On the finite element approximation for non-stationary saddle-point problems

Abstract

In this paper, we present a numerical analysis of the hydrostatic Stokes equations, which are linearization of the primitive equations describing the geophysical flows of the ocean and the atmosphere. The hydrostatic Stokes equations can be formulated as an abstract non-stationary saddle-point problem, which also includes the non-stationary Stokes equations. We first consider the finite element approximation for the abstract equations with a pair of spaces under the discrete inf–sup condition. The aim of this paper is to establish error estimates for the approximated solutions in various norms, in the framework of analytic semigroup theory. Our main contribution is an error estimate for the pressure with a natural singularity term $t^{-1}$, which is induced by the analyticity of the semigroup. We also present applications of the error estimates for the finite element approximations of the non-stationary Stokes and the hydrostatic Stokes equations.

1 Introduction

Let $\varOmega = (0,1)^2 \times (-D,0) \subset \mathbb {R}^3$ be a (shallow) box domain with $D>0$. We consider the (non-stationary) hydrostatic Stokes equations

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t - \Delta u + \nabla _H p = 0, &{} \text {in } \varOmega \times (0,T), \\ {{\mathrm{div}}}_H \bar{v} = 0, &{} \text {in } \varOmega \times (0,T), \\ u(0) = u_0, &{} \text {in } \varOmega \end{array}\right. } \end{aligned}$$

(1)

for an unknown velocity $u :\varOmega \rightarrow \mathbb {R}^2$ and pressure $p :G:=(0,1)^2 \rightarrow \mathbb {R}$, where $\nabla _H p = (\partial _x p, \partial _y p)^T$, ${{\mathrm{div}}}_H v = \partial _x v_1 + \partial _y v_2$, and $\bar{v} = \int _{-D}^0 v(\cdot ,z)dz$. We impose the boundary conditions

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _z u = 0, &{} \text {on } \varGamma _u := G \times \{ 0 \}, \\ u = 0, &{} \text {on } \varGamma _b := G \times \{ -D \}, \\ u \text {and} p \text {are periodic} &{} \text {on } \varGamma _l := \partial G \times (-D,0). \end{array}\right. } \end{aligned}$$

(2)

These are the linearized equations of the primitive equations (without the Coriolis force) described as

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u + (U \cdot \nabla ) u - \Delta u + \nabla _H p = 0, &{} \text {in } \varOmega \times (0,T), \\ \partial _z p = 0, &{} \text {in } \varOmega \times (0,T), \\ {{\mathrm{div}}}U = 0, &{} \text {in } \varOmega \times (0,T), \\ u(0) = u_0, &{} \text {in } \varOmega \end{array}\right. } \end{aligned}$$

(3)

with boundary conditions

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _z u = 0, u_3 = 0, &{} \text {on } \varGamma _u, \\ U = 0, &{} \text {on } \varGamma _b, \\ \text {{ U} and { p} are periodic} &{} \text {on } \varGamma _l, \end{array}\right. } \end{aligned}$$

where $U = (u, u_3) :\varOmega \rightarrow \mathbb {R}^3$. The primitive equations are derived from the Navier–Stokes equations under the assumption that the vertical motion is much smaller than the horizontal motion, and were first introduced by Lions et al. [21,22,23]. This model is considered to describe the geophysical flows of the ocean and the atmosphere.

In this paper, we consider the finite element approximation of the hydrostatic Stokes problem (1) and (2) as follows. Find $u_h :(0,T) \rightarrow V_h$ and $p_h :(0,T) \rightarrow Q_h$ satisfying the variational equation

$$\begin{aligned} {\left\{ \begin{array}{ll} (u_{h,t}, v_h)_\varOmega + (\nabla u_h, \nabla v_h)_\varOmega - ({{\mathrm{div}}}_H \bar{v}_h, p_h)_G = 0, &{} \forall v_h \in V_h, \\ ({{\mathrm{div}}}_H \bar{u}_h, q_h)_G = 0, &{} \forall q_h \in Q_h, \\ (u_h(0), v_h)_\varOmega = (u_0,v_h)_\varOmega , &{} \forall v_h \in V_{h,\sigma }, \end{array}\right. } \end{aligned}$$

where $V_h \subset H^1(\varOmega )^2$ and $Q_h \subset L^2_0(G)$ are finite-dimensional subspaces with suitable boundary conditions, the bracket $(\cdot , \cdot )_X$ expresses the $L^2$-inner product over a domain X, and $V_{h,\sigma } = \{ v_h \in V_h \mid ({{\mathrm{div}}}_H \bar{v}_h, q_h)_G = 0, \forall q_h \in Q_h \}$. The precise definitions are given in Sect. 4. The aim of this paper is to establish error estimates for $u_h$ and $p_h$ in the framework of analytic semigroup theory, as preliminaries for the numerical analysis of the primitive equations. Although there are several results available on the finite element method for the steady hydrostatic Stokes equations (e.g., [12,13,14,15]), there are no results for the non-stationary case, to the best of our knowledge.

As with the Navier–Stokes equations, the primitive equations are widely used in numerical computations for atmospheric and oceanic phenomena. Finite element approximations and error estimates are presented in [3, 4, 6, 16] for the steady primitive equations and in [5, 17, 18] for non-stationary problems. In [18] and [17], error estimates for various fully discretized schemes are provided. These estimates have exponential growth in time T (i.e., $e^{cT}$), since their arguments are based on the discrete Gronwall inequality. Therefore, these results are time-local estimates in a sense.

We are interested in time-global error estimates for finite element approximations of the hydrostatic Stokes and the primitive equations. In contrast to the three-dimensional Navier–Stokes equations, it is known that the three-dimensional primitive equations are globally well-posed in the $L^p$-settings (see [2] for $p=2$, and [19] for $p \in (1,\infty )$). In their proofs, the analyticity of the hydrostatic Stokes semigroup plays a crucial role. We can thus expect that the analytic semigroup approach presents an efficient method for the numerical analysis of the primitive equations. Indeed, for the two-dimensional Navier–Stokes equations, time-global error estimates have been obtained via the analytic semigroup approach in [24]. Their results are based on the error estimates for the non-stationary Stokes equations established in [25].

