times Q_h$ of $(u, p)$ by

$$\begin{aligned} \fancyscript{A}_h((\hat{u}_h, \hat{p}_h), (v_h, q_h)) = \fancyscript{A}((u, p), (v_h, q_h)), \quad \forall (v_h, q_h) \in V_h\times Q_h. \end{aligned}$$

(13)

Hypothesis 3

The solution $(u, p)$ of (2) satisfies $u\in H^1(0,T; V\cap H^2(\varOmega )^d) \cap Z^2$ and $p\in H^1(0,T; Q \cap H^1(\varOmega ) )$.

Theorem 2

(Error estimate) Suppose Hypotheses 1–3 hold. Assume condition (6). Suppose $f_h=f$ and that $u_h^0$ is the first component of the Stokes projection of $(u^0, 0)$ by (13). Let $(u_h,p_h)$ be the solution of (9). Then, there exists a constant $c_*$ of (10) such that

$$\begin{aligned} \Vert u_h-u\Vert _{l^\infty (H^1)}, \ \ \Bigl \Vert \overline{D}_{\varDelta t}u_h - \frac{\partial {u}}{\partial {t}} \Bigr \Vert _{l^2(L^2)},\ \ \Vert p_h-p\Vert _{l^2(L^2)} \nonumber \\ \le c_*(\varDelta t \Vert u \Vert _{Z^2} + h \Vert (u, p) \Vert _{H^1(0,T; H^2\times H^1)} ). \end{aligned}$$

(14)

Remark 4

In the case of the inhomogeneous Dirichlet boundary condition $u=u_b$ in place of (1c), similar stability and convergence results are obtained if there exists a function $\tilde{u}_b\in H^1(H^2(\varOmega )^d)\cap Z^2$ such that $\tilde{u}_b|_{\varGamma }=u_b$.

4 Proofs of Theorems 1 and 2

This section is devoted to the proofs of Theorems 1 and 2.

4.1 Preliminaries

We prepare six lemmas and a proposition to be used in the proofs. We omit the proofs of the first five lemmas only by referring to the related bibliography.

Lemma 1

(Discrete Gronwall’s inequality, [34, Lemma 4.6]) Let $a_1$ be a non-negative number, $\varDelta t$ be a positive number, and $\{x^{n}\}_{n\ge 0}, \{y^{n}\}_{n\ge 1}$ and $\{b^{n}\}_{n\ge 1}$ be non-negative sequences. Suppose

$$\begin{aligned} \overline{D}_{\varDelta t} x^{n} + y^{n} \le a_1 x^{n-1} + b^{n},\quad \forall n\ge 1. \end{aligned}$$

Then, it holds that

$$\begin{aligned} x^{n} + \varDelta t\sum _{i=1}^ny^{i} \le \exp ( a_1n\varDelta t ) \Bigl ( x^{0} + \varDelta t\sum _{i=1}^nb^{i}\Bigr ),\quad \forall n\ge 1. \end{aligned}$$

Lemma 2

(Korn’s inequality, [12]) Let $\varOmega$ be a bounded domain with a Lipschitz-continuous boundary. Then, we have the following.

• (i) There exists a positive constant $\alpha _1$ such that

  $$\begin{aligned} (\Vert D(v)\Vert _0^2+\Vert v\Vert _0^2)^{1/2} \ge \alpha _1 \Vert v\Vert _1,\quad \forall v\in H^1(\varOmega )^d. \end{aligned}$$

• (ii) The norms $\Vert D(\cdot )\Vert _0$ and $\Vert \cdot \Vert _1$ are equivalent in $H^1_0(\varOmega )^d$.

Lemma 3

([5, 9, 10]) There exist linear operators $\varPi _h: H^2(\varOmega ) \rightarrow X_h$ and $\varPi _h^C: L_0^2(\varOmega ) \rightarrow Q_h$, which satisfy

$$\begin{aligned} \Vert \varPi _h v - v\Vert _1&\le \alpha _2 h \Vert v\Vert _2,\quad \forall v \in H^2(\varOmega )^d,\end{aligned}$$

(15a)

$$\begin{aligned} \Vert \varPi _h^C q - q\Vert _0&\le \alpha _3 h \Vert q\Vert _1,\quad \forall q \in L_0^2(\varOmega )\cap H^1(\varOmega ), \end{aligned}$$

(15b)

where $\alpha _2$ and $\alpha _3$ are positive constants independent of $h$.

