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Convergence of meshfree collocation methods for fully nonlinear parabolic equations

Yumiharu Nakano

TL;DR

It is proved that the convergence of meshfree collocation methods for the terminal value problems of fully nonlinear parabolic partial differential equations in the framework of viscosity solutions can be proved.

Abstract

We prove the convergence of meshfree collocation methods for the terminal value problems of fully nonlinear parabolic partial differential equations in the framework of viscosity solutions, provided that the basis function approximations of the terminal condition and the nonlinearities are successful at each time step. A numerical experiment with a radial basis function demonstrates the convergence property.

Convergence of meshfree collocation methods for fully nonlinear parabolic equations

TL;DR

It is proved that the convergence of meshfree collocation methods for the terminal value problems of fully nonlinear parabolic partial differential equations in the framework of viscosity solutions can be proved.

Abstract

We prove the convergence of meshfree collocation methods for the terminal value problems of fully nonlinear parabolic partial differential equations in the framework of viscosity solutions, provided that the basis function approximations of the terminal condition and the nonlinearities are successful at each time step. A numerical experiment with a radial basis function demonstrates the convergence property.

Paper Structure

This paper contains 4 sections, 5 theorems, 85 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Generalization of Kansa's method
  3. Convergence
  4. A numerical example

Key Result

Theorem 3.8

Suppose that Assumptions assum:3.1-assum:3.4, assum:3.6 hold. Then we have

Figures (1)

  • Figure 4.1: The analytical solution (left) and the numerical solution (right) with $h=10^{-2}$ and $N=25$ uniformly spaced points.

Theorems & Definitions (18)

  • Example 2.1
  • Remark 2.2
  • Example 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 3.3
  • Example 3.6
  • Theorem 3.8
  • Remark 3.9
  • Lemma 3.10
  • ...and 8 more