In order to derive error estimates for the finite element approximation, we formulate the hydrostatic Stokes equations as an abstract evolution problem. We define

$$\begin{aligned} V = \{ v \in H^1(\varOmega )^2 \mid v|_{\varGamma _b} = 0,\;v\;\text {is periodic on}\,\varGamma _{l} \}. \end{aligned}$$

(4)

Additionally, let $Q = L^2_0(G)$ and $H=L^2(\varOmega )^2$. Then, a weak form of the hydrostatic Stokes equations (2) can be given as a non-stationary saddle-point problem as follows. Find $u :(0,T) \rightarrow V$ and $p :(0,T) \rightarrow Q$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} (u_t(t), v)_H + a(u(t),v) + b(v,p(t)) = 0 , &{} \forall v \in V, \\ b(u(t),q) = 0, &{} \forall q \in Q, \end{array}\right. } \end{aligned}$$

(5)

where

$$\begin{aligned} a(u,v) = \iiint _\varOmega \nabla u : \nabla v \,dxdydz, \quad b(v,q) = - \iint _G ({{\mathrm{div}}}_H \bar{v}) q \,dxdy \end{aligned}$$

for $u,v \in V$ and $q \in Q$. Once we write the hydrostatic Stokes equations as above, we can use the same arguments for error estimates as is the case for the usual Stokes problem in [25]. Then, we can obtain the following error estimates for the velocity:

$$\begin{aligned} \Vert \nabla (u(t) - u_h(t)) \Vert _{L^2(\varOmega )}&\le C h t^{-1} \Vert u_0\Vert _{L^2(\varOmega )} , \\ \Vert u(t) - u_h(t) \Vert _{L^2(\varOmega )}&\le C h^2 t^{-1} \Vert u_0\Vert _{L^2(\varOmega )}, \end{aligned}$$

under the assumptions stated in Sect. 2 (Assumptions 1 and 2). In particular, the discrete inf–sup condition (11) plays an important role as in the stationary case.

According to [25], we can also obtain an error estimate for the pressure. The estimate presented in [25] is

$$\begin{aligned} \Vert p(t) - p_h(t) \Vert _{L^2} \le Ch (t^{-1} + t^{-3/2}) \Vert u_0\Vert _{L^2} \end{aligned}$$

for the two-dimensional Stokes equations

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t - \Delta u + \nabla p = 0, \\ {{\mathrm{div}}}u = 0, \\ u(0) = u_0 \end{array}\right. } \end{aligned}$$

(6)

with the Dirichlet boundary condition. However, the singularity $t^{-3/2}$ is unnatural from the viewpoint of analytic semigroup theory. Indeed, an optimal order error estimate should be of the form

$$\begin{aligned} \Vert p(t) - p_h(t) \Vert _{L^2} \le C h \Vert \nabla p(t) \Vert _{L^2}, \end{aligned}$$

and (6) yields

$$\begin{aligned} \Vert \nabla p(t) \Vert _{L^2} \le \Vert u_t(t) \Vert _{L^2} + \Vert \Delta u(t) \Vert _{L^2} \le C t^{-1} \Vert u_0\Vert _{L^2}. \end{aligned}$$

In this paper, we first address the abstract problem (5) for bilinear forms $a :V \times V \rightarrow \mathbb {R}$ and $b :V \times Q \rightarrow \mathbb {R}$ defined on Hilbert spaces V and Q, and its Galerkin approximation

$$\begin{aligned} {\left\{ \begin{array}{ll} (u_{h,t}(t), v_h)_H + a(u_h(t),v_h) + b(v_h,p_h(t)) = 0 , &{} \forall v_h \in V_h, \\ b(u_h(t),q_h) = 0, &{} \forall q_h \in Q_h, \end{array}\right. } \end{aligned}$$

(7)

for appropriate finite-dimensional subspaces $V_h \subset V$ and $Q_h \subset Q$. The main contribution of this paper is to derive error estimates for the approximation problem (7), both for the velocity u and the pressure p. In particular, we remove the term $t^{-3/2}$ from the error estimate for the pressure. Consequently, our results are a generalization and modification of the results in [25]. After deriving the error estimates for (7), we apply the results to error estimates for the finite element approximation of the hydrostatic Stokes equations.

Here, we present the idea of the proof for error estimates in the case of the hydrostatic Stokes equations. Our strategy is similar to that of [25]. Namely, we first rewrite the error in terms of the contour integral of the resolvent and then we reduce the error estimate for the velocity to that of the resolvent problem. The key idea is to establish the $V'$-error estimate for the resolvent problem as well as the $H^1$- and $L^2$-error estimates, which are already addressed in [25]. Here, $V'$ denotes the dual space of V defined by (4). This estimate coincides with the $H^{-1}$-error estimate if we impose the Dirichlet boundary condition. Then, we can obtain the $V'$-error estimate for the time derivative of the velocity of the form

$$\begin{aligned} \Vert u_t(t) - u_{h,t}(t) \Vert _{V'} \le Cht^{-1} \Vert u_0\Vert _{L^2(\varOmega )}. \end{aligned}$$

Finally, we can establish an error estimate for the pressure without the term $t^{-3/2}$, with the aid of the discrete inf–sup condition. We shall perform the above procedure in the abstract setting.

The rest of the paper is organized as follows. In Sect. 2, we introduce an abstract saddle-point problem and its Galerkin approximation, as well as the notation and assumptions in Sect. 2.1. After that, we introduce the resolvent problems and present some preliminary results in Sect. 2.2. Our main results are presented in Sect. 3. As mentioned above, we derive the error estimate in the dual norm for the resolvent problem, and then we establish the error estimate for the evolution equation. We apply these results for the Stokes and the hydrostatic Stokes equations in Sect. 4. For the usual Stokes equations (Sect. 4.1), error estimates for the velocity are already available. However, the estimate for the pressure presented here is strictly sharper than that of [25]. The error estimates for the non-stationary hydrostatic Stokes equations will be presented in Sect. 4.2. Finally, we present our conclusions and areas for future works in Sect. 5.