Remark 5

$\varPi _h$ is nothing but the Lagrange interpolation operator and $\varPi _h^C$ is the Clément interpolation operator.

Lemma 4

([16, eq. (3.6)]) Let $\{\fancyscript{T}_h\}_{h\downarrow 0}$ be a regular family of triangulations of $\bar{\varOmega }$. Then, there exists a positive constant $\alpha _4$ independent of $h$ such that

$$\begin{aligned} |q_h|_h \le \alpha _4 \Vert q_h\Vert _0,\quad \forall q_h\in Q_h. \end{aligned}$$

(16)

Lemma 5

([16, Lemma 3.2]) There exists a positive constant $\alpha _5=c(\nu )$ independent of $h$ such that for any $h$

$$\begin{aligned} \inf _{(u_h,p_h)\in V_h\times Q_h}\sup _{(v_h,q_h)\in V_h\times Q_h} \frac{\fancyscript{A}_h((u_h,p_h), (v_h,q_h))}{\Vert (u_h,p_h)\Vert _{V\times Q} \Vert (v_h,q_h)\Vert _{V\times Q}}\ge \alpha _5. \end{aligned}$$

(17)

Remark 6

The conventional inf-sup condition [17] requires a positive constant $\beta ^*$ independent of $h$ such that

$$\begin{aligned} \inf _{q_h\in Q_h}\sup _{v_h\in V_h} \frac{b(v_h, q_h)}{\Vert v_h\Vert _1 \Vert q_h\Vert _0} \ge \beta ^*. \end{aligned}$$

Although the condition does not hold true for the pair $V_h$ and $Q_h$ of the P1/P1 finite element spaces, $\fancyscript{A}_h$ satisfies the stability inequality (17) for this pair.

The Stokes projection (13) has the following property, which is essentially proved in [6].

Proposition 2

Suppose $(u, p)\in (V \cap H^2(\varOmega )^d)\times (Q \cap H^1(\varOmega ))$. Then, there exists a positive constant $\alpha _6=c(1/\nu , \nu )$ independent of $h$ such that for any $h$ the Stokes projection $(\hat{u}_h, \hat{p}_h)$ of $(u, p)$ by (13) satisfies

$$\begin{aligned} \Vert \hat{u}_h - u\Vert _1,\ \ \Vert \hat{p}_h - p\Vert _0,\ \ |\hat{p}_h - p|_h \le \alpha _6 h \Vert (u,p)\Vert _{H^2\times H^1}. \end{aligned}$$

(18)

Proof

Let $\varPi _h$ and $\varPi _h^\mathrm{C}$ be the operators in Lemma 3. From (13) and Lemmas 3, 4 and 5 it holds that

$$\begin{aligned}&\Vert (\hat{u}_h-\varPi _hu,\hat{p}_h-\varPi _h^\mathrm{C}p)\Vert _{V\times Q}\\&\quad \le \frac{1}{\alpha _5}\sup _{(v_h,q_h)\in V_h\times Q_h} \frac{\fancyscript{A}_h((\hat{u}_h-\varPi _hu,\hat{p}_h-\varPi _h^\mathrm{C}p), (v_h,q_h))}{\Vert (v_h,q_h)\Vert _{V\times Q}}\\&\quad = \frac{1}{\alpha _5} \sup _{(v_h,q_h)\in V_h\times Q_h} \frac{\fancyscript{A}_h((u-\varPi _hu,p-\varPi _h^\mathrm{C}p), (v_h,q_h))+\fancyscript{C}_h(p,q_h)}{\Vert (v_h,q_h)\Vert _{V\times Q}}\\&\quad \le c(1/\nu , \nu ) \bigl (\Vert (u-\varPi _hu, p-\varPi _h^\mathrm{C}p)\Vert _{V\times Q}+|p|_h\bigr ) \\&\quad \le c^\prime (1/\nu , \nu ) h \Vert (u, p)\Vert _{H^2\times H^1}, \end{aligned}$$

which implies (18). $\square$

Lemma 6

Assume Hypothesis 1 and condition (6). Then, for any $n\in \{0,\ldots ,N_T\}$ it holds that

$$\begin{aligned} \Vert v - v \circ X_1^n \Vert _0 \le c_0 \varDelta t \Vert v\Vert _1,\quad \forall v \in V. \end{aligned}$$

