A High Order HDG Method for Stokes Flow in Curved Domains

Abstract

We propose and analyze a high order hybridizable discontinuous Galerkin (HDG) method for the Stokes equations in a curved domain. It is based on approximating the domain by a polyhedral computational subdomain where an HDG solution is computed. To obtain a high order approximation of the Dirichlet boundary data in the computational domain, we employ a transferring technique based on integrating the approximation of the gradient. In addition, we first seek for a discrete pressure having zero-mean in the computational domain and then the zero-mean condition in the entire domain is recovered by a post-process that involves an extrapolation of the discrete pressure. We prove that the method provides optimal order of convergence for the approximations of the pressure, the velocity and its gradient, that is, order $h^{k+1}$ if the local discrete spaces are constructed using polynomials of degree k and the meshsize is h. We present numerical experiments validating the method.

Appendices

Appendix

HDG Implementation

The HDG method (6) can be implemented similarly to the standard HDG method for polygonal domains [33]. The boundary data $\widetilde{{\varvec{g}}}_h$ defined in (3) produces a slight modification in the global system, as we will see. We first construct the local matrices associated to an element $K\in {\mathsf {T}}_h$ and the boundary data will be imposed after the global matrix is assembled. To fix ideas we briefly describe the two dimensional case. Let $K\in {\mathsf {T}}_h$ with edges $e_1$, $e_2$ and $e_3$, $N_{\mathrm {L}}=\dim (\text {P}_k(K))$, $N_{{\varvec{u}}}=\dim (\varvec{{\mathcal {P}}}_k(K))$, $N_p=\dim ({\mathcal {P}}_k(K))$ and $N_{\widehat{{\varvec{u}}}}=\dim (\varvec{{\mathcal {P}}}_k(e_1))=$$\dim (\varvec{{\mathcal {P}}}_k(e_2))=$$\dim (\varvec{{\mathcal {P}}}_k(e_3))$. Let $\{\varPsi _j\}_{j=1}^{N_\mathrm {L}}$, $\{\varvec{\psi }_j\}_{j=1}^{N_{{\varvec{u}}}}$, $\{\psi _j\}_{j=1}^{N_p}$ and $\{\varvec{\xi }^{e_l}_j\}_{j=1}^{N_{\widehat{{\varvec{u}}}}}$ ($l=1$, 2 and 3) the basis function of $\text {P}_k(K)$, $\varvec{{\mathcal {P}}}_k(K)$, ${\mathcal {P}}_k(K)$ and ${\mathcal {P}}_k(e_l)$, respectively. In our numerical experiments we considered Dubiner basis [28] for the polynomial spaces in K and Legendre basis for the polynomial space on the edges. We write now the unknowns restricted to K as linear combination of the basis functions, i.e., $\mathrm {L}_h\vert _K = \sum _{j=1}^{N_{\mathrm {L}}} \alpha _j^K \varPsi _j$, ${\varvec{u}}_h\vert _K = \sum _{j=1}^{N_{{\varvec{u}}}} \beta _j^K \varvec{\psi }_j$, ${\widetilde{p}}_h\vert _K= \sum _{j=1}^{N_p} \gamma _j^K \psi _j$ and $\widehat{{\varvec{u}}}_h\vert _{e_l}= \sum _{j=1}^{N_{\widehat{{\varvec{u}}}}} \lambda _j^{e_l} \varvec{\xi }^{e_l}_j$. We decompose ${\widetilde{p}}_h\vert _K = \overline{p_h}^{K}+\rho _h$, where $\overline{p_h}^{K} \in {\mathcal {P}}_0(K)$ and $\rho _h\in {\mathcal {P}}_k(K)\cap L^2_0(K)$. Since Dubiner basis are orthogonal on K and $\psi _1$ is a constant, then $\rho _h = \sum _{j=2}^{N_p} \gamma _j^K \psi _j$. Thus, the system associated to (6a)–(6c) is of the form

$$\begin{aligned} {\mathbf {A}}^K \begin{bmatrix} \varvec{\alpha }^K\\ \varvec{\beta }^K\\ \varvec{\gamma }^K \end{bmatrix} ={\mathbf {E}}^K \begin{bmatrix} \varvec{\lambda }^{e_1}\\ \varvec{\lambda }^{e_2}\\ \varvec{\lambda }^{e_3} \end{bmatrix} + {\mathbf {F}}^K, \quad \text {where}\, {\mathbf {A}}^K\in {\mathbb {R}}^{N_T\times N_T}, {\mathbf {E}}^K\in {\mathbb {R}}^{N_T \times 3N_{\widehat{{\varvec{u}}}}}, \end{aligned}$$