2 Preliminaries

2.1 Notation and assumptions

Throughout this paper, except for in the last section, the symbols H, V, and Q denote Hilbert spaces with dense and continuous injections $V \hookrightarrow H \hookrightarrow V'$, where $V'$ is the dual space of V, and $a :V \times V \rightarrow \mathbb {C}$ and $b :V \times Q \rightarrow \mathbb {C}$ are continuous bilinear forms. We assume that a is symmetric for simplicity, and that a is coercive and b satisfies the inf–sup condition:

$$\begin{aligned} {\text {Re}}\, a(v,v)\ge & {} \alpha \Vert v\Vert _V, \quad \forall v \in V, \nonumber \\ \sup _{v \in V} \frac{|b(v,q)|}{\Vert v\Vert _V}\ge & {} \beta _1 \Vert q\Vert _Q, \quad \forall q \in Q, \end{aligned}$$

(8)

for some positive constants $\alpha$ and $\beta _1$. We consider the following abstract non-stationary Stokes problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} (u_t(t), v)_H + a(u(t),v) + b(v, p(t)) = 0, &{} \forall v \in V, \\ b(u(t), q) = 0, &{} \forall q \in Q, \\ u(0) = u_0, \end{array}\right. } \end{aligned}$$

(9)

for $t \in (0,T)$, where $u_0 \in H_\sigma := \overline{V_\sigma }^{\Vert \cdot \Vert _H}$ and

$$\begin{aligned} V_\sigma := \{ v \in V \mid b(v,q) = 0, \ \forall q \in Q \}. \end{aligned}$$

We define a linear operator A on $H_\sigma$ associated with the bilinear form a by

$$\begin{aligned} {\left\{ \begin{array}{ll} D(A) = \{ u \in V_\sigma \mid \exists w \in H_\sigma \text { s.t. } (w,v)_H = a(u,v), \ \forall v \in V_\sigma \}, \\ (Au, v)_H = a(u,v), \qquad \forall u \in D(A), \quad \forall v \in V_\sigma , \end{array}\right. } \end{aligned}$$

(10)

which is the abstract version of the Stokes operator. By coercivity (8) and the usual semigroup theory (e.g., see [26]), the operator $-A$ generates an analytic contraction semigroup $e^{-tA}$ on $H_\sigma$. Thus, choosing $v \in V_\sigma$ as a test function in (9), we can construct a mild solution by $u(t) = e^{-tA}u_0$ for $t > 0$. Moreover, owing to the inf–sup condition and the closed range theorem (see, e.g., [9]), we can find $p(t) \in Q$ that satisfies

$$\begin{aligned} b(v,p(t)) = -(u_t(t), v)_H + a(u(t), v), \quad \forall v \in V, \end{aligned}$$

for almost all $t \in (0,T)$. The uniqueness of these solutions is clear. Finally, we define a linear operator $B :D(B) \subset Q \rightarrow H$ associated with b by

$$\begin{aligned} {\left\{ \begin{array}{ll} D(B) = \{ q \in Q \mid \exists w \in H \text { s.t. } (w,v)_H = b(v,q), \ \forall v \in V \}, \\ (Bq, v)_H = b(v,q), \qquad \forall q \in D(B), \quad \forall v \in V. \end{array}\right. } \end{aligned}$$

We next consider the Galerkin approximation for (9). Let $V_h \subset V$ and $Q_h \subset Q$ be finite-dimensional subspaces. We assume that they have the following properties:

Assumption 1

• (A-1) [discrete inf–sup condition] There exists $\beta _2 > 0$ such that

  $$\begin{aligned} \sup _{v_h \in V_h} \frac{|b(v_h,q_h)|}{\Vert v_h\Vert _V} \ge \beta _2 \Vert q_h\Vert _Q, \quad \forall q_h \in Q_h, \end{aligned}$$

  (11)

  uniformly in $h>0$.

• (A-2) [approximation property (1)] For each $v \in D(A)$, we can find $v_h \in V_h$ satisfying

  $$\begin{aligned} \Vert v_h \Vert _V\le & {} C \Vert v\Vert _V, \nonumber \\ \Vert v-v_h \Vert _H\le & {} C h \Vert v\Vert _V, \nonumber \\ \Vert v-v_h \Vert _V\le & {} C h \Vert Av\Vert _H, \end{aligned}$$

  (12)

  where C is independent of h and v.

• (A-3) [approximation property (2)] For each $q \in D(B)$, we can find $q_h \in Q_h$ satisfying

  $$\begin{aligned} \Vert q-q_h \Vert _Q \le C h \Vert Bq\Vert _H, \end{aligned}$$

  where C is independent of h and q.

$\square$

Remark 1

The assumption (A-2) includes the condition on “elliptic regularity”. For example, let $\varOmega \subset \mathbb {R}^2$ be a polygonal domain, $H=L^2(\varOmega )$, $V = H^1_0(\varOmega )$, and $V_h$ be the conforming $P^1$-finite element space with respect to a shape-regular triangulation of $\varOmega$. Then, for every $v \in H^2(\varOmega )$, we can construct $v_h \in V_h$ with the error estimate

$$\begin{aligned} \Vert v-v_h\Vert _{H^1_0} \le Ch \Vert v\Vert _{H^2}. \end{aligned}$$

However, if A is the Laplace operator defined by

$$\begin{aligned} (Au, v)_{L^2} = (\nabla u, \nabla v)_{L^2}, \quad u,v \in H^1_0(\varOmega ), \end{aligned}$$

then the error estimate (12) does not hold in general. Indeed, (12) requires that the condition $D(A) = H^2(\varOmega ) \cap H^1_0(\varOmega )$ is satisfied, or equivalently,

$$\begin{aligned} \Vert v \Vert _{H^2} \le C \Vert Av\Vert _{L^2} \end{aligned}$$

for all $v \in D(A)$, which is not true for non-convex polygonal domains (see [11]). $\square$

We also introduce the discrete “solenoidal” space $V_{h,\sigma }$, defined by

$$\begin{aligned} V_{h,\sigma } := \{ v_h \in V_h \mid b(v_h,q_h) = 0, \ \forall q_h \in Q_h \}. \end{aligned}$$

(13)

Note that $V_{h,\sigma } \not \subset V_\sigma$ in general. We now formulate the Galerkin (semi-discrete) approximation for the problem (9) as follows: find $u_h(t) \in V_h$ and $q_h(t) \in Q_h$ that satisfy

$$\begin{aligned} {\left\{ \begin{array}{ll} (u_{h,t}(t), v_h)_H + a(u_h(t),v_h) + b(v_h, p_h(t)) = 0, &{} \forall v_h \in V_h, \\ b(u_h(t), q_h) = 0, &{} \forall q_h \in Q_h, \\ u_h(0) = P_{h,\sigma } u_0, \end{array}\right. } \end{aligned}$$

(14)

where $P_{h,\sigma } :H \rightarrow V_{h,\sigma }$ is the orthogonal projection. We define the discrete “Stokes operator” $A_h$ by

$$\begin{aligned} (A_h u_h, v_h)_H = a(u_h, v_h), \quad \forall u_h, v_h \in V_{h,\sigma }. \end{aligned}$$

Then, as in the continuous case, the operator $-A_h$ generates an analytic contraction semigroup $e^{-tA_h}$ on $H_{h,\sigma } := (V_{h,\sigma }, \Vert \cdot \Vert _H)$, and thus we can construct a unique solution $(u_h, p_h)$ of the Eq. (14) due to the discrete inf–sup condition (11).

