An a-posteriori error estimate for $$hp$$ -adaptive DG methods for elliptic eigenvalue problems on anisotropically refined meshes

Abstract

(2013) 'An a-posteriori error estimate for hp-adaptive DG methods for elliptic eigenvalue problems on anisotropically rened meshes.', Computing., 95 (1 Supplement). S319-S341.Further information on publisher's website:http://dx.doi.org/10.1007/s00607-012-0261-5Publisher's copyright statement:The nal publication is available at Springer via http://dx.doi.org/10.1007/s00607-012-0261-5Additional information: 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-prot 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. Abstract We prove an a-posteriori error estimate for an hp-adaptive discontinuous Galerkin method for the numerical solution of elliptic eigenvalue problems with discontinuous coefficients on anisotropically refined rectangular elements. The estimate yields a global upper bound of the errors for both the eigenvalue and the eigenfunction and lower bound of the error for the eigenfunction only. The anisotropy of the underlying meshes is incorporated in the upper bound through an alignment measure. We present a series of numerical experiments to test the flexibility and robustness of this approach within a fully automated hp-adaptive refinement algorithm.

An Anisotropic Recovery-Based Error Estimator for Adaptive Discontinuous Galerkin Methods

“…7, output of the DRAMA procedure. The highly stretched triangles allow the sharp detection of all the layers with a considerably low number of elements, in particular when compared with the grids in figures 16 and 17 in [30]. Actually, from a qualitative viewpoint, we detect very elongated elements capturing the boundary layers and the diagonal solution drop.…”

“…We consider the same setting as in [30,Section 5.3] where problem ( 1) is discretized in = (−1, 1) 2 , for μ = 2.5e−4, β = (− sin(π/6), cos(π/6)) T , σ = 0, and f = 0. The problem is supplemented by the non-homogeneous Dirichlet boundary condition We run DRAMA algorithm with the input parameters TOL = 2.5e−3, MTOL = 3e−3, kmax = 20 and by selecting a structured 80 × 80 initial mesh T 0 h .…”

“…In particular, we adopt the symmetric Interior Penalty DG (IPDG) formulation, developed by various authors in the late seventies [2,3,19,50]. Since the IPDG method is stable in all regimes, including in the advection-dominated one, we are allowed to start the automatic mesh adaptive procedure from a relatively coarse (non-resolving) mesh (see, e.g., [9,30,34]). The mesh adaptation is driven by a recovery-type a posteriori error estimator through a metric-based approach [28].…”

“…We can categorize these works according to the adopted error analysis. For instance, in [7,22,35], the authors resort to a residual-based error estimator for the isotropic case, whereas an anisotropic counterpart is analyzed in [9,30]. As an alternative, in [29,46], a goal-oriented error analysis for a DG approximation is developed.…”

A painless automatic hp-adaptive strategy for elliptic problems

“…Note that the implementational effort is not accounted for in that survey. Finally, we also mention the family of hp-adaptive Discontinuous Galerkin (DG) methods, [37][38][39][40][41][42][43][44][45], which require from a specific implementation that is not always easily transferable to commercial FEM codes.…”