A Nonlinear Discretization Theory for Meshfree Collocation Methods applied to Quasilinear Elliptic Equations

TL;DR

This paper develops a nonlinear discretization framework for meshfree collocation methods applied to quasilinear elliptic PDEs, ensuring stability and convergence with oversampling and extending previous linear results.

Contribution

It generalizes meshfree collocation methods to quasilinear problems, establishing uniform stability and convergence using oversampling and nonlinear stability analysis.

Findings

Proved uniform stability for discretizations of quasilinear elliptic problems.

Established convergence of meshfree methods for boundary value problems.

Extended analysis to bifurcation and center manifolds in elliptic PDEs.

Abstract

We generalize our earlier results concerning meshfree collocation methods for semilinear elliptic second order problems to the quasilinear case. The stability question, however, is treated differently, namely by extending a paper on uniformly stable discretizations of well-posd linear problems to the nonlinear case. These two ingredients allow a proof that all well-posed quasilinear elliptic second-order problems can be discretized in a uniformly stable way by using sufficient oversampling, and then the error of the numerical solution behaves like the error obtainable by direct approximation of the true solution by functions from the chosen trial space, up to a factor induced by being forced to use a H\"older-type theory for the nonlinear PDE. We apply our general technique to prove convergence of meshfree methods for quasilinear elliptic equations with Dirichlet and non-Dirichlet boundary conditions. This is achieved for bifurcation and center manifolds of elliptic partial differential equations and their numerical methods as well.