A radial basis function Galerkin method for inhomogeneous nonlocal diffusion

Abstract

Abstract We introduce a meshfree discretization for a nonlocal diffusion problem using a localized basis of radial basis functions. Our method consists of a conforming radial basis of local Lagrange functions for a variational formulation of a volume constrained nonlocal diffusion equation. We also establish an L 2 error estimate on the local Lagrange interpolant. The stiffness matrix is assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, sparse, symmetric positive definite stiffness matrix. We demonstrate that both the continuum and discrete problems are well-posed and present numerical results for the convergence behavior of the radial basis function method. We explore approximating the solution to inhomogeneous differential equations by solving inhomogeneous nonlocal integral equations using the proposed radial basis function method.