(19)

Proof

Let any $n\in \{0,\ldots ,N_T\}$ be fixed. Let $y(x,s) \equiv x-s w^n(x)\varDelta t$ for $s \in [0,1]$ and $J_s(x) \equiv \det (\partial y/\partial x)$ be the Jacobian. It holds that

$$\begin{aligned} J_s(x) \ge 1/2,&\quad \forall s\in [0,1],\\ v(x) - v \circ X_1^{n}(x) = \bigl [ v\bigl (y(x,s)\bigr ) \bigr ]_{s=1}^{0}&= \varDelta t \int _0^{1} \bigl [ \{w^{n}(x) \cdot \nabla \}v \bigr ]\bigl ( y(x,s) \bigr ) ds. \end{aligned}$$

Changing the variable from $x$ to $y$ and using the above evaluation of the Jacobian, we have

$$\begin{aligned} \Vert v - v \circ X_1^n\Vert _0&\le c_0 \varDelta t \int _0^1 \bigl \Vert \nabla v\bigl ( y(\cdot ,s) \bigr ) \bigr \Vert _0 ds \le c_0 \sqrt{2} \varDelta t \int _0^1 \Vert \nabla v\Vert _0 ds \le c_0^\prime \varDelta t \Vert v\Vert _1, \end{aligned}$$

which completes the proof. $\square$

4.2 Proof of Theorem 1

From (9) with $v_h=0\in V_h$ and (11), it holds that for $n=0,\ldots , N_T$

$$\begin{aligned} b\left( u^n_h, q_h\right) -\fancyscript{C}_h\left( p^n_h, q_h\right) = 0, \quad \forall q_h\in Q_h, \end{aligned}$$

which gives for $n=1,\ldots , N_T$

$$\begin{aligned} b\left( \overline{D}_{\varDelta t}u_h^n, q_h\right) -\fancyscript{C}_h\left( \overline{D}_{\varDelta t}p^n_h, q_h\right) = 0, \quad \forall q_h\in Q_h. \end{aligned}$$

(20)

Substituting $(\overline{D}_{\varDelta t}u_h^n, 0) \in V_h\times Q_h$ into $(v_h, q_h)$ in (9) and using (20) with $q_h = -p_h^n$, we have for $n=1,\ldots ,N_T$

$$\begin{aligned} \Bigl ( \frac{u^n_h - u^{n-1}_h \circ X_1^n}{\varDelta t}, \overline{D}_{\varDelta t}u_h^n \Bigr ) + a\left( u^n_h, \overline{D}_{\varDelta t}u_h^n\right) +\fancyscript{C}_h\left( \overline{D}_{\varDelta t}p^n_h, p^n_h\right) + d^n\big (u_h^{n-1}, \overline{D}_{\varDelta t}u_h^n\big ) \nonumber \\ = \left( f_h^n, \overline{D}_{\varDelta t}u_h^n\right) . \end{aligned}$$

(21)

We evaluate each term in (21) as follows.

$$\begin{aligned} \Bigl ( \frac{u^n_h - u^{n-1}_h \circ X_1^n}{\varDelta t}, \overline{D}_{\varDelta t}u_h^n \Bigr )&= \Bigl ( \overline{D}_{\varDelta t}u_h^n + \frac{1}{\varDelta t}( u_h^{n-1} - u_h^{n-1}\circ X_1^n), \overline{D}_{\varDelta t}u_h^n\Bigr ) \nonumber \\&= \Vert \overline{D}_{\varDelta t}u_h^n\Vert _0^2 + \frac{1}{\varDelta t} \bigl ( u_h^{n-1} - u_h^{n-1}\circ X_1^n, \overline{D}_{\varDelta t}u_h^n\bigr ) \nonumber \\&\ge \Vert \overline{D}_{\varDelta t}u_h^n\Vert _0^2 - \frac{1}{\varDelta t} \Vert u_h^{n-1} - u_h^{n-1}\circ X_1^n\Vert _0 \Vert \overline{D}_{\varDelta t}u_h^n\Vert _0 \nonumber \\&\ge \Vert \overline{D}_{\varDelta t}u_h^n\Vert _0^2 - c_0 \Vert u_h^{n-1}\Vert _1\Vert \overline{D}_{\varDelta t}u_h^n\Vert _0 \quad (\text{ by } (19)) \nonumber \\&\ge \Vert \overline{D}_{\varDelta t}u_h^n\Vert _0^2 - \Bigl ( c_0^2 \Vert u_h^{n-1}\Vert _1^2 + \frac{1}{4}\Vert \overline{D}_{\varDelta t}u_h^n\Vert _0^2 \Bigr ) \nonumber \\&\ge \frac{3}{4} \Vert \overline{D}_{\varDelta t}u_h^n\Vert _0^2 - c_0^\prime \Vert D\bigl ( u_h^{n-1} \bigr ) \Vert _0^2,\end{aligned}$$

