Performance of nonconforming spectral element method for Stokes problems

Abstract

A nonconforming spectral element method for the Stokes problem on nonsmooth domains has been proposed in Mohapatra et al. (J Comput Appl Math 372:112696, 2020). The main focus of this article is to study the performance of this method for Stokes problems on smooth curvilinear domains and Stokes problem with mixed boundary conditions. Various test cases are considered including the generalized Stokes problem in ${\mathbb {R}}^{2}$ and ${\mathbb {R}}^{3}$ to verify the exponential accuracy of the method.

Appendix

Here, we state the regularity estimate for the generalized Stokes problem (1)–(3) and define the jumps in ${\mathbf {u}},p$ and ${\mathbf {u}}_{x_{k}}$ in different Sobolev norms. Finally we state the stability estimate.

1.1 A1. Regularity estimate

The following fundamental regularity estimate is based on ADN (Agmin–Douglis–Nirenberg) theory (Agmon et al. 1964).

Let $\Omega$ be an open bounded subset of class $C^{r},r=\text {max}(m+2,2)$. For ${\mathbf {u}}\in {\mathbf {W}}^{1,2}(\Omega )$, $p\in L^{2}(\Omega )$ being solutions of the generalized Stokes equations (1)–(3) and for ${\mathbf {f}}\in {\mathbf {W}}^{m,2}(\Omega )$, $h\in W^{m+1,2}(\Omega )$ and ${\mathbf {g}}\in {\mathbf {W}}^{m+\frac{3}{2},2}(\Gamma )$, then ${\mathbf {u}}\in {\mathbf {W}}^{m+2,2}(\Omega )$, $p\in W^{m+1,2}(\Omega )$ and there exists a constant $C_{0}(\alpha ,m,\Omega )$ such that

$$\begin{aligned} ||u||_{{\mathbf {W}}^{m+2,2}(\Omega )}+||p||_{W^{m+1,2}(\Omega )/{\mathbb {R}}}\le C_{0}\left( ||{\mathbf {f}}||_{{\mathbf {W}}^{m,2}(\Omega )}+||h||_{W^{m+1,2}(\Omega )}+||{\mathbf {g}}||_{{\mathbf {W}}^{m+\frac{3}{2},2}(\Gamma )}\right) . \end{aligned}$$

(8)

1.2 A2. Stability estimate

Since the approximation is nonconforming, to enforce the continuity along the inter element boundaries we introduce the jumps in ${\mathbf {u}},{\mathbf {u}}_{x_{1}},{\mathbf {u}}_{x_{2}}$ and p in suitable Sobolev norms. Let the edge $\gamma _{s}$ be common to the adjacent elements $\Omega _{l}\,\,\text {and}\,\,\Omega _{m}.$ Assume that edge $\gamma _{s}$ is the image of $\eta =1$ under the map $M_{l}$ which maps S to $\Omega _{l}$ and also the image of $\eta =-1$ under the map $M_{m}$ which maps S to $\Omega _{m}$. Then the jumps along the inter-element boundaries are defined as

$$\begin{aligned}&\left\| [{\mathbf {u}}]\right\| _{0,\gamma _{s}}^{2}=\left\| \hat{{\mathbf {u}}}_{m}(\xi ,-1)-\hat{{\mathbf {u}}}_{l}(\xi ,1)\right\| _{0,I}^{2},\\&\left\| [{\mathbf {u}}_{x_{k}}]\right\| _{\frac{1}{2},\gamma _{s}}^{2}=\Vert ({\hat{{\mathbf {u}}}_{m})}_{x_{k}}(\xi ,-1)-{(\hat{{\mathbf {u}}}_{l})}_{x_{k}}(\xi ,1)\Vert _{\frac{1}{2},I}^{2},\\&\left\| [p]\right\| _{\frac{1}{2},\gamma _{s}}^{2}=\left\| {\hat{p}}_{m}(\xi ,-1)-{\hat{p}}_{l}(\xi ,1)\right\| _{\frac{1}{2},I}^{2}. \end{aligned}$$

Here, $I=(-1,1).$ The expressions on the right hand side in the above equation are given in the transformed coordinates $\xi$ and $\eta .$

Let us consider the boundary condition. Let $\gamma _{s}\subseteq \partial \Omega \cap \Omega _{l}$ be the image of $\xi =1$ under the map** $M_{l}$ which maps S to $\Omega _{l}.$ Then

$$\begin{aligned} ||{\mathbf {u}}_{l}||_{\frac{3}{2},\gamma _{s}}^{2}=||\hat{{\mathbf {u}}}_{l}(1,\eta )||_{\frac{3}{2},I}^{2}. \end{aligned}$$

Let ${\mathbf {u}},p\in \Pi ^{L,W}$. We now define the quadratic form

$$\begin{aligned} {\mathcal {V}}^{L,W}({\mathbf {u}},p)&=\sum _{l=1}^{L}||{\mathcal {L}}({\mathbf {u}}_{l},p_{l})||_{0,\Omega _{l}}^{2}+\sum _{l=1}^{L}||{\mathcal {D}}{\mathbf {u}}_{l}||_{1,\Omega _{l}}^{2}\nonumber \\&\quad +\sum _{\gamma _{s}\subseteq {\bar{\Omega }}\setminus \partial \Omega }\left( ||[{\mathbf {u}}]||_{0,\gamma _{s}}^{2}+\sum _{k=1}^{2}||[{\mathbf {u}}_{x_{k}}]||_{\frac{1}{2},\gamma _{s}}^{2}+||[p]||_{\frac{1}{2},\gamma _{s}}^{2}\right) +\sum _{\gamma _{s}\subseteq \partial \Omega \cap \Omega _{l}}||{\mathbf {u}}_{l}||_{\frac{3}{2},\gamma _{s}}^{2}. \end{aligned}$$

(9)

Then, we have the following result.

Theorem

For W large enough, there exists a constant $C>0$ such that the estimate

$$\begin{aligned} {\mathcal {U}}^{L,W}({\mathbf {u}},p)\le C(\ln W)^{2}{\mathcal {V}}^{L,W}({\mathbf {u}},p) \end{aligned}$$

(10)

holds. Where ${\mathcal {U}}^{L,W}({\mathbf {u}},p)$ is defined in Sect. 2 (see (7)).

The proof of this one is very similar to Theorem 4.1 in (Mohapatra et al. 2020).