Fortin operators for DPG advection discretizations

Abstract

We construct Fortin operators for discontinuous Petrov-Galerkin (DPG) discretizations of the advection equation with a piecewise constant divergence-free advection vector $β$, on simplicial meshes of any spatial dimension $N$ and for any polynomial degree $k \ge 1$ of the trial space. A minimal test space is built on each element from facet and interior bubbles and later augmented. The Fortin operator is shown to be bounded, uniformly over shape-regular mesh families, in the natural $β$-weighted broken test graph norm built on $L_q$ for every $1 < q < \infty$, where $q$ is the exponent conjugate to the trial exponent $p$. A non-characteristic facet condition is assumed when $k \ge 2$, while the lowest-order case requires no such condition and admits characteristic facets. As applications we prove that the practical fully discrete residual minimization method is quasioptimal in the DPG energy norm for every $1 < p < \infty$, with a quasioptimality constant governed solely by the Fortin operator, that its computable residual is a globally reliable and efficient a posteriori error estimator, and that augmenting the test space and changing the test norm improves the results in the lowest-order case.