Solving Dirichlet Boundary-value Problems on Curved Domains by Extensions from Subdomains

Abstract

We present a technique for numerically solving Dirichlet boundary-value problems for second-order elliptic equations on domains $\Omega$ with curved boundaries. This is achieved by using suitably defined extensions from polyhedral subdomains $\textsf{D}_h$; the problem of dealing with curved boundaries is thus reduced to the evaluations of simple line integrals. The technique is independent of the representation of the boundary, of the space dimension, and allows for the use of only polyhedral elements. We apply this technique to the hybridizable discontinuous Galerkin method and provide extensive numerical experiments showing that the convergence properties of the resulting method are the same as those for the case in which $\Omega=\textsf{D}_h$ whenever the distance of $\textsf{D}_h$ to $\partial\Omega$ is of order h. In particular, we find that when using polynomial approximations of degree k on the polyhedral subdomains $\textsf{D}_h$, both the scalar and vector unknowns converge with the optimal order of $k+1$ in the whole domain $\Omega$ for $k\ge0$. Moreover, we also find that for $k\ge1$, both a postprocessing of the scalar unknown in the polyhedral subdomain $\textsf{D}_h$ as well as its extension from it converge with order $k+2$. The extension also converges with order 2 for $k=0$.