Locally conservative discontinuous Petrov–Galerkin finite elements for fluid problems

Abstract

We develop a locally conservative formulation of the discontinuous Petrov–Galerkin finite element method (DPG) for convection–diffusion type problems using Lagrange multipliers to exactly enforce conservation over each element. We provide a proof of convergence as well as extensive numerical experiments showing that the method is indeed locally conservative. We also show that standard DPG, while not guaranteed to be conservative, is nearly conservative for many of the benchmarks considered. The new method preserves many of the attractive features of DPG, but turns the normally symmetric positive-definite DPG system into a saddle point problem.