Minimum residual discretization of a semilinear elliptic problem

TL;DR

This paper introduces a least-squares penalization approach to extend the DPG method for semilinear elliptic problems, enabling effective approximation and error estimation.

Contribution

It develops a novel least-squares penalization technique that allows standard DPG methods to handle semilinear elliptic problems with nonlinear constraints.

Findings

The method achieves a Cea estimate for the approximation error.

Numerical results demonstrate the scheme's effectiveness on uniform and adaptive meshes.

Abstract

We propose a least-squares penalization as a means to extend the discontinuous Petrov-Galerkin (DPG) method with optimal test functions to a class of semilinear elliptic problems. The nonlinear contributions are replaced with independent unknowns so that standard DPG techniques apply to the then linear problem with non-trivial kernel. The nonlinear relations are added as least-squares constraints. Assuming solvability of the semilinear problem and an Aubin-Nitsche-type approximation property for the primal variable, we prove a Cea estimate for the approximation error in canonical norms. Numerical results with uniform and adaptively refined meshes illustrate the performance of the scheme.