Demkowicz was supported in part by the Department of Energy [National Nuclear Security Adminis- tration] under Award Number [DE-FC52-08NA28615], and by a research contract with Boeing. Gopalakrishnan was supported in part by the National Science Foundation under grant DMS-0713833.
Numerical Methods for Partial Differential Equations
Galerkin methods, Diffusion processes, Reaction-diffusion equations
We lay out a program for constructing discontinuous Petrov–Galerkin (DPG) schemes having test function spaces that are automatically computable to guarantee stability. Given a trial space, a DPG discretization using its optimal test space counterpart inherits stability from the well posedness of the undiscretized problem. Although the question of stable test space choice had attracted the attention of many previous authors, the novelty in our approach lies in the fact we identify a discontinuous Galerkin (DG) framework wherein test functions, arbitrarily close to the optimal ones, can be locally computed. The idea is presented abstractly and its feasibility illustrated through several theoretical and numerical examples.
Demkowicz, Leszek and Gopalakrishnan, Jay, "A Class of Discontinuous Petrov–Galerkin Methods. II. Optimal Test Functions" (2010). Mathematics and Statistics Faculty Publications and Presentations. 51.