2.2 Finite element method for resolvent problems

We show error estimates for (14) via the resolvent estimates, as originally shown in [25] for the non-stationary Stokes problem. Let $\varGamma = \{ re^{\pm i(\pi - \delta )} \in \mathbb {C}\mid r \in [0,\infty ) \}$ be a path for $\delta \in (0,\pi /2)$, which is oriented so that the imaginary part increases along $\varGamma$. Then, since the semigroups $e^{-tA}$ and $e^{-tA_h}$ are analytic, we can write

$$\begin{aligned} u(t) - u_h(t) = \frac{1}{2\pi i} \int _\varGamma e^{t\lambda } \left[ (\lambda + A)^{-1} - (\lambda + A_h) P_{h,\sigma } \right] u_0 d\lambda . \end{aligned}$$

(15)

Therefore, the error estimate for $u_h$ is reduced to that of resolvent problems:

$$\begin{aligned} \lambda (w,v)_H + a(w,v) = (g,v)_H, \quad \forall v \in V_\sigma , \end{aligned}$$

(16)

and

$$\begin{aligned} \lambda (w_h,v_h)_H + a(w_h,v_h) = (g,v_h)_H, \quad \forall v_h \in V_{h,\sigma }, \end{aligned}$$

(17)

for given $g \in H$ and $\lambda \in \Sigma _\delta := \{ z \in \mathbb {C}\setminus \{0\} \mid |\arg z | < \pi - \delta \}$ with an arbitrarily fixed $\delta \in (0,\pi /2)$. Owing to the closed range theorem, the problem (16) is equivalent to the following equation for w and $\pi$:

$$\begin{aligned} {\left\{ \begin{array}{ll} \lambda (w,v)_H + a(w,v) + b(v,\pi ) = (g,v)_H, &{} \forall v \in V, \\ b(w,q) = 0, &{} \forall q \in Q, \end{array}\right. } \end{aligned}$$

(18)

and (17) is also equivalent to the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \lambda (w_h,v_h)_H + a(w_h,v_h) + b(v_h,\pi _h) = (g,v_h)_H, &{} \forall v \in V_h, \\ b(w_h,q_h) = 0, &{} \forall q_h \in Q_h, \end{array}\right. } \end{aligned}$$

(19)

where $w_h \in V_h$ and $\pi _h \in Q_h$ are unknown functions. We assume that Eq. (18) admits the following estimate.

Assumption 2

(A-4) For each $g \in H$ and $\lambda \in \Sigma _\delta$, Eq. (18) has a unique solution $(w,\pi ) \in V_\sigma \times Q$, which admits the regularity $w \in D(A)$ and $\pi \in D(B)$. Moreover, the following resolvent estimate holds:

$$\begin{aligned} |\lambda | \Vert w\Vert _H + |\lambda |^{1/2} \Vert w\Vert _V + \Vert Aw\Vert _H + \Vert B\pi \Vert _H \le C \Vert g\Vert _H, \end{aligned}$$

where C is independent of g and $\lambda$. $\square$

It is known that assumptions (A-1)–(A-4) allow us to obtain error estimates for the velocity.

Theorem 2.1

Let $\delta \in (0,\pi /2)$, and suppose that assumptions (A-1)–(A-4) hold. Then, we have

$$\begin{aligned} \left\| \left[ (\lambda + A)^{-1}P_\sigma - (\lambda + A_h)^{-1} P_{h,\sigma } \right] g \right\| _V&\le C h \Vert g\Vert _H, \end{aligned}$$

(20)

$$\begin{aligned} \left\| \left[ (\lambda + A)^{-1}P_\sigma - (\lambda + A_h)^{-1} P_{h,\sigma } \right] g \right\| _H&\le C h^2 \Vert g\Vert _H, \end{aligned}$$

(21)

for any $g \in H$ and $\lambda \in \Sigma _\delta$, where $P_\sigma :H \rightarrow H_\sigma$ is the orthogonal projection and C is independent of h, g, and $\lambda$. $\square$

An outline of the proof is given below. We refer the reader to [25, Theorems 3.2 and 4.2] for the detailed proof.

Proof

Fix $g \in H$, $\delta \in (0,\pi /2)$, and $\lambda \in \Sigma _\delta$ arbitrarily. Let $(w,\pi )$ and $(w_h,\pi _h)$ be the solutions of (18) and (19), respectively. Choose $v_h \in V_h$ and $q_h \in Q_h$ arbitrarily and consider $w_h - v_h$ and $\pi _h - q_h$. Substituting $\phi _h \in V_h$ and $\psi _h \in Q_h$ into (18) and (19), we have

$$\begin{aligned} a_\lambda (w_h-v_h,\phi _h) + b(\phi _h,\pi _h-q_h)= & {} a_\lambda (w-v_h,\phi _h) + b(\phi _h,\pi -q_h) \nonumber \\= & {} : {}_{V_h'}\langle F_h, \phi _h \rangle _{V_h}, \quad \forall \phi _h \in V_h, \end{aligned}$$

(22)

and

$$\begin{aligned} b(w_h-v_h,\psi _h) = b(w-v_h,\psi _h) =: {}_{Q_h'}\langle G_h, \psi _h \rangle _{Q_h}, \quad \forall \psi _h \in Q_h, \end{aligned}$$

(23)

where

$$\begin{aligned} a_\lambda (u,v) = \lambda (u,v)_H + a(u,v) \end{aligned}$$

for $u,v \in V$. By the elementary inequality $|s\lambda + t| \ge \sin (\delta /2)(s|\lambda |+t)$ for $\lambda \in \Sigma _{\pi -\delta }$ and $s,t \ge 0$, we can obtain

$$\begin{aligned} |a_\lambda (v,v)| \ge \alpha _1 \Vert v\Vert _{H^1(\varOmega )}^2 \end{aligned}$$

for all $v \in V$, where $\alpha _1>0$ is independent of $\lambda$. Therefore, Eqs. (22) and (23), the discrete inf–sup condition (A-1), and the generalized Lax–Milgram theorem (e.g., see [9, Theorem 2.34]) yield

$$\begin{aligned} |\lambda |^{1/2} \Vert w_h-v_h\Vert _H + \Vert w_h-v_h\Vert _V + \Vert \pi _h-q_h\Vert _Q \le C \left( \Vert F_h\Vert _{V_h'} + \Vert G_h\Vert _{Q_h'} \right) \end{aligned}$$

