This work was partly supported by the NSF under grants DMS-1211635 and DMS-1014817
Mathematics of Computation
Polynomials, Mathematical statistics, Instrumental variables (Statistics)
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.
© Copyright 2014, American Mathematical Society
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Published as: Gopalakrishnan, J., & Qiu, W. (2014). An analysis of the practical DPG method. Mathematics of Computation, 83(286), 537-552.