DPG Methods for a Fourth-Order div Problem

Abstract

We study a fourth-order div problem and its approximation by the discontinuous Petrov-Galerkin method with optimal test functions. We present two variants, based on first and second-order systems. In both cases, we prove well-posedness of the formulation and quasi-optimal convergence of the approximation. Our analysis includes the fully-discrete schemes with approximated test functions, for general dimension and polynomial degree in the first-order case, and for two dimensions and lowest-order approximation in the second-order case. Numerical results illustrate the performance for quasi-uniform and adaptively refined meshes.