(24)

for some constant $C>0$, which is independent of h due to (A-1). From the definition of $F_h$ and $G_h$, we have

$$\begin{aligned} \Vert F_h\Vert _{V_h'} + \Vert G_h\Vert _{Q_h'} \le C \left( |\lambda |^{1/2} \Vert w-v_h\Vert _H + \Vert w-v_h\Vert _V + \Vert \pi -q_h\Vert _Q \right) , \end{aligned}$$

which implies, together with (24), that

$$\begin{aligned}&|\lambda |^{1/2} \Vert w-w_h\Vert _H + \Vert w-w_h\Vert _V + \Vert \pi -\pi _h\Vert _Q\\&\quad \le C \left( |\lambda |^{1/2} \Vert w-v_h\Vert _H + \Vert w-v_h\Vert _V + \Vert \pi -q_h\Vert _Q \right) . \end{aligned}$$

Therefore, assumptions (A-2)–(A-4) lead to

$$\begin{aligned} |\lambda |^{1/2} \Vert w-w_h\Vert _H + \Vert w-w_h\Vert _V + \Vert \pi -\pi _h\Vert _Q \le C h \Vert g\Vert _H, \end{aligned}$$

(25)

which implies (21).

The error estimate (20) is demonstrated by the standard duality argument. Consider the dual problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \lambda (\phi , \zeta )_H + a(\phi , \zeta ) + b(\phi , \eta ) = (\phi , w-w_h)_H, &{} \forall \phi \in V, \\ b(\zeta , \psi ) = 0, &{} \forall \psi \in Q. \end{array}\right. } \end{aligned}$$

(26)

The solution $(\zeta ,\eta ) \in V \times Q$ has the estimate

$$\begin{aligned} |\lambda | \Vert \zeta \Vert _H + |\lambda |^{1/2} \Vert \zeta \Vert _V + \Vert A\zeta \Vert _H + \Vert B\eta \Vert _H \le C \Vert v-v_h\Vert _H, \end{aligned}$$

(27)

from assumption (A-4). Substituting $\phi = w-w_h$ into (26) and recalling Eqs. (18) and (19), we have

$$\begin{aligned} \Vert w-w_h\Vert _H^2 \le Ch \Vert g\Vert _H \times \left( |\lambda |^{1/2} \Vert \zeta -\zeta _h\Vert _H + \Vert \zeta -\zeta _h\Vert _V + \Vert \eta -\eta _h\Vert _Q \right) \end{aligned}$$

for any $\zeta _h \in V_h$ and $\eta _h \in Q_h$. Hence, together with (A-2), (A-3), and (27), we obtain (20). $\square$

3 Abstract results

This section is devoted to the error estimates for the abstract Stokes problems.

Theorem 3.1

Let $u_0 \in H_\sigma$ and let (u, p) and $(u_h,p_h)$ be the solutions of (9) and (14), respectively. Assume that (A-1)–(A-4) hold. Then, we have the following error estimates:

$$\begin{aligned} \Vert u(t) - u_h(t) \Vert _V\le & {} C h t^{-1} \Vert u_0\Vert _H, \end{aligned}$$

(28)

$$\begin{aligned} \Vert u(t) - u_h(t) \Vert _H\le & {} C h^2 t^{-1} \Vert u_0\Vert _H, \end{aligned}$$

(29)

$$\begin{aligned} \Vert u_t(t) - u_{h,t}(t) \Vert _{V'}\le & {} C h t^{-1} \Vert u_0\Vert _H, \end{aligned}$$

(30)

$$\begin{aligned} \Vert p(t) - p_h(t) \Vert _Q\le & {} C h t^{-1} \Vert u_0\Vert _H, \end{aligned}$$

(31)

for all $t \in (0,T)$, where each constant C depends only on the constants appearing in assumptions (A-1)–(A-4), but is independent of h, $u_0$, t, and T. $\square$

Remark 2

The error estimates for the velocity (29) and (30) are given in [25, Theorems 3.1 and 4.1]. Although there is an error estimate for the pressure in [25], our result, (28), is strictly sharper. Indeed, it was shown that the estimate

$$\begin{aligned} \Vert p(t) - p_h(t) \Vert _Q \le C h \left( t^{-1} + t^{-3/2} \right) \Vert u_0\Vert _H \end{aligned}$$

holds in [25, Theorem 5.1]. $\square$

The estimates (29) and (30) are the consequence of the resolvent estimates (21) and (20), and the estimate for the pressure (28) is obtained from the discrete inf–sup condition (11). To show (31), we need another resolvent estimate.

Lemma 3.1

Let $\delta \in (0,\pi /2)$. Suppose that assumptions (A-1)–(A-4) hold. Then, we have

$$\begin{aligned} \left\| \left[ (\lambda + A)^{-1}P_\sigma - (\lambda + A_h)^{-1} P_{h,\sigma } \right] g \right\| _{V'} \le C h |\lambda |^{-1} \Vert g\Vert _H, \end{aligned}$$

for any $g \in H$ and $\lambda \in \Sigma _\delta$, where C is independent of h, g, and $\lambda$. $\square$

Proof

Fix $g \in H$, $\delta \in (0,\pi /2)$, and $\lambda \in \Sigma _\delta$ arbitrarily, and let $(w,\pi )$ and $(w_h,\pi _h)$ be the solutions of (18) and (19), respectively. It is sufficient to show that

$$\begin{aligned} (w-w_h,v)_H \le C h |\lambda |^{-1} \Vert g\Vert _H \Vert v\Vert _V \end{aligned}$$

(32)

holds for arbitrary $v \in V$, since $\Vert F \Vert _{V'} = \sup _{v \in V} (F,v)_H / \Vert v\Vert _V$ for $F \in H \hookrightarrow V'$. Fix $v \in V$ and choose $v_h \in V_h$ as in (A-2). Then,

$$\begin{aligned} (w-w_h,v)_H = (w-w_h,v-v_h)_H + (w-w_h,v_h)_H =: I_1 + I_2. \end{aligned}$$

