An analysis of the practical DPG method

Abstract

In this work we give a complete error analysis of the Discontinuous Petrov Galerkin (DPG) method, accounting for all the approximations made in its practical implementation. Specifically, we consider the DPG method that uses a trial space consisting of polynomials of degree $p$ on each mesh element. Earlier works showed that there is a "trial-to-test" operator $T$, which when applied to the trial space, defines a test space that guarantees stability. In DPG formulations, this operator $T$ is local: it can be applied element-by-element. However, an infinite dimensional problem on each mesh element needed to be solved to apply $T$. In practical computations, $T$ is approximated using polynomials of some degree $r>p$ on each mesh element. We show that this approximation maintains optimal convergence rates, provided that $r\ge p+N$, where $N$ is the space dimension (two or more), for the Laplace equation. We also prove a similar result for the DPG method for linear elasticity. Remarks on the conditioning of the stiffness matrix in DPG methods are also included.

Mathematics of Computation, 2012

Cited by in corpus (1)

• Analysis of the Discontinuous Petrov-Galerkin Method with Optimal Test Functions for the Reissner-Mindlin Plate Bending Model