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Extended HDG methods for second order elliptic interface problems. (English) Zbl 1452.65339

Summary: 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 \(k\) (\(k\geq 1\)) 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.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs

Software:

MIBPB; IIMPACK
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Full Text: DOI arXiv

References:

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