(22a)

$$\begin{aligned} a(u^n_h, \overline{D}_{\varDelta t}u_h^n)&= \overline{D}_{\varDelta t} \Bigl ( \frac{1}{2} a(u^n_h, u^n_h) \Bigr ) + \frac{\varDelta t}{2} a( \overline{D}_{\varDelta t}u_h^n, \overline{D}_{\varDelta t}u_h^n )\nonumber \\&= \overline{D}_{\varDelta t} \bigl ( \nu \Vert D(u^n_h)\Vert _0^2 \bigr ) + \nu \varDelta t \Vert D\bigl ( \overline{D}_{\varDelta t}u_h^n \bigr )\Vert _0^2 \nonumber \\&\ge \overline{D}_{\varDelta t} \bigl ( \nu \Vert D(u^n_h)\Vert _0^2 \bigr ),\end{aligned}$$

(22b)

$$\begin{aligned} {{\fancyscript{C}}_h} \left( \overline{D}_{\varDelta t}p^n_h, p^n_h\right)&= \overline{D}_{\varDelta t} \Bigl ( \frac{1}{2} \fancyscript{C}_h(p^n_h, p^n_h) \Bigr ) +\frac{\varDelta t}{2} \fancyscript{C}_h\left( \overline{D}_{\varDelta t}p_h^n, \overline{D}_{\varDelta t}p_h^n \right) \nonumber \\&= \overline{D}_{\varDelta t} \Bigl ( \frac{\delta _0}{2} |p^n_h|_h^2 \Bigr ) + \frac{\delta _0\varDelta t}{2} | \overline{D}_{\varDelta t}p_h^n |_h^2 \nonumber \\&\ge \overline{D}_{\varDelta t} \Bigl ( \frac{\delta _0}{2} |p^n_h|_h^2 \Bigr ),\end{aligned}$$

(22c)

$$\begin{aligned} -d^n(u_h^{n-1}, \overline{D}_{\varDelta t}u_h^n)&= -(\lambda ^n u_h^{n-1}, \overline{D}_{\varDelta t}u_h^n) \nonumber \\&\le c_1 \Vert u_h^{n-1}\Vert _0^2 + \frac{1}{8}\Vert \overline{D}_{\varDelta t}u_h^n\Vert _0^2 \nonumber \\&\le c_1^\prime \Vert D(u_h^{n-1})\Vert _0^2 + \frac{1}{8}\Vert \overline{D}_{\varDelta t}u_h^n\Vert _0^2, \end{aligned}$$

(22d)

$$\begin{aligned} \left( f_h^n, \overline{D}_{\varDelta t}u_h^n\right)&\le 2\Vert f_h^n\Vert _0^2 + \frac{1}{8} \Vert \overline{D}_{\varDelta t}u_h^n\Vert _0^2, \end{aligned}$$

(22e)

where Lemma 2 has been used for (22a) and (22d). Combining (22) with (21), we have for $n = 1,\ldots , N_T$

$$\begin{aligned}&\overline{D}_{\varDelta t} \Bigl ( \nu \Vert D(u_h^n)\Vert _0^2 + \frac{\delta _0}{2} |p^n_h|_h^2 \Bigr ) + \frac{1}{2} \Vert \overline{D}_{\varDelta t}u_h^n\Vert _0^2 \le (c_0^\prime + c_1^\prime ) \Vert D( u_h^{n-1}) \Vert _0^2 + 2\Vert f_h^n\Vert _0^2. \end{aligned}$$

(23)

Hence, the first three inequalities of (12) are obtained by applying Lemma 1 to (23).

