Pseudo and anisotropic MFS for Laplace equation and optimal sources using maximal projection method with a substitution function

Abstract

The paper creates two new families of fundamental solutions for the 3D Laplace equation, presented into two parts. For the first part in terms of a planar line as a new coordinate the derived 2D like fundamental solution has a logarithmic singularity, which results in a method of pseudo fundamental solutions. We propose two methods to determine the optimal values of the offset parameter used to locate the source points. In the second part, an anisotropic distance function r(g) in terms of a symmetric non-negative anisotropic metric tensor is introduced, which satisfies a certain quadratic matrix equation, and then ln r(g) is proved to be a new fundamental solution. Using a unit orientation vector we can derive the metric tensor in closed-form, and prove that it is a singular projection operator. Given the unit orientation vector satisfying a cone condition, a method of anisotropic fundamental solutions is developed. They are distinct from the traditional 3D MFS. Owing to a weaker singularity than that of 1/r appeared in the 3D MFS, the method of pseudo fundamental solutions and the method of anisotropic fundamental solutions outperform the 3D MFS. Some numerical experiments explore the performance of these two novel methods.