Successive finite element methods for Stokes equations

High order finite element methods for incompressible Stokes problems have been developed well in 2-dimensional domain and analyzed along to the inf-sup condition [5, 7, 10]. They, however, endure their large numbers of degrees of freedom. To be worse in case of Scott-Vogelius, instability appears on pressure, if the mesh bears singular vertices.

Recently, we found a so called sting function having an interesting quadrature rule. In the Scott-Vogelius finite element method, it causes the discrete Stokes problem to be singular on the presence of exactly singular vertices. Even on nearly singular vertices, the pressure solution is easy to be spoiled. To overcome the problem based on understanding the causes, we did a new error analysis in a successive way and restored the ruined pressure by simple post-process driven from it [9].

In this paper, we will suggest a new finite element method to calculate a pressure solution at low cost, utilizing the successive way in the precedented new error analysis. In our method, the characteristics of a sting function depicted in Figure 1 will also play a key role.

We will solve first the decoupled equation for velocity in the $P^{4}$ divergence-free space inherited from the $\mathcal{C}^{1}$-Argyris $P^{5}$ stream function space. It is simpler and smaller than the divergence-free subspace of the $P^{4}$-$P^{3}$ Scott-Vogelius finite element space.

The main stage of our method is the successive 5 steps calculating four kinds of local $P^{3}$-pressures for triangles, regular vertices, nearly singular vertices and singular corners, respectively, as well as one $P^{0}$-pressure in the last step. They are successive in the sense that each step needs the calculated in the previous step.

Superposition of all the calculated reaches at the final $P^{3}$-pressure which shows the optimal order of convergence same as a $P^{3}$-projection of the continuous pressure. The chief time cost of the new method is on solving two linear systems for the $P^{4}$-velocity and $P^{0}$-pressure, respectively.

The paper is organized as follows. In the next section, the detail on finding a $P^{4}$-velocity will be offered. After short review of nearly singular vertices in Section 3, we will introduce a new basis for $P^{3}$-pressures consisting of non-sting, sting and constant functions and decompose the space of $P^{3}$-pressures in Section 4. Then, we will devote Section 5 to describing the successive 5 steps to find a $P^{3}$-pressure solution. Finally, a numerical test will be presented in the last section.

Throughout the paper, for a set $S\subset\mathbb{R}^{2}$, standard notations for Sobolev spaces are employed and $L_{0}^{2}(S)$ is the space of all $f\in L^{2}(S)$ whose integrals over $S$ vanish. We will use $\|\star\|_{m,S}$, $|\star|_{m,S}$ and $(\cdot,\cdot)_{S}$ for the norm, seminorm for $H^{m}(S)$ and $L^{2}(S)$ inner product, respectively. If $S=\Omega$, it may be omitted in the indices. Denoting by $P^{k}$, the space of all polynomials on $\mathbb{R}^{2}$ of order less than or equal $k$, $f\big{|}_{S}\in P^{k}$ will mean that $f$ coincides with a polynomial in $P^{k}$ on $S$.

Let $\Omega$ be a simply connected polygonal domain in $\mathbb{R}^{2}$ and $\{\mathcal{T}_{h}\}_{h>0}$ a regular family of triangulations of $\Omega$. Denote by $\mathcal{P}_{h}^{k}(\Omega)$, discrete polynomial spaces:

In this paper, we will approximate a pair of velocity and pressure $(\mathbf{u},p)\in[H_{0}^{1}(\Omega)]^{2}\times L_{0}^{2}(\Omega)$ which satisfies an incompressible Stokes problem:

for a given body force $\mathbf{f}\in[L^{2}(\Omega)]^{2}$.

