Discontinuous Galerkin methods on hp-anisotropic meshes I: a priori error analysis

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.

A Class of Domain Decomposition Preconditioners for hp-Discontinuous Galerkin Finite Element Methods

“…Finally, we assume that the following bounded local variation property holds (cf. [25,26]): for any pair of elements…”

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

“…Let now u hp ∈ S p (T ) be the discontinuous Galerkin approximation obtained by (12). Moreover, let f hp and a hp denote piece polynomial approximations in S p (T ) and S p (T ) 2 to the right-hand side f and the flow field a, respectively.…”

“…Let u be the solution of (1) and u hp ∈ S p (T ) its DG approximation obtained by (12). Let the error estimator η and the data approximation error Θ be defined by (22).…”

“…We shall use these auxiliary forms to express both the continuous form A in (3) and the discontinuous Galerkin form A hp in (12). Indeed, we have…”

“…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.…”