(39)

${\mathbf {F}}^K\in {\mathbb {R}}^{N_T}$ and $N_T:=N_{\mathrm {L}}+N_{{\varvec{u}}}+N_p$. On the other hand, the local contribution of the global equation (6g) can be written in the following matrix form

$$\begin{aligned} {\mathbf {B}}^{K} \begin{bmatrix} \varvec{\alpha }^K\\ \varvec{\beta }^K\\\ \varvec{\gamma }^K \end{bmatrix} -{\mathbf {H}}^{K}\overline{p_h}^K = \varvec{\Lambda }^{K} \begin{bmatrix} \varvec{\lambda }^{e_1}\\ \varvec{\lambda }^{e_2}\\ \varvec{\lambda }^{e_3} \end{bmatrix}, \, \text {where}\,{\mathbf {B}}^{K}\in {\mathbb {R}}^{3N_{\widehat{{\varvec{u}}}}\times N_T}, {\mathbf {H}}^{K}\in {\mathbb {R}}^{3N_{\widehat{{\varvec{u}}}}} \end{aligned}$$

(40)

and $\varvec{\Lambda }^{K}\in {\mathbb {R}}^{3N_{\widehat{{\varvec{u}}}}\times 3N_{\widehat{{\varvec{u}}}}}$. Since the matrix on the left hand side of (39) is invertible, we express the unknowns $\varvec{\alpha }^K$, $\varvec{\beta }^K$ and $\varvec{\gamma }^K$ in terms of $\varvec{\lambda }^{e_l}$ and substitute them in (40) to obtain a local system of the form

$$\begin{aligned} {\mathbf {M}}^{K} \begin{bmatrix} \varvec{\lambda }^{e_1}\\ \varvec{\lambda }^{e_2}\\ \varvec{\lambda }^{e_3} \end{bmatrix} + {\mathbf {H}}^K\overline{p_h}^K ={\mathbf {b}}^{K}, \quad \hbox {where }{\mathbf {M}}^K\in {\mathbb {R}}^{N_{\widehat{{\varvec{u}}}} \times N_{\widehat{{\varvec{u}}}}}\hbox { and }{\mathbf {b}}^K\in {\mathbb {R}}^{N_{\widehat{{\varvec{u}}}}}. \end{aligned}$$

(41)

Now, let $N_e$ and $N_K$ be the number of edges and elements of the triangulation. We denote by $\varvec{\lambda }^l$ ($l=1$, ..., $N_e$) the global unknowns associated to the lth edge. We define the global vectors $\varvec{\lambda }=[(\varvec{\lambda }^1)^t, \ldots , (\varvec{\lambda }^{N_e})^t]^t$ and $\varvec{\sigma }=[{\overline{p}}_h^{K_1},\ldots ,{\overline{p}}_h^{K_{N_K}}]^t$. Thus, once the above local matrices are computed, we assembly on the edges of the triangulation and obtain the global system of the form

$$\begin{aligned} \begin{bmatrix} {\mathbf {M}}&{\mathbf {H}} \end{bmatrix} \begin{bmatrix} \varvec{\lambda }\\ \varvec{\sigma } \end{bmatrix}= {\mathbf {b}}, \quad \text {where} \quad {\mathbf {M}}\in {\mathbb {R}}^{(N_eN_{\widehat{{\varvec{u}}}})\times (N_eN_{\widehat{{\varvec{u}}}})}, {\mathbf {H}}\in {\mathbb {R}}^{(N_eN_{\widehat{{\varvec{u}}}})\times N_K} \end{aligned}$$

(42)

and ${\mathbf {b}}\in {\mathbb {R}}^{(N_eN_{\widehat{{\varvec{u}}}})}$. In addition, equation (6f) can be written as $[|K_1| \quad \ldots \quad |K_{N_K}|] \varvec{\sigma }=0$, which, together with (42) and the boundary condition, fully determines $\varvec{\lambda }$ and $\varvec{\sigma }$.

