A Nitsche-eXtended finite element method for distributed optimal control problems of elliptic interface equations

Wang, Tao; Yang, Chaochao; Xie, Xiaoping

doi:10.48550/arxiv.1810.02271

Cited by 1 publication

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“…This method enriches the standard linear finite element space with additional cut basis functions and generalizes the results in [1,5]. Recently, the Nitsche-XFEM was extended to the discretization of optimal control problems of elliptic interface equations ( [53,56]). We note that the technique of using cut basis functions as the enrichment was also applied in [43,51,52,54] to develop interface-unfitted discontinuous Galerkin methods for the elliptic interface problems.…”

Section: Introductionmentioning

confidence: 85%

Extended HDG methods for second order elliptic interface problems

Han

Chen

Wang

et al. 2019

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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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