Numerical solution of two-dimensional elliptic PDEs with nonlocal boundary conditions

Abstract

In the present paper, two numerical methods are analyzed for the solution of two-dimensional Poisson equation with two different types of nonlocal boundary conditions. The first numerical method is a collocation method based on Haar wavelet whereas the second numerical method is a meshless method based on different types of radial basis functions (RBFs). A two-point boundary condition and an integral boundary condition are the two types of nonlocal boundary conditions considered in the present work. For the collocation method based on Haar wavelet a new approach is formulated which involves the approximation of a fourth order mixed derivative by a Haar expansion which is integrated subsequently to get wavelet approximation of the solution. For the meshless method based on RBFs, the algorithm is implemented using two different splitting schemes (with and without shape parameter splitting) for numerical solution of the model. The comparative analysis of the meshless methods with and without shape parameter splitting scheme is performed between themselves as well as with the Haar wavelet. Accuracy and efficiency wise performance is confirmed through application of the algorithms on the benchmark tests.