Distributed source scheme for Poisson equation using finite element method

Abstract

The finite-element method (FEM) is one of the most versatile numerical techniques to solve partial differential equations (PDEs). The approximations in FEM result in numerous errors and various techniques are available to reduce the errors. Distributed Source Scheme (DSS) is a recently proposed method to reduce error around a single source of field which obeys the Poisson equation using the 3-D finite-difference method (FDM). DSS achieves this feat by introducing pseudo sources around the singular source. In this article, the proposed method is studied for single and bulk sources using 2-D FEM along with experimental validation. The numerical solution from FEM is compared with the approximate solution from integral equations over the whole computational domain for the error analysis. The error due to boundary values are isolated by considering the solution of integral equations on boundary elements in FEM. The maximum error in the numerical solution of a single source is reduced from 4.96% to 3.28×10−4% using DSS and to 0.55% using Truncated Distributed Source Scheme (TDSS) with a truncation number of 5. The mean difference in numerical and approximate potential at the boundaries of bulk sources is reduced from 1.21×105 Volt to 2.69×104 Volt using DSS.