A Residual Minimization approach for Nonlinear Partial Differential Equations set in Banach spaces
This work addresses the challenge of solving nonlinear PDEs in Banach spaces for computational mathematics and engineering applications, presenting an incremental improvement with a new formulation for residual minimization.
The authors tackled the numerical solution of nonlinear PDEs in Banach spaces by proposing a residual-minimization strategy that approximates solutions through variational residual minimization in a discrete dual norm, with the p-Laplacian as a model problem and numerical experiments supporting the theory.
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.