Three-dimensional Haar wavelet method for singularly perturbed elliptic boundary value problems on non-uniform meshes

Abstract

Recently, the two-dimensional elliptic singularly perturbed boundary value problems have received attention. These problems have not been much explored numerically. A highly accurate numerical scheme on different non-uniform meshes is suggested to solve such problems. In particular, the Haar wavelet method on a special type of non-uniform mesh and Shishkin mesh is proposed; because of the use of the block pulse function, it is easy to derive and handle the operator matrices. Also, it has been shown that Shiskin mesh provides better results. The use of the piecewise-uniform Shishkin mesh with the Haar wavelet scheme contains a novelty in itself. Through rigorous analysis, the method is shown as first-order convergent in L2-norm. The theoretical results are confirmed by computational results obtained in the maximum norm and L2-norm for two test problems. From the comparative results provided in tables, it is worth noting that the Shishkin mesh is an excellent choice to solve these problems instead of a particular type of non-uniform mesh for the proposed scheme. This can be justified because the grid points defined by the Shishkin mesh are adequately distributed in the layer and non-layer regions.