A High Order HDG Method for Curved-Interface Problems Via Approximations from Straight Triangulations
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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 fit 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 \(O(h^2)\).
KeywordsDiscontinuous Galerkin High order Curved boundary Curved interface
W. Qiu is partially supported by the GRF of Hong Kong (Grant Nos. 9041980 and 9042081) and a Grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11302014). M. Solano is partially supported by Conicyt-Chile through Grant Fondecyt-1160320, BASAL Project CMM, Universidad de Chile and by Conicyt project Anillo ACT1118 (ANANUM). P. Vega acknowledges the Scholarship Program of Conicyt-Chile. The authors would like to thank the anonymous referees for their constructive criticism, which resulted in a better version of this paper.