Nonlinear discontinuous Petrov-Galerkin methods

TL;DR

This paper introduces nonlinear discontinuous Petrov-Galerkin methods for convex nonlinear problems, providing theoretical analysis, error estimates, and numerical validation for a minimal residual approach with guaranteed error control.

Contribution

It develops a nonlinear dPG framework with a mixed and weighted least-squares formulation, including a priori and a posteriori error estimates, and demonstrates its effectiveness through numerical examples.

Findings

Quasi-optimal a priori error estimates obtained.

Reliable and efficient a posteriori error estimates established.

Numerical examples confirm the method's error control and uniqueness properties.

Abstract

The discontinuous Petrov-Galerkin method is a minimal residual method with broken test spaces and is introduced for a nonlinear model problem in this paper. Its lowest-order version applies to a nonlinear uniformly convex model example and is equivalently characterized as a mixed formulation, a reduced formulation, and a weighted nonlinear least-squares method. Quasi-optimal a priori and reliable and efficient a posteriori estimates are obtained for the abstract nonlinear dPG framework for the approximation of a regular solution. The variational model example allows for a built-in guaranteed error control despite inexact solve. The subtle uniqueness of discrete minimizers is monitored in numerical examples.