DPG methods for a fourth-order div problem

TL;DR

This paper develops and analyzes DPG methods for a fourth-order div problem, providing proofs of well-posedness and convergence, and demonstrating effectiveness through numerical experiments.

Contribution

It introduces two variants of DPG methods for the problem, including analysis for fully-discrete schemes with general dimensions and polynomial degrees.

Findings

Proved well-posedness of the formulations.

Established quasi-optimal convergence of the approximations.

Numerical results confirm the method's effectiveness on various meshes.

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.