A Nonlinear Discretization Theory for Meshfree Collocation Methods applied to Quasilinear Elliptic Equations
For researchers in numerical analysis and PDEs, this provides a rigorous convergence framework for meshfree methods on quasilinear elliptic equations, though the result is incremental as it generalizes prior work.
The authors extend a stability theory for meshfree collocation methods from semilinear to quasilinear elliptic PDEs, proving that well-posed quasilinear problems can be uniformly stably discretized with sufficient oversampling, and the error is bounded by the best approximation error times a Hölder-type factor. The theory is applied to Dirichlet and non-Dirichlet boundary conditions, including bifurcation and center manifolds.
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 Hölder-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.