hp-Adaptive composite discontinuous Galerkin methods for elliptic eigenvalue problems on complicated domains

Abstract

Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. In this paper we develop the a posteriori error estimation of hpadaptive discontinuous Galerkin composite finite element methods (DGFEMs) for the discretization of second-order elliptic eigenvalue problems. DGFEMs allow for the approximation of problems posed on computational domains which may contain local geometric features. The dimension of the underlying composite finite element space is independent of the number of geometric features. This is in contrast with standard finite element methods, as the minimal number of elements needed to represent the underlying domain can be very large and so the dimension of the finite element space. Computable upper bounds on the error for both eigenvalues and eigenfunctions are derived. Numerical experiments highlighting the practical application of the proposed estimators within an automatic hp-adaptive refinement procedure will be presented.

Higher order stable generalized finite element method for the elliptic eigenvalue and source problems with an interface in 1D

“…To our best knowledge, most of the work on XFEM/GFEM is for source problems or interface source problems and there is no work on these methods for solving differential eigenvalue problems. Traditionally, differential eigenvalue problems are solved by using standard FEMs (c.f., [8,11,16,17,[36][37][38]42]), isogeometric analysis (c.f., [19,29,30]), discontinuous Galerkin methods (c.f., [4,26,27]), etc. We are not going to expand the literature review here and only mention the most-recent quadrature rule blending techniques developed in [3] for FEMs and in [13,15,20,21,39] for isogeometric analysis.…”

Spectral approximation of elliptic operators by the Hybrid High-Order method

“…We observe a good agreement with the convergence order predicted by Corollary 4.6, that is, the convergence order for the eigenfunctions in the H 1 -seminorm is indeed h k+1 . 1.34e-7 6.01 1.62e-6 6.05 8.63e-6 6.05 3.34e-5 6.12 32 2.09e-9 6.00 2.51e-8 6.01 1.34e-7 6.01 5.14e-7 6.02 64 3.26e-11 6.00 3.92e-10 6.00 2.09e-9 6.00 8.01e-9 6.00 Table 2: Unit square, relative eigenvalue errors, η = 1.…”

An adaptive composite discontinuous Galerkin method for elliptic problems on complicated domains with discontinuous coefficients

“…In recent years, a new discontinuous Galerkin method referred to as Discontinuous Galerkin Composite Finite Element Method (DGCFEM) [4], has been developed for the numerical solution of partial differential equations on complicated domains characterized by small geometric details or holes. Subsequently, in [17,16] adaptivity was added to the method and later in [14] the method was extended to eigenvalue problems. What all these incarnations of DGCFEM have in common is the focus on domains with geometries difficult to resolve using standard FEMs.…”