A high order HDG method for curved-interface problems via approximations from straight triangulations. (English) Zbl 1371.65121
Summary: 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 (HDG) 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 \(O(h^2)\).
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
Keywords:
high order; curved boundary; curved interface; hybridizable discontinuous Galerkin method; numerical result; convergenceReferences:
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