An analysis of the practical DPG method (Q2871175)

The paper considers a general abstract problem of finding \(u\in U\) such that NEWLINE\[NEWLINE b(u,v) = l(v) \quad \forall v \in V, NEWLINE\]NEWLINE where \(U\) and \(V\) are Hilbert spaces and \(b\) and \(l\) corresponding bilinear and linear forms. This problem is discretized by the discontinuous Petrov-Galerkin (DPG) method and an abstract error estimate based on Babuška's theory is proved. This abstract result is then applied to two specific examples: the Laplace equation and linear elasticity. In both cases, the DPG approximation is rigorously formulated, abstract assumptions are verified, and a priori error estimates yielding the convergence rates are presented. The condition number of the corresponding stiffness matrices is provided. The paper does not contain any numerical experiment. It extends previous results of the first author, namely those by \textit{L. Demkowicz} and \textit{J. Gopalakrishnan} [SIAM J. Numer. Anal. 49, No. 5, 1788--1809 (2011; Zbl 1237.65122)] and by \textit{J. Bramwell} et al. [Numer. Math. 122, No. 4, 671--707 (2012; Zbl 1283.74094)].