A Meshless Particle Method for Poisson and Diffusion Problems with Discontinuous Coefficients and Inhomogeneous Boundary Conditions

Abstract

We present a meshless particle method for Poisson and diffusion problems on domains with discontinuous coefficients and possibly inhomogeneous boundary conditions. The method is based on a domain-decomposition approach with suitable interface and boundary conditions between regions of different diffusivities, and on using discretization-corrected particle strength exchange operators [B. Schrader, S. Reboux, and I. F. Sbalzarini, J. Comput. Phys., 229 (2010), pp. 4159--4182]. We propose and compare two methods: The first one is based on an immersed interface approach, where interfaces are determined implicitly using a simplified phase-field equation. The second method uses a regularization technique to transform inhomogeneous interface or boundary conditions to homogeneous ones with an additional continuous volume contribution. After presenting the methods, we demonstrate their capabilities and limitations on several one-dimensional and three-dimensional test cases with Dirichlet and Neumann boundary conditions, and both regular and irregular particle distributions.