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Article overview
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Extended HDG methods for second order elliptic interface problems | | Yihui Han
; Huangxin Chen
; Xiao-Ping Wang
; Xiaoping Xie
; | | Date: |
22 Oct 2019 | | Abstract: | 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. | | Source: | arXiv, 1910.9769 | | Services: | Forum | Review | PDF | Favorites | |
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