Weifeng Qiu, Manuel Solano, Patrick Vega:
A high order HDG method for curved-interface problems via approximations from straight triangulations
We propose a novel technique to solve elliptic problems involving a non-polygonal interface/boundary. It is based on a high order hybridizable discontinuous Galerkin method where the mesh does not exactly t the domain. We first study the case of a curved-boundary value problem with mixed boundary conditions since it is crucial to understand the applicability of the technique to curved interfaces. The Dirichlet data is approximated by using the transferring technique developed in a previous paper. The treatment of the Neumann data is new. We then extend these ideas to curved interfaces. We provide numerical results showing that, in order to obtain optimal high order convergence, it is desirable to construct the computational domain by interpolating the boundary/interface using piecewise linear segments. In this case the distance of the computational domain to the exact boundary is only O(h^2).
This preprint gave rise to the following definitive publication(s):
Weifeng QIU, Manuel SOLANO, Patrick VEGA: A high order HDG method for curved-interface problems via approximations from straight triangulations. Journal of Scientific Computing, vol. 69, 3, pp. 1384-1407, (2016).