Since a standard energy method yields $\Vert w\Vert _H+\Vert w_h\Vert _H \le C|\lambda |^{-1}\Vert g\Vert _H$, we have

$$\begin{aligned} |I_1| \le C |\lambda |^{-1}\Vert g\Vert _H \cdot h \Vert v\Vert _V \end{aligned}$$

from assumption (A-2). Moreover, Eqs. (18) and (19) imply

$$\begin{aligned} \lambda I_2 = -a(w-w_h,v_h) - b(v_h,\pi -\pi _h). \end{aligned}$$

Together with (25), we have

$$\begin{aligned} |I_2| \le Ch|\lambda |^{-1} \Vert g\Vert _H, \end{aligned}$$

which gives (32). Hence we can complete the proof. $\square$

Proof of Theorem 3.1

(1) Proof of (29) and (30). The derivation of these estimates was originally presented in [25]. Indeed, one can check (29) and (30) directly from Eq. (15) and Theorem 2.1.

(2) Proof of (31). From (15) and Lemma 3.1, we have

$$\begin{aligned} \Vert u_t(t) - u_{h,t}(t) \Vert _{V'} \le \int _\varGamma |\lambda e^{t\lambda }| \cdot Ch |\lambda |^{-1} \Vert u_0\Vert _H |d\lambda | \le C h t^{-1} \Vert u_0\Vert _H, \end{aligned}$$

where $\varGamma = \partial \Sigma _\delta$ for an arbitrary $\delta \in (0,\pi /2)$.

(3) Proof of (28). Fix $q_h \in Q_h$ arbitrarily. Then, by the discrete inf–sup condition (11), we have

$$\begin{aligned} \beta _2 \Vert p_h(t) - q_h \Vert _Q \le \sup _{v_h \in V_h} \frac{b(v_h,p_h(t) - q_h)}{\Vert v_h\Vert _V} . \end{aligned}$$

The Eqs. (9) and (14) yield

$$\begin{aligned} b(v_h, p_h(t) - q_h) = b(v_h,p(t)-q_h) + (u_t(t)-u_{h,t}(t),v_h)_H + a(u(t)-u_h(t),v_h), \end{aligned}$$

which leads to

$$\begin{aligned} \Vert p_h(t) - q_h \Vert _Q \le C \left( \Vert p(t)-q_h\Vert _Q + ht^{-1}\Vert u_0\Vert _H \right) \Vert v_h\Vert _V \end{aligned}$$

from (29) and (31). Therefore, noting $p-p_h = (p-q_h) + (q_h-p_h)$, we have

$$\begin{aligned} \Vert p(t)-p_h\Vert _Q \le C \Vert p(t)-q_h\Vert _Q + Cht^{-1}\Vert u_0\Vert _H. \end{aligned}$$

Finally, choosing $q_h \in Q_h$ as in assumption (A-3), we obtain

$$\begin{aligned} \Vert p(t)-q_h\Vert _Q \le Ch\Vert Bp(t)\Vert _H \le C h t^{-1} \Vert u_0\Vert _H, \end{aligned}$$

since $Bp = -u_t - Au$ and $u(t) = e^{-tA}u_0$. This completes the proof. $\square$

Remark 3

Throughout this section, we have considered the homogeneous problem (9). We now consider the inhomogeneous problem

$$\begin{aligned} {\left\{ \begin{array}{ll} (u_t,v)_H + a(u,v) + b(v,p) = \langle f,v \rangle _{V,V'}, &{} \forall v \in V, \\ b(u,q) = 0, &{} \forall q \in Q, \end{array}\right. } \end{aligned}$$

with an external force $f :(0,T) \rightarrow V'$. If $f \in C^\theta ([0,T]; H)$ for some $\theta \in (0,1]$, then we have

$$\begin{aligned} \Vert u(t)-u_h(t) \Vert _H + h \Vert u(t)-u_h(t)\Vert _V \le Ch \left( t^{-1} \Vert u_0 \Vert _H + t^\theta |f|_{C^\theta ([0,T];H)} + \Vert f(t)\Vert _H \right) \end{aligned}$$

by the same argument as in [10, §5]. However, we cannot extend this result to the $V'$-error estimate and the pressure estimate at present. $\square$

4 Applications

In this section, we apply Theorem 3.1 to the non-stationary Stokes and the hydrostatic Stokes problem. Hereafter, $L^2(\varOmega )$ and $H^s(\varOmega )$ denote the Lebesgue and Sobolev spaces, respectively.

4.1 Non-stationary Stokes equation

Let $\varOmega \subset \mathbb {R}^d$ ($d=2,3$) be a convex polygonal or polyhedral domain. We consider the non-stationary Stokes equations in $\varOmega$ with the homogeneous Dirichlet boundary condition:

$$\begin{aligned} {\left\{ \begin{array}{ll} (u_t(t), v) + (\nabla u(t), \nabla v) - ({{\mathrm{div}}}v, p(t)) = 0, &{} \forall v \in H^1_0(\varOmega )^d, \\ ({{\mathrm{div}}}u(t), q) = 0, &{} \forall q \in L^2_0(\varOmega ), \\ u(0) = u_0, \end{array}\right. } \end{aligned}$$

(33)

where $u_0 \in L^2_\sigma (\varOmega )$. Here, we use the usual notation

$$\begin{aligned} H^1_0(\varOmega )&= \{ v \in H^1(\varOmega ) \mid v|_{\partial \varOmega } = 0 \}, \\ L^2_0(\varOmega )&= \{ q \in L^2(\varOmega ) \mid \textstyle \int _\varOmega q = 0 \}, \\ H^1_{0,\sigma }(\varOmega )&= \{ v \in H^1_0(\varOmega )^d \mid {{\mathrm{div}}}v = 0 \}, \\ L^2_\sigma (\varOmega )&= \overline{H^1_{0,\sigma }(\varOmega )}^{\Vert \cdot \Vert _{L^2}}, \end{aligned}$$

and let $(\cdot , \cdot )$ denote the $L^2$-inner product over $\varOmega$. Let $H=L^2(\varOmega )^d$, $V=H^1_0(\varOmega )^d$, $Q=L^2_0(\varOmega )$, $a(u,v)=(\nabla u, \nabla v)$, and $b(v,q)=-({{\mathrm{div}}}v, q)$. Then, $H_\sigma = L^2_\sigma (\varOmega )$ and $V_\sigma = H^1_{0,\sigma }(\varOmega )$. We also define the operators A and B as in Sect. 2. The domain of A coincides with $H^2(\varOmega ) \cap V_\sigma$ and the estimate

$$\begin{aligned} \Vert v \Vert _{H^2} \le C \Vert Av\Vert _{L^2}, \quad \forall v \in D(A), \end{aligned}$$

(34)

holds since $\varOmega$ is a convex polygon or polyhedron (see [20] for the 2D case and [8] for the 3D case). Hence, assumption (A-4) holds.