Now we prove the last inequality of (12). From Lemmas 4, 5 and 6 it holds that

$$\begin{aligned} \Vert p_h^n\Vert _0&\le \Vert (u_h^n,p_h^n)\Vert _{V\times Q} \le \frac{1}{\alpha _5} \sup _{(v_h,q_h)\in V_h\times Q_h} \frac{\fancyscript{A}_h((u_h^n,p_h^n), (v_h,q_h))}{\Vert (v_h,q_h)\Vert _{V\times Q}} \nonumber \\&= \frac{1}{\alpha _5} \sup _{(v_h,q_h)\in V_h\times Q_h} \frac{(f_h^n, v_h) - \frac{1}{\varDelta t}(u^n_h - u^{n-1}_h \circ X_1^n, v_h) - d^n(u_h^{n-1}, v_h)}{\Vert (v_h,q_h)\Vert _{V\times Q}} \nonumber \\&\le \frac{1}{\alpha _5} \Bigl \{ \Vert f_h^n\Vert _0 + \Vert \overline{D}_{\varDelta t}u^n_h\Vert _0 + \frac{1}{\varDelta t} \Vert u_h^{n-1}-u_h^{n-1}\circ X_1^n \Vert _0 + \Vert \lambda ^nu_h^{n-1}\Vert _0\Bigr \}\nonumber \\&\le \frac{1}{\alpha _5} \bigl \{ \Vert f_h^n\Vert _0 + \Vert \overline{D}_{\varDelta t}u^n_h\Vert _0 + (c_0+c_1) \Vert u_h^{n-1}\Vert _1 \bigr \},\nonumber \end{aligned}$$

which yields the last inequality of (12) by the first and second inequalities of (12).

4.3 Proof of Theorem 2

Let $(\hat{u}_h, \hat{p}_h)(t)\in V_h\times Q_h$ be the Stokes projection of $(u, p)(t)\in H^2(\varOmega )^d\times H^1(\varOmega )$ by (13) and set

$$\begin{aligned} e_h^n\equiv u_h^n-\hat{u}_h^n,\quad \epsilon _h^n\equiv p_h^n-\hat{p}_h^n,\quad \eta (t)\equiv (u-\hat{u}_h)(t). \end{aligned}$$

From (2), (9), (13) and identity

$$\begin{aligned} e_h^n = \eta ^n - u^n + u^n_h, \end{aligned}$$

it holds that for any $(v_h, q_h)\in V_h\times Q_h$

$$\begin{aligned}&\Bigl (\frac{e_h^n-e_h^{n-1}\circ X_1^n}{\varDelta t}, v_h\Bigr ) + a(e_h^n, v_h)+b(v_h, \epsilon _h^n)+b(e_h^n, q_h) -\fancyscript{C}_h (\epsilon _h^n, q_h) + d^n(e_h^{n-1},v_h)\nonumber \\&=\Bigl (\frac{\eta ^n - \eta ^{n-1} \circ X_1^n}{\varDelta t} - \frac{u^n - u^{n-1} \circ X_1^n}{\varDelta t}, v_h\Bigr ) + \Bigl ( \frac{Du^n}{Dt}, v_h \Bigr ) + d^n ( \eta ^{n-1} +u^n - u^{n-1}, v_h )\nonumber \\&= \Bigl ( \frac{\eta ^n - \eta ^{n-1} \circ X_1^n}{\varDelta t} + \Bigl (\frac{Du^n}{Dt} - \frac{u^n - u^{n-1} \circ X_1^n}{\varDelta t}\Bigr ) + \lambda ^n (\eta ^{n-1} + u^n - u^{n-1}), v_h \Bigr ) \nonumber \\&= (\tilde{f}_h^n, v_h), \end{aligned}$$

(24)

where

$$\begin{aligned} \tilde{f}_h^n&\equiv \frac{\eta ^n - \eta ^{n-1} \circ X_1^n}{\varDelta t} + \Bigl ( \frac{Du^n}{Dt} - \frac{u^n - u^{n-1} \circ X_1^n}{\varDelta t} \Bigr ) + \lambda ^n (\eta ^{n-1} + u^n - u^{n-1}). \end{aligned}$$

We evaluate $\Vert \tilde{f}_h\Vert _{l^2(L^2)}$. It holds that

$$\begin{aligned} \Vert \tilde{f}_h^n\Vert _0&\le \Bigl \Vert \frac{\eta ^n \!-\! \eta ^{n-1} \circ X_1^n}{\varDelta t} \Bigr \Vert _0 + \Bigl \Vert \frac{Du^n}{Dt} - \frac{u^n - u^{n-1} \circ X_1^n}{\varDelta t} \Bigr \Vert _0 + \Vert \lambda ^n (\eta ^{n-1} + u^n - u^{n-1})\Vert _0\nonumber \\&\equiv I_1^n + I_2^n + I_3^n. \end{aligned}$$

