Extended HDG methods for second order elliptic interface problems
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In this paper, we propose two arbitrary order eXtended hybridizable Discontinuous Galerkin (X-HDG) methods for second order elliptic interface problems in two and three dimensions. The first X-HDG method applies to any piecewise C^2 smooth interface. It uses piecewise polynomials of degrees k(k>= 1) and k-1 respectively for the potential and flux approximations in the interior of elements inside the subdomains, and piecewise polynomials of degree k for the numerical traces of potential on the inter-element boundaries inside the subdomains. Double value numerical traces on the parts of interface inside elements are adopted to deal with the jump condition. The second X-HDG method is a modified version of the first one and applies to any fold line/plane interface, which uses piecewise polynomials of degree k-1 for the numerical traces of potential. The X-HDG methods are of the local elimination property, then lead to reduced systems which only involve the unknowns of numerical traces of potential on the inter-element boundaries and the interface. Optimal error estimates are derived for the flux approximation in L^2 norm and for the potential approximation in piecewise H^1 seminorm without requiring "sufficiently large" stabilization parameters in the schemes. In addition, error estimation for the potential approximation in L^2 norm is performed using dual arguments. Finally, we provide several numerical examples to verify the theoretical results.
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