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An Analysis of the Practical DPG Method

Thursday, August 18, 3:30PM – 5PM
POB 6.304

Dr. Weifeng (Frederick) Qiu IMA, U. Minnesota

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 ≥ 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.

References [1] Jay Gopalakrishnan, and Weifeng Qiu. An analysis of the practical DPG method, IMA preprint 2374, submitted.

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