Let $y(x, s)\equiv x-(1-s)w^n(x) \varDelta t$ and $t(s)\equiv t^{n-1} +s\varDelta t~(s\in [0,1])$. From Proposition 2 the terms $I_i^n~(i = 1, 2, 3)$ are evaluated as

$$\begin{aligned} I_1^n&\le \Vert \overline{D}_{\varDelta t}\eta ^n\Vert _0 + \frac{1}{\varDelta t}\Vert \eta ^{n-1}- \eta ^{n-1}\circ X_1^n\Vert _0 \nonumber \\&\le \frac{1}{\sqrt{\varDelta t}} \Vert \eta \Vert _{H^1(t^{n-1},t^n; L^2)} + c_0 \Vert \eta ^{n-1}\Vert _1\quad (\text{ by } (19)) \nonumber \\&\le \alpha _6 h \Bigl ( \frac{1}{\sqrt{\varDelta t}} \Vert (u, p) \Vert _{H^1(t^{n-1}, t^n; H^2\times H^1 )} + c_0 \Vert (u, p)^{n-1}\Vert _{H^2\times H^1} \Bigr ), \\ I_2^n&= \Bigl \Vert \Bigl \{ \Bigl ( \frac{\partial {}}{\partial {t}} + w^n(\cdot )\cdot \nabla \Bigr ) u \Bigr \} (\cdot , t^n) - \int _0^1 \Bigl \{ \Bigl ( \frac{\partial {}}{\partial {t}} + w^n(\cdot )\cdot \nabla \Bigr ) u \Bigr \} \bigl ( y(\cdot , s), t(s) \bigr ) ds \Bigr \Vert _0 \\&\le \varDelta t \int _0^1ds \int _{s}^1 \Bigl \Vert \Bigl \{ \Bigl ( \frac{\partial {}}{\partial {t}} + w^n(\cdot )\cdot \nabla \Bigr )^2 u \Bigr \} \bigl ( y(\cdot ,s_1), t(s_1) \bigr ) \Bigr \Vert _0 ds_1 \\&\le c_0 \sqrt{\varDelta t} \Vert u \Vert _{Z^2(t^{n-1}, t^n)}\quad \text{(by } \det ( \partial y(\cdot , s)/\partial x ) \ge 1/2,~\forall s\in [0,1]\text{) },\\ I_3^n&\le c_1 (\Vert \eta ^{n-1}\Vert _0 + \Vert u^n- u^{n-1} \Vert _0) \nonumber \\&\le c_1 \bigl ( {\alpha _6 h} \Vert (u, p)^{n-1}\Vert _{H^2\times H^1} + \sqrt{\varDelta t} \Vert u \Vert _{H^1(t^{n-1},t^n; L^2)} \bigr ), \end{aligned}$$

which imply

$$\begin{aligned} \Vert \tilde{f}_h\Vert _{l^2(L^2)}&\le c_2 ( \varDelta t\Vert u\Vert _{Z^2} + h \Vert (u,p)\Vert _{H^1(H^2\times H^1)}). \end{aligned}$$

(25)

From the definitions of $(u_h^0,p_h^0)$ and $({\hat{u}}_h^0,{\hat{p}}_h^0)$ it holds that for any $q_h \in Q_h$

$$\begin{aligned} b(e_h^0,q_h) - \fancyscript{C}_h(\epsilon _h^0, q_h)&= b(u_h^0, q_h) - \fancyscript{C}_h(p_h^0, q_h) - \bigl \{ b(\hat{u}_h^0, q_h) - \fancyscript{C}_h(\hat{p}_h^0, q_h) \bigr \} \\&= b(u^0, q_h) - b(u^0, q_h) =0. \end{aligned}$$

Applying Theorem 1 to (24), we obtain

$$\begin{aligned}&\Vert e_h\Vert _{l^\infty (H^1)},\ \ \Vert \overline{D}_{\varDelta t}e_h \Vert _{l^2(L^2)},\ \ \Vert \epsilon _h \Vert _{l^2(L^2)} \nonumber \\&\quad \le c_*\bigl (\Vert e_h^0\Vert _1 + |\epsilon _h^0|_h + \varDelta t \Vert u \Vert _{Z^2}+ h \Vert (u,p)\Vert _{H^1(H^2\times H^1)}\bigr ) \end{aligned}$$

