Ultraweak formulation of linear PDEs in nondivergence form and DPG approximation

TL;DR

This paper introduces an ultraweak formulation for linear PDEs in nondivergence form with Cordes condition coefficients, and develops DPG methods with error estimators, validated through numerical experiments.

Contribution

It presents a novel ultraweak formulation for nondivergence form PDEs and develops DPG methods with stability and error analysis.

Findings

DPG methods are effective for nondivergence PDEs.

A posteriori error estimators guide adaptive mesh refinement.

Numerical experiments confirm theoretical results.

Abstract

We develop and analyze an ultraweak formulation of linear PDEs in nondivergence form where the coefficients satisfy the Cordes condition. Based on the ultraweak formulation we propose discontinuous Petrov--Galerkin (DPG) methods. We investigate Fortin operators for the fully discrete schemes and provide a posteriori estimators for the methods under consideration. Numerical experiments are presented in the case of uniform and adaptive mesh-refinement.