A stable RBF partition of unity local method for elliptic interface problems in two dimensions

Abstract

The interface problems are faced with multiple connected domains and consequently, their solutions or derivatives might be discontinuous. This paper proposes the use of collocation based radial basis function partition of unity method (RBF-PUM) for solving two-dimensional elliptic interface problems. The RBF-PUM is a local method that allows overcoming the high computational cost associated with the global RBF methods. In the RBF-PUM, the domain is split into overlapping patches forming a covering of it. However, this method suffers from instability when the RBF shape parameter ε tends to zero. To overcome this issue, we use the RBF-QR algorithm which offers stable computations for all values of ε and provides higher accuracy. To obtain the appropriate solution in the vicinity of the interface, the domain decomposition technique is used. In this technique, the approximation in each subdomain is built separately, and proper jump conditions are then imposed across the interface. We illustrate how to apply the proposed method to Sturm-Liouville, Sturm-Liouville eigenvalue and elastostatic interface problems. The proposed method in dealing with arbitrary interfaces within different domain sizes is validated. We present some numerical examples in which the results are compared with exact solutions and those provided by other numerical methods.