A Residual Minimization approach for Nonlinear Partial Differential Equations set in Banach spaces

TL;DR

This paper introduces a residual-minimization method for solving nonlinear PDEs in Banach spaces, providing a framework that includes error estimation and adaptive mesh refinement, demonstrated on the p-Laplacian.

Contribution

It develops a residual minimization strategy with a saddle-point formulation and a Newton solver, enabling efficient and adaptive solutions for nonlinear PDEs in Banach spaces.

Findings

The method provides a natural a posteriori error estimator.

The Newton iteration guarantees solvability at each step.

Numerical experiments validate the approach on the p-Laplacian.

Abstract

In this work, we propose and analyze a residual-minimization strategy for the numerical solution of nonlinear PDEs posed in Banach spaces. Given a finite-dimensional trial space and a suitably enriched discrete test space (of higher dimension than the trial space), we approximate the solution by minimizing the variational residual in a discrete dual norm. This minimization is equivalent to a nonlinear saddle-point formulation for the discrete solution in the trial space together with a residual representative in the test space. The latter provides a natural a posteriori error estimator, enabling automatic mesh adaptivity. To solve the resulting nonlinear saddle-point problem, we propose a Newton iteration whose linearized saddle-point system is symmetric, thereby guaranteeing solvability at each step. We take the $p$-Laplacian as a model problem and support the theoretical developments with representative numerical experiments, using standard $H^1$-conforming piecewise linear functions for the trial space, and lowest-order Crouzeix--Raviart functions for the test space.