(26)

from (25). Since $(u_h^0, p_h^0)$ and $(\hat{u}_h^0, \hat{p}_h^0)$ are the Stokes projections of $(u^0, 0)$ and $(u^0, p^0)$ by (13), respectively, it holds that

$$\begin{aligned} \Vert e_h^0\Vert _1&= \Vert u_h^0 - \hat{u}_h^0\Vert _1 \le \Vert u_h^0 - u^0\Vert _1 +\Vert u^0 - \hat{u}_h^0\Vert _1 \le c \alpha _6 h \Vert (u^0,p^0)\Vert _{H^2\times H^1},\end{aligned}$$

(27a)

$$\begin{aligned} |\epsilon _h^0|_h&= |p_h^0-\hat{p}_h^0|_h \le | p_h^0 - 0 |_h + |\hat{p}_h^0-p^0|_h + |p^0|_h \le c \alpha _6 h \Vert (u^0,p^0) \Vert _{H^2\times H^1}.\qquad \end{aligned}$$

(27b)

Combining (27) with (26), we obtain (14). $\square$

5 Numerical Results

In this section two- and three-dimensional test problems are computed by scheme (9). Theorems 1 and 2 are valid for any positive number $\delta _0$ in (8). Here we concentrate on recognizing the theoretical convergence order numerically and simply fix $\delta _0 = 1$.

Quadrature formulae of degree five for $d=2$ (seven points) and $3$ (fifteen points) [31] are employed for computation of the integral

$$\begin{aligned} \int _K u^{n-1}_h\circ X_1^n(x) v_h(x)\ dx \end{aligned}$$

appearing in scheme (9).

In the following examples, $N$ is the division number of each side of the domain (square or cube) and we (re)define $h\equiv 1/N$. The time increment $\varDelta t$ is set to be $\varDelta t=h$ in order to keep the same orders of both $\varDelta t$ and $h$, since the error estimates in Theorem 2 are of $O(\varDelta t+h)$.

In scheme (9) the initial function $u_h^0$ is chosen as the first component of the Stokes projection of $(u^0, 0)$ by (13). Let $(u_h,p_h)$ be the solution of scheme (9) and $(u,p)$ be each analytical solution set in the examples. We define $Err$ by

$$\begin{aligned} Err \equiv \frac{\Vert u_h-\varPi _hu\Vert _{l^2(H^1)} + \Vert p_h-\varPi _hp\Vert _{l^2(L^2)}}{\Vert \varPi _hu\Vert _{l^2(H^1)} + \Vert \varPi _hp\Vert _{l^2(L^2)}} \end{aligned}$$

(28)

as the relative error between $(u_h,p_h)$ and $(u,p)$, where the same symbol $\varPi _h$ has been used as its scalar version, i.e., $\varPi _h: H^2(\varOmega )\rightarrow M_h$. The system of linear equations in the coefficients of $u_h^n$ and $p_h^n$ of scheme (9) is solved by MINRES.

Example 1

(2D) Let $d=2$. In problem (1) we set $\varOmega =(0, 1)^2, T=1$, four values of $\nu$,

$$\begin{aligned} \nu =10^{-k},\quad k=1,\ldots ,4, \end{aligned}$$

and functions $w$ and $\lambda$,

$$\begin{aligned} w (x,t)&= \Bigl ( - \frac{\partial {\psi }}{\partial {x_2}} ,\ \frac{\partial {\psi }}{\partial {x_1}} \Bigr )(x,t),\quad \lambda (x,t) = \left( \begin{array}{ll} \sin \pi (x_1+t) &{} \sin \pi (x_2+t)\\ \cos \pi (x_2+t) &{} \sin \pi (x_1+x_2+t) \end{array} \right) , \end{aligned}$$

where $\psi$ is a function defined by

$$\begin{aligned} \psi (x,t)&= \frac{\sqrt{3}}{2\pi } \sin ^2 (\pi x_1) \sin ^2 (\pi x_2) \sin \{ \pi (x_1+x_2 + t) \}. \end{aligned}$$

The functions $f$ and $u^0$ are given so that the exact solution is

$$\begin{aligned} (u, p)(x,t) = \bigl ( w(x,t), \sin \{\pi (x_1 + 2 x_2 + t)\} \bigr ). \end{aligned}$$

The solution is normalized so that $\Vert u\Vert _{C^0(L^\infty )}=\Vert p\Vert _{C^0(L^\infty )}=1$.

