Operator splitting method for solving anisotropic problem

Abstract

The electroencephalography is a non-invasive technique to study electrical brain activity. The electrical brain activity is a complex process of electrical propagation because the brain structure is an incredibly complex structure. This complex structure leads to different conductivity property in term of its magnitude and orientation, called anisotropic conductivity. Using Maxwell’s equations, the electrical brain activity has been studied intensively. For simplification, the quasistatic Maxwell’s equations are used to model the electrical brain activity and it leads to deal with a Poisson’s equation. In this research, a feasibility study of using Operator Splitting Method (OSM) to solve anisotropic 2-Dimensional (2D) Poisson’s equation is performed. A freeware of finite element method (FEM) is employed to build matrices used in the OSM algorithm. The OSM algorithm which is written in Matlab is then tested to solve anisotropic 2D Laplace’s equation and anisotropic Poisson’s equation with dipolar source. Some numerical experiments have been performed to test the performance of the OSM algorithm. The OSM solution of anisotropic 2D Laplace’s equation coincide with the exact and direct numerical solution of the problem. For anisotropic 2D Poisson’s equation with dipolar source, some similar results has been obtained too. The pattern of the OSM solutions are similar to the pattern of direct numerical solutions of the problem. The results arise a hope to attempt implementing the OSM algorithm for more complex problem such as a realistic human head model.