Next, we consider the finite element approximation of (33). Let $\mathcal {T}_h$ be a conforming triangulation (or tetrahedralization) of $\varOmega$ with parameter $h=\max _{K \in \mathcal {T}_h} {\text {diam}} K$. We define the pair of approximation spaces $(V_h, Q_h)$ as the $P^1$-bubble-$P^1$ element (MINI element) or $P^2$-$P^1$ element (Taylor–Hood) with respect to $\mathcal {T}_h$. For the precise definition, see, e.g., [9]. We set $V_{h,\sigma }$ as in (13). If $\mathcal {T}_h$ is shape-regular and quasi-uniform, then we can check that assumptions (A-1)–(A-3) hold, together with (34) (see, e.g., [1, 9]).

The approximation scheme is as follows. Find $(u_h(t), p_h(t) \in V_h \times Q_h$ which satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} (u_{h,t}(t), v_h) + (\nabla u_h(t), \nabla v_h) - ({{\mathrm{div}}}v_h, p_h(t)) = 0, &{} \forall v_h \in V_h, \\ ({{\mathrm{div}}}u_h(t), q_h) = 0, &{} \forall q_h \in Q_h, \\ u_h(0) = P_{h,\sigma } u_0, \end{array}\right. } \end{aligned}$$

(35)

where $P_{h,\sigma }$ is the $L^2$-projection onto $V_{h,\sigma }$, as in (14). Then, since we have already checked that assumptions (A-1)–(A-4) hold, we can state the following error estimates.

Theorem 4.1

Let $\varOmega$ be a convex polygonal or polyhedral domain, $\mathcal {T}_h$ be a shape-regular and quasi-uniform triangulation of $\varOmega$, and $(V_h,Q_h)$ be the pair of finite elements mentioned above. Let (u, p) and $(u_h,p_h)$ be the solutions of (33) and (35), respectively, for the initial value $u_0 \in L^2_\sigma (\varOmega )$. Then, we have the following error estimates:

$$\begin{aligned} \Vert u(t) - u_h(t) \Vert _{H^1}&\le C h t^{-1} \Vert u_0\Vert _{L^2},\\ \Vert u(t) - u_h(t) \Vert _{L^2}&\le C h^2 t^{-1} \Vert u_0\Vert _{L^2}, \\ \Vert u_t(t) - u_{h,t}(t) \Vert _{H^{-1}}&\le C h t^{-1} \Vert u_0\Vert _{L^2}, \\ \Vert p(t) - p_h(t) \Vert _{L^2}&\le C h t^{-1} \Vert u_0\Vert _{L^2}, \end{aligned}$$

for all $t > 0$, where each constant C is independent of h, $u_0$, t, and T. $\square$

Remark 4

For the MINI element, the convergence rate is optimal. However, for the Taylor–Hood element, it is not. For optimal order error estimates with higher order elements, our argument requires higher regularity for the solution of the resolvent problem Eq. (18), which is impossible when $g \in H$ has no more regularity. We leave the investigation of the optimal order error estimates for higher order elements as an area for future work. $\square$

4.2 Non-stationary hydrostatic Stokes equation

The second example is the hydrostatic Stokes problem, which is a linearized form of the primitive equations. Let $G=(0,1) \subset \mathbb {R}^2$ and $\varOmega = G \times (-D,0) \subset \mathbb {R}^3$ with $D>0$. The unknown functions of the hydrostatic Stokes equations are the horizontal velocity $u :\varOmega \times (0,T) \rightarrow \mathbb {R}^2$ and the surface pressure $p :G \times (0,T) \rightarrow \mathbb {R}$. The equations are given by

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t - \Delta u + \nabla _H p = 0, &{} \text {in } \varOmega \times (0,T), \\ {{\mathrm{div}}}_H \bar{u} = 0, &{} \text {in } \varOmega \times (0,T), \\ u(\cdot , 0) = u_0, &{} \text {in } \varOmega , \end{array}\right. } \end{aligned}$$

(36)

with boundary conditions

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _z u = 0, &{} \text {on } \varGamma _u := G \times \{ 0 \}, \\ u = 0, &{} \text {on } \varGamma _b := G \times \{ -D \}, \\ \text {{ u} is periodic} &{} \text {on } \varGamma _l := \partial G \times (-D,0), \end{array}\right. } \end{aligned}$$

(37)

where $\nabla _H q= (\partial _x q, \partial _y q)^T$, ${{\mathrm{div}}}_H v = \partial _x v_1 + \partial _y v_2$, and $\bar{v}(x,y) = \int _{-D}^{0} v(x,y,z) dz$.

Let $H = L^2(\varOmega )^2$, $V = \{ v \in H^1(\varOmega )^2 \mid v|_{\varGamma _b} = 0 \text { and } v|_{\varGamma _l} \text { is periodic} \}$, $Q=L^2_0(G)$, $a(u,v) = (\nabla u, \nabla v)_\varOmega$ for $u,v \in V$, and $b(v,q) = -({{\mathrm{div}}}_H \bar{v}, q)_G$ for $(v,q) \in V \times Q$, where $(\cdot , \cdot )_X$ denotes the $L^2$-inner product over a domain X. Then, the weak formulation of the problem (36) and (37) is described by Eq. (9), i.e.,

$$\begin{aligned} {\left\{ \begin{array}{ll} (u_t(t), v)_\varOmega + (\nabla u, \nabla v)_\varOmega - ({{\mathrm{div}}}_H \bar{v}, p)_G = 0, &{} \forall v \in V, \\ ({{\mathrm{div}}}_H \bar{u}, q)_G = 0, &{} \forall q \in Q. \end{array}\right. } \end{aligned}$$

Thus, we can construct a finite element scheme for the hydrostatic Stokes equations. Let $\mathcal {T}_h$ be a tetrahedralization (a set of open tetrahedra) of $\varOmega$ with $h = \max _{K \in \mathcal {T}_h} {\text {diam}} K$, and let $\tilde{\mathcal {T}}_h$ be the triangulation of $\varGamma _u$ induced by $\mathcal {T}_h$. Namely,

$$\begin{aligned} \tilde{\mathcal {T}}_h = \{ T \subset \varGamma _u \mid \exists K \in \mathcal {T}_h \text { s.t. } \overline{T} = \varGamma _u \cap \partial K \}, \end{aligned}$$

where $\overline{T}$ and $\overline{G}$ are the closures in $\mathbb {R}^2$. We suppose that

• $V_h$ is the space of $P^2$-finite elements or $P^1$-bubble finite elements with respect to $\mathcal {T}_h$, and each $v_h \in V_h$ vanishes on $\varGamma _b$ and is periodic on $\varGamma _l$,

• $Q_h$ is the space of $P^1$-finite elements with respect to $\tilde{\mathcal {T}}_h$ and $\int _G q_h = 0$ for each $q_h \in Q_h$.