We set $N =16, 32, 64, 128$ and $256$. The left of Fig. 1 shows a sample mesh ($N=16$). The right of Fig. 1 exhibits the graphs of $Err$ versus $h\ (=\varDelta t)$ in logarithmic scale. We can see that $Err$ is almost of first order in $h$ and $\varDelta t$ for all $\nu$.

Fig. 1

A sample mesh (left, $N=16$) and graphs of $Err$ versus $h$ (right) for Example 1

Example 2

(3D) Let $d=3$. In problem (1) we set $\varOmega =(0, 1)^3$ and functions $w$ and $\lambda$,

$$\begin{aligned} w (x,t)&= \mathrm{rot}\, \varPsi (x, t),\\ \lambda (x,t)&= \left( \begin{array}{lll} \sin \pi (x_1+t) &{} \sin \pi (x_2+t) &{} \sin \pi (x_3+t)\\ \cos \pi (x_2+t) &{} \sin \pi (x_1+x_2+t) &{} \sin \pi (x_2+x_3+t)\\ \cos \pi (x_3+t) &{} \cos \pi (x_2+x_3+t) &{} \sin \pi (x_1+x_2+x_3+t) \end{array} \right) , \end{aligned}$$

where $\varPsi$ is a function defined by

$$\begin{aligned} \varPsi _1(x,t)&\equiv \frac{8\sqrt{3}}{27\pi }\sin (\pi x_1) \sin ^2 (\pi x_2) \sin ^2 (\pi x_3) \sin \{\pi (x_2+x_3+t)\},\\ \varPsi _2(x,t)&\equiv \frac{8\sqrt{3}}{27\pi }\sin ^2 (\pi x_1) \sin (\pi x_2) \sin ^2 (\pi x_3) \sin \{\pi (x_3+x_1+t)\},\\ \varPsi _3(x,t)&\equiv \frac{8\sqrt{3}}{27\pi }\sin ^2 (\pi x_1) \sin ^2 (\pi x_2) \sin (\pi x_3) \sin \{\pi (x_1+x_2+t)\}. \end{aligned}$$

Constants $T$ and $\nu$ are the same as those in Example 1. The functions $f$ and $u^0$ are given so that the exact solution is

$$\begin{aligned} (u, p)(x,t) = \bigl ( w(x,t), \sin \{\pi (x_1 + 2 x_2 + x_3 + t)\} \bigr ). \end{aligned}$$

The solution is normalized so that $\Vert u\Vert _{C^0(L^\infty )}=\Vert p\Vert _{C^0(L^\infty )}=1$.

We set $N=8, 16, 32$ and $64$. The left of Fig. 2 shows a sample mesh ($N=8$). The right of Fig. 2 exhibits graphs of $Err$ versus $h\ (= \varDelta t)$ in logarithmic scale. We can see that $Err$ is of better order than first one for $\nu =10^{-1}$ and is almost of first order for $\nu =10^{-k}~(k=2, 3, 4)$ in $h$ and $\varDelta t$.

Fig. 2

A sample mesh (left, $N=8$) and graphs of $Err$ versus $h$ (right) for Example 2

These results for both Examples 1 and 2 are consistent with Theorem 2.

6 Conclusions

A combined finite element scheme with a characteristics method and Brezzi–Pitkäranta’s stabilization method for the Oseen equations has been studied. Stability (Theorem 1) and convergence (Theorem 2) results with the optimal error estimates for the velocity and the pressure have been proved. The scheme has the advantages of both of the characteristics method and Brezzi–Pitkäranta’s stabilization method, i.e., robustness for convection-dominated problems, symmetry of the resulting matrix and the small number of degrees of freedom. In order to construct the initial approximate velocity we have also introduced a stabilized Stokes projection, which works well in the analysis without any loss of convergence order. The theoretical convergence order has been recognized in two- and three-dimensional test problems in Examples 1 and 2, respectively. To devise general higher-order stabilized characteristics schemes is a future work. A corresponding stabilized characteristics scheme for the Navier–Stokes equations will be studied in a forthcoming paper [25].