We proceed now to impose the boundary condition. Let $e\in {\mathcal {E}}_h^\partial$ and $K^e$ the triangle where it belongs. We assume that the only boundary edge of K is e and denote by $e_1$ and $e_2$ the other two edges. If K had two boundary edges the procedure is analogous. From (6c) and (38) we have that

$$\begin{aligned} \varvec{\Lambda }^e \varvec{\lambda }^e = {\mathbf {b}}^{{\varvec{g}}} - {\mathbf {J}}^e \varvec{\alpha }^{K^e}, \end{aligned}$$

where

$$\begin{aligned} \displaystyle {\mathbf {b}}^{{\varvec{g}}}:=\bigg \{\int _e \int _0^{l({\varvec{x}})}{\varvec{g}}(\bar{{\varvec{x}}}({\varvec{x}}))\cdot \varvec{\xi }^e_i({\varvec{x}})\,ds\,d{\varvec{x}}\bigg \}_{i=1}^{N_{\widehat{{\varvec{u}}}}} \end{aligned}$$

and

$$\begin{aligned} \displaystyle {\mathbf {J}}^e:=\bigg \{\int _e \int _0^{l({\varvec{x}})}\varvec{\varPsi }_j({\varvec{x}}+s{\varvec{t}}({\varvec{x}})){\varvec{t}}({\varvec{x}})\cdot \varvec{\xi }^e_i({\varvec{x}})\,ds\,d{\varvec{x}}\bigg \}_{i=1}^{N_{\widehat{{\varvec{u}}}}}. \end{aligned}$$

Here, we see how the basis functions $\varvec{\varPsi }_j$ (defined in K) are being evaluated (extrapolated) along the path ${\varvec{x}}+s{\varvec{t}}({\varvec{x}}).$ From (39) we write $\varvec{\alpha }^{K}$ in terms of $\varvec{\lambda }^{e_1}$, $\varvec{\lambda }^{e_2}$ and $\varvec{\lambda }^{e}$, say $\varvec{\alpha }^{K} = {\mathbf {K}} \begin{bmatrix} \varvec{\lambda }^{e_1}\\ \varvec{\lambda }^{e_2}\\ \varvec{\lambda }^e \end{bmatrix} +{\mathbf {r}}$, and obtain the system

$$\begin{aligned} \begin{bmatrix} \begin{bmatrix} \varvec{\Theta }_{N_{\widehat{{\varvec{u}}}}\times 3N_{\widehat{{\varvec{u}}}}}&\varvec{\Theta }_{N_{\widehat{{\varvec{u}}}}\times 3N_{\widehat{{\varvec{u}}}}}&\varvec{\Lambda }^e \end{bmatrix} + {\mathbf {J}}^e {\mathbf {K}} \end{bmatrix} \begin{bmatrix} \varvec{\lambda }^{e_1}\\ \varvec{\lambda }^{e_2}\\ \varvec{\lambda }^e \end{bmatrix} ={\mathbf {b}}^{{\varvec{g}}}-{\mathbf {J}}^e{\mathbf {r}}. \end{aligned}$$

(43)

Finally, we substitute the rows in (42) associated to the global degrees of freedom of the edge e by the equations in (43). This resulting global system is not symmetric (unless ${\mathbf {J}}^e=\varvec{\Theta }$). If $\varOmega$ were polygonal, then ${\mathbf {J}}^e=\varvec{\Theta }$ and the (43) becomes $\varvec{\Lambda }^e \varvec{\lambda }^e={\mathbf {b}}^{{\varvec{g}}}$ which is one of the ways to impose Dirichlet boundary conditions.

Integration Over Curved Regions

In order to compute the approximation $\overline{p_h}^{{\mathsf {D}}_h}$ and errors in ${\mathsf {D}}_h^c$, integrals over ${\widetilde{K}}_{ext}^e$ must be calculated. We write ${\widetilde{K}}_{ext}^e$ as a union of three disjoint regions $T_1$, $T_2$ and S as Fig. 8 shows. The same quadrature rules considered in the calculation of the local matrices of the method are used now in the triangles $T_1$ and $T_2$. Now, let $e_S$ and $\varGamma _S$ denote the straight and curved part of the boundary of S, resp. For a point ${\varvec{x}}$ in $\varGamma _h$, we construct $\sigma _{{\varvec{n}}_S}({\varvec{x}})$, where ${\varvec{n}}_S$ is the unit normal of $e_S$ pointing to $\varGamma _S$. Thus, for a smooth enough function q defined on S, we approximate

$$\begin{aligned} \int _S q({\varvec{x}}) d{\varvec{x}}= \int _{e_S} \int _0^{l({\varvec{x}})} q({\varvec{x}}+t{\varvec{n}}_S) {\varvec{n}}_S \,dt \,d{\varvec{x}} \end{aligned}$$

using one-dimensional quadrature on both integrals.

Fig. 8

Examples of ${\widetilde{K}}_{ext}^e$ divided in triangles $T_1$ and $T_2$, and a region S (dashed area)