Let $A_{h}^{5}$ be a space of $\mathcal{C}^{1}$-Argyris triangle elements [2, 4] such that

where

Defining a divergence-free space $V_{h}$ as

we can solve $\mathbf{u}_{h}\in V_{h}$ satisfying

A vertex $\mathbf{V}$ is called exactly singular if the union of all edges sharing $\mathbf{V}$ belongs to the union of two lines. To be precise, let $K_{1},K_{2},\cdots,K_{J}$ be all $J$ triangles sharing $\mathbf{V}$ and denote by $\theta(K_{k})$, the angle of $K_{k}$ at $\mathbf{V}$, $k=1,2,\cdots,J$. Define

Then $\Upsilon(\mathbf{V})=\{\pi\}\mbox{ or }\emptyset$ if and only if $\mathbf{V}$ is exactly singular.

Since $\{\mathcal{T}_{h}\}_{h>0}$ is regular, there exists $\vartheta>0$ such that

which depends on the shape regularity parameter $\sigma$ of $\{\mathcal{T}_{h}\}_{h>0}$. Set

then call a vertex $\mathbf{V}$ to be nearly singular if $\Upsilon(\mathbf{V})=\emptyset$ or

otherwise regular. From the following lemma [9], we note that each interior nearly singular vertex is isolated from others.

For each triangle $K\in\mathcal{T}_{h}$, define

In the remaining of the paper, we will use the following notations:

• $C_{\sigma}$ : a generic constant which depends only the shape regularity parameter $\sigma$ of $\{\mathcal{T}_{h}\}_{h>0}$,

• $\mathcal{K}(\mathbf{V})$ : the union of all triangles in $\mathcal{T}_{h}$ sharing a vertex $\mathbf{V}$ as in Figure 3-(b),

• $\mathcal{V}_{h}$ : the set of all vertices in $\mathcal{T}_{h}$,

• $|K|,|E|$: the area and length of a triangle $K$ and an edge $E$, respectively.

In this section, we will find the following discrete pressures:

in a successive way consisting of the following 5 steps.

Step 1.

Calculate a non-sting pressure $p_{h}^{K}\in\mathcal{N}_{h}(K)$ using $\mathbf{u}_{h},\mathbf{f}$ for each triangle $K\in\mathcal{T}_{h}$ and superpose them to form

$$p_{h}^{\mathcal{N}}=\sum p_{h}^{K}.$$

Step 2.

Calculate a sting pressure $p_{h}^{\mathbf{V}_{r}}\in\mathcal{S}_{h}(\mathbf{V}_{r})$ using $p_{h}^{\mathcal{N}},\mathbf{u}_{h},\mathbf{f}$ for each regular vertex $\mathbf{V}_{r}$ and superpose them to form

$$\dot{p_{h}}=p_{h}^{\mathcal{N}}+\sum p_{h}^{\mathbf{V}_{r}}.$$

Step 3.

Calculate a sting pressure $p_{h}^{\mathbf{V}_{s}}\in\mathcal{S}_{h}(\mathbf{V}_{s})$ using $\dot{p_{h}},\mathbf{u}_{h},\mathbf{f}$ for each nearly singular vertex $\mathbf{V}_{s}$ except corners and superpose them to form

$$\ddot{p_{h}}=\dot{p_{h}}+\sum p_{h}^{\mathbf{V}_{s}}.$$

Step 4.

Calculate a sting pressure $p_{h}^{\mathbf{V}_{c}}\in\mathcal{S}_{h}(\mathbf{V}_{c})$ using $\ddot{p_{h}},\mathbf{u}_{h},\mathbf{f}$ for each nearly singular corner $\mathbf{V}_{c}$ and superpose them to form

$$\dddot{p_{h}}=\ddot{p_{h}}+\sum p_{h}^{\mathbf{V}_{c}}.$$

Step 5.

Calculate a piecewise constant pressure $p_{h}^{\mathpzc{C}}\in\mathcal{P}_{h}^{0}(\Omega)$ using $\dddot{p_{h}},\mathbf{u}_{h},\mathbf{f}$.

The detail of Step $i$ above will be given in Section 5.$i$ below, $i=1,2,3,4,5$.

If we define $p_{h}\in\mathcal{P}_{h}^{3}(\Omega)$ as

it shows the optimal order of convergence as in the following theorem.