Discontinuous Galerkin methods on hp-anisotropic meshes II: a posteriori error analysis and adaptivity

Abstract

We consider the a priori error analysis of hp-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with nonnegative characteristic form under weak assumptions on the mesh design and the local finite element spaces employed. In particular, we prove a priori hp-error bounds for linear target functionals of the solution, on (possibly) anisotropic computational meshes with anisotropic tensor-product polynomial basis functions. The theoretical results are illustrated by a numerical experiment.

An optimization-based framework for anisotropic simplex mesh adaptation

“…2 We have also experimented with solving the p þ 1 local dual problems as done in [12] for quadrilateral elements, but numerically observed no quantifiable difference in the quality of the error estimate and hence the adaptation efficiency.…”

“…For anisotropic adaptation on quadrilateral meshes, a common approach is to combine a fixed-fraction marking strategy with a competitive anisotropy selection based on the minimum error-to-dof configuration [12,13]. Unfortunately, the anisotropic error samples collected on a simplex can be noisy and can compromise the quality of adaptation decision, as documented in [16,17].…”

“…Both Georgoulis et al [12] and Ceze and Fidkowski [13] used local solves to guide their anisotropy decisions on quadrilateral 1 meshes. However, the perspective set forth in those works is that of steepest descent in the discrete space, in which the local solves are used to guide the sequence of anisotropic subdivision of quadrilateral elements, which permit orthogonal directional decomposition.…”

High–Order hp–Adaptive Discontinuous Galerkin Finite Element Methods for Compressible Fluid Flows

“…In the h-version setting, we again exploit the algorithm outlined in Section 5. For the case when an element has been selected for polynomial enrichment we consider the p-version counterpart of Algorithm 5.1 and solve local problems based on increasing the polynomial degrees anisotropically in one direction at a time by one degree, or isotropically by one degree; see [7] for details. …”

An a-posteriori error estimate forhp-adaptive DG methods for convection–diffusion problems on anisotropically refined meshes

“…In all the tests, our new hp-version anisotropic refinement strategy outperforms similar strategies based on isotropic mesh refinement by orders of magnitude. Let us also point out that in [12,13], a duality-based a-posteriori approach was successfully proposed and studied for hp-adaptive DG methods for convection-diffusion problems on anisotropically refined meshes and with anisotropically enriched elemental polynomial degrees.…”