A meshless finite difference method for elliptic interface problems based on pivoted QR decomposition

Abstract

We propose to solve elliptic interface problems by a meshless finite difference method, where the second order elliptic operator and jump conditions are discretized with the help of the QR decomposition of an appropriately rescaled multivariate Vandermonde matrix with partial pivoting. A prescribed consistency order is achieved on irregular nodes with small influence sets, which allows to place the nodes directly on the unfitted interface and leads to sparse system matrices with the density of nonzero entries comparable to the density of the system matrices arising from the mesh-based finite difference or finite element methods. Numerical experiments on a number of standard test problems with known solutions demonstrate convergence orders up to O(h6) for both the approximate solution and its gradient, and a robust performance of the method in the case when the interface is known inaccurately.