Then, we can introduce the finite element approximations for (36) and (37) as follows. Find $u_h :(0,T) \rightarrow V_h$ and $p_h :(0,T) \rightarrow Q_h$ which satisfy

$$\begin{aligned} {\left\{ \begin{array}{ll} (u_{h,t}(t), v_h)_\varOmega + (\nabla u_h, \nabla v_h)_\varOmega - ({{\mathrm{div}}}_H \bar{v}_h, p_h)_G = 0, &{} \forall v_h \in V_h, \\ ({{\mathrm{div}}}_H \bar{u}_h, q_h)_G = 0, &{} \forall q_h \in Q_h, \\ (u_h(0), v_h)_\varOmega = (u_0, v_h)_\varOmega , &{} \forall v_h \in V_{h,\sigma }, \end{array}\right. } \end{aligned}$$

(38)

where $V_{h,\sigma }$ is defined by (13).

In order to discuss the error estimates in the framework of Theorem 3.1, we should check the conditions (A-1)–(A-4). Let A be the operator defined by (10), which is called the hydrostatic Stokes operator in the present case. Then, it is known that $0 \in \rho (A)$, A generates a bounded analytic semigroup on $H_\sigma$, and A satisfies the regularity property

$$\begin{aligned} \Vert v \Vert _{H^2(\varOmega )} \le C \Vert Av\Vert _{L^2(\varOmega )}, \quad \forall v \in D(A). \end{aligned}$$

We refer to [19, Theorem 3.1] for the proof. Therefore, we can check that conditions (A-2)–(A-4) hold.

Finally, we confirm (A-1) holds by introducing a prismatic mesh.

Definition 1

We say that a tetrahedralization $\mathcal {T}_h$ of $\varOmega$ is prismatic if the following condition holds: for each $K \in \mathcal {T}_h$, there exists $T \in \tilde{\mathcal {T}}_h$ such that

$$\begin{aligned} K \subset P_T := \{ (x,y,z) \in \varOmega \mid (x,y,0) \in T \}, \end{aligned}$$

where $\tilde{\mathcal {T}}_h$ is the triangulation of $\varGamma _u$ induced by $\mathcal {T}_h$. $\square$

We can construct such a mesh by the following procedure.

• 1. Triangulate the surface $\varGamma _u$ and denote the triangulation by $\tilde{\mathcal {T}}_h$.

• 2. Construct a prism $P_T$ in $\varOmega$ for each $T \in \tilde{\mathcal {T}}_h$.

• 3. Decompose each prism $P_T$ into tetrahedra so that the set of tetrahedra becomes a conforming tetrahedralization of $\varOmega$.

In [7], it is proved that the pair $(V_h,Q_h)$ mentioned above satisfies the discrete inf–sup condition (11), provided that the mesh is prismatic. Indeed, if the mesh is prismatic, then we can extend a function $q_h \in Q_h$ naturally to a piecewise linear function over the mesh $\mathcal {T}_h$, and thus we can use the usual inf–sup condition for the MINI element or Taylor–Hood element. Hence, we can confirm (A-1) holds.

Therefore, we can apply Theorem 3.1 and obtain the following error estimates.

Theorem 4.2

Let $\varOmega$, $\mathcal {T}_h$, and $(V_h,Q_h)$ be as described above. Assume that the mesh $\mathcal {T}_h$ is shape-regular, quasi-uniform, and prismatic. Let (u, p) and $(u_h,p_h)$ be the solutions of (36) and (38), respectively, for the initial value $u_0 \in L^2(\varOmega )^2$ satisfying ${{\mathrm{div}}}_H \bar{u_0} = 0$ in the distributional sense. Then, we have the following error estimates:

$$\begin{aligned} \Vert u(t) - u_h(t) \Vert _{H^1(\varOmega )}&\le C h t^{-1} \Vert u_0\Vert _{L^2(\varOmega )},\\ \Vert u(t) - u_h(t) \Vert _{L^2(\varOmega )}&\le C h^2 t^{-1} \Vert u_0\Vert _{L^2(\varOmega )}, \\ \Vert u_t(t) - u_{h,t}(t) \Vert _{V'}&\le C h t^{-1} \Vert u_0\Vert _{L^2(\varOmega )}, \\ \Vert p(t) - p_h(t) \Vert _{L^2(G)}&\le C h t^{-1} \Vert u_0\Vert _{L^2(\varOmega )}, \end{aligned}$$

for all $t > 0$, where each constant C is independent of h, $u_0$, t, and T, and $V'$ is the dual space of $V = \{ v \in H^1(\varOmega )^2 \mid v|_{\varGamma _b} = 0 \text { and } v|_{\varGamma _l} \text { is periodic} \}$. $\square$

5 Concluding remarks

In the present paper, we considered the abstract non-stationary saddle-point problem (9) and its finite element approximation (14). Our main contribution (Theorem 3.1) is the derivation of error estimates for the velocity and the pressure in various norms. In particular, the error estimate for the pressure with the optimal singularity (i.e., the term $t^{-1}$) is a new result. We then applied this result to establish error estimates for the finite element approximation for the non-stationary Stokes and the hydrostatic Stokes equations. However, as mentioned in Remark 3, we have not obtained the error estimates for the pressure for inhomogeneous problems. Moreover, the convergence rate is not optimal for finite elements of higher degree (Remark 4). Furthermore, we should consider a numerical analysis for the primitive equations (3) in the framework of analytic semigroup theory, as performed in [24] for the two-dimensional Navier–Stokes equations. These problems remain an area for future work.