A Feasibility Study: Operator Splitting for Solving Anisotropic Problem

Abstract

The electroencephalography (EEG) is a non-invasive technique to study electrical brain activity (while the brain is performing a cognitive task). 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 properties in terms of its magnitude and orientation, called anisotropic conductivity. Using Maxwell's equations, 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 a new method, called Operator Splitting Method (OSM), to solve anisotropic 2-Dimensional (2D) Poisson’s equation is performed. Freeware of the finite element method (FEM), FreeFEM++, 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 the dipolar source. Afterward, the OSM solutions are validated by using an exact solution and a direct numerical solution. By using L2-Error Norm, the convergence rate of the OSM algorithm is then analyzed. Some numerical experiments have been performed to test the performance of the OSM algorithm. The OSM solution of anisotropic 2D Laplace’s equation coincides with the exact and direct numerical solution of the problem. For anisotropic 2D Poisson’s equation with dipolar source, some similar results have been obtained too. The pattern of the OSM solutions is similar to the pattern of direct numerical solutions of the problem. The results arise a hope to attempt to implement the OSM algorithm for more complex problems such as a realistic human head model.