A High Order Mixed-FEM for Diffusion Problems on Curved Domains

Abstract

We propose and analyze a high order mixed finite element method for diffusion problems with Dirichlet boundary condition on a domain $\Omega$ with curved boundary $\Gamma$. The method is based on approximating $\Omega$ by a polygonal subdomain $\mathrm {D}_{h}$, with boundary $\Gamma _h$, where a high order conforming Galerkin method is considered to compute the solution. To approximate the Dirichlet data on the computational boundary $\Gamma _h$, we employ a transferring technique based on integrating the extrapolated discrete gradient along segments joining $\Gamma _h$ and $\Gamma$. Considering general finite dimensional subspaces we prove that the resulting Galerkin scheme, which is ${\mathbf {H}}(\mathrm {div}\,; \mathrm {D}_{h})$-conforming, is well-posed provided suitable hypotheses on the aforementioned subspaces and integration segments. A feasible choice of discrete spaces is given by Raviart–Thomas elements of order $k\ge 0$ for the vectorial variable and discontinuous polynomials of degree k for the scalar variable, yielding optimal convergence if the distance between $\Gamma _h$ and $\Gamma$ is at most of the order of the meshsize h. We also approximate the solution in $\mathrm {D}_{h}^{c}\,{:}{=}\,\Omega \backslash \overline{\mathrm {D}_{h}}$ and derive the corresponding error estimates. Numerical experiments illustrate the performance of the scheme and validate the theory.

Appendices

Appendix

In this section we use the equivalence of the the norms ${\vert \vert \vert \cdot \vert \vert \vert }_e$ and $\Vert \cdot \Vert _{0,{\widetilde{K}}_{ext}^{e}}$ (cf. Lemma 3.4) to provide an estimate of ${\widetilde{C}}_{ext}^{e}$ defined in (2.14).

Appendix A: Estimates of ${\widetilde{C}}_{ext}^{e}$

The following result extends the estimation in [15, Lemma A.1] to the case when the norm $\Vert \cdot \Vert _{0,{\widetilde{K}}_{ext}^{e}}$ is considered.

Lemma A.1

Let e be any edge in ${\mathcal {E}}_{h}^{\partial }$. Let ${\mathcal {L}}$ be the line segment with endpoints given by the center of the biggest ball contained in $K^{e}$, and the point of the set ${\widetilde{K}}_{ext}$ where the polynomial p achieves its maximum. Suppose that assumption (A.1) holds. Assume further that ${\mathcal {L}}$ is contained in interior of the closure of the set $K^{e}\cup {\widetilde{K}}_{ext}^{e}$, denoted by $B^{e}$. Then, for any $p\in \mathrm {P}_{l}(B^{e})$ we have

$$\begin{aligned} \Vert p\Vert _{0,{\widetilde{K}}_{ext}^{e}}\le C({\widetilde{r}}_{e})^{1/2}(l+1)^{2}\eta _{e}^{l}\Vert p\Vert _{0,K^{e}}, \end{aligned}$$

where ${\widetilde{r}}_{e}\,{:}{=}\,{\widetilde{H}}_{e}/h_{e}^{\perp }$ and $\eta _{e}\,{:}{=}\,1+2\gamma _{K^e} {\widetilde{r}}_{e}+2\Big (\gamma _{K^e}{\widetilde{r}}_{e}(1+\gamma _{K^e} {\widetilde{r}}_{e})\Big )^{1/2}$. Here the constant C solely depends on the shape-regularity constant $\gamma _{K^e}$.

Proof

We begin by noting that ${\mathcal {L}}$ can be subdivided as

$$\begin{aligned} I_{int}^{e}\,{:}{=}\, \Big \{{\mathbf {x}}\in {\mathcal {L}}:\;{\mathbf {x}}\cap K^{e}\ne \emptyset \Big \}{\quad \hbox {and}\quad }I_{ext}^{e}\,{:}{=}\, \Big \{{\mathbf {x}}\in {\mathcal {L}}:\;{\mathbf {x}}\cap {\widetilde{K}}_{ext}^{e}\ne \emptyset \Big \}, \end{aligned}$$

from which

$$\begin{aligned} \displaystyle \Vert p\Vert _{0,{\widetilde{K}}_{ext}^{e}}^{2}\le |{\widetilde{K}}_{ext}^{e}|\max _{{\mathbf {x}}\in {\widetilde{K}}_{ext}^{e}}|p({\mathbf {x}})|\le |{\widetilde{K}}_{ext}^{e}|\Vert p\Vert _{\mathrm {L}^{\infty }(I_{ext}^{e})}^{2}\le C{h_{K^{e}}{\widetilde{H}}_{e}}\Vert p\Vert _{\mathrm {L}^{\infty }(I_{ext}^{e})}^{2}, \end{aligned}$$

owing to the relation $|{\widetilde{K}}_{ext}^{e}|\le C{h_{K^{e}}{\widetilde{H}}_{e}}$. Next, we proceed as in [15, Lemma A.1] and prove that $\Vert p\Vert _{\mathrm {L}^{\infty }(I_{ext}^{e})}\le \eta _{e}^{l}\Vert p\Vert _{\mathrm {L}^{\infty }(I_{int}^{e})}$. In fact, in virtue of [11, Lemma 4.3], this is fulfilled by observing that

$$\begin{aligned} \displaystyle \frac{|I_{ext}^{e}|}{|I_{int}^{e}|}\le \frac{|I_{ext}^{e}|}{\rho _{K^{e}}} \le \frac{{\widetilde{H}}_{e}}{\rho _{K^{e}}} \le \gamma _{K^e}\frac{{\widetilde{H}}_{e}}{h_{K^{e}}}\le \gamma _{K^e}{\widetilde{r}}_{e}, \end{aligned}$$

where $\rho _{K^{e}}$ is the radius of the biggest ball contained in $K^{e}$, since $h_{e}^{\perp }\le h_{K^{e}}$ and $h_{K^{e}}\le \gamma _{K^e} \rho _{K^{e}}$. In addition, by standard scaling arguments there holds

$$\begin{aligned} \Vert p\Vert _{\mathrm {L}^{\infty }(I_{int}^{e})}\le \Vert p\Vert _{\mathrm {L}^{\infty }(K^{e})}\le C {\left( h_{K^{e}}\right) ^{-1}}(l+1)^{2}\Vert p\Vert _{0,K^{e}}. \end{aligned}$$

Finally, the proof is completed by noting that $h_{K^{e}}^{-1}\le \left( h_{e}^{\perp }\right) ^{-1}\le {\widetilde{r}}_{e}/{\widetilde{H}}_{e}$. $\square$

The previous result, together with the estimates in Lemma 3.4, implies that

$$\begin{aligned} {\widetilde{C}}_{ext}^{e} \le {C}(C_1^e)^{-1}C_{2}^e(l+1)^{2}\eta _{e}^{l}. \end{aligned}$$