A mixed method for Dirichlet problems with radial basis functions

TL;DR

This paper introduces a simplified radial basis function method for solving Dirichlet problems, employing a Lagrangian multiplier to effectively handle boundary conditions, confirmed by numerical experiments.

Contribution

It proposes a novel, simplified discretization approach using radial basis functions combined with a Lagrangian multiplier for boundary conditions.

Findings

Numerical experiments confirm the theoretical accuracy.

The method simplifies previous radial basis function approaches.

Effective handling of boundary conditions with Lagrangian multipliers.

Abstract

We present a simple discretization by radial basis functions for the Poisson equation with Dirichlet boundary condition. A Lagrangian multiplier using piecewise polynomials is used to accommodate the boundary condition. This simplifies previous attempts to use radial basis functions in the interior domain to approximate the solution and on the boundary to approximate the multiplier, which technically requires that the mesh norm in the interior domain is significantly smaller than that on the boundary. Numerical experiments confirm theoretical results.