Error analysis of a Galerkin method to solve the forward problem in MEG using the boundary element method

Sources of brain activity, e.g. epileptic foci, can be localized with Magnetoencephalography (MEG) measurements by recording the magnetic field outside the head. For a successful surgery a very high localization accuracy is needed. The most often used conductor model in the source localization is an analytic sphere, which is not always adequate, and thus a realistically shaped conductor model is needed. In this paper we examine a Galerkin method with linear basis functions to solve the forward problem in MEG using the boundary element method. Its accuracy is compared to the collocation method with constant and linear basis functions. The accuracies are determined for a unit sphere for which analytic solutions are available. The Galerkin method gives a clear improvement in the accuracy of the forward problem especially for the tangential component of the magnetic field. At realistic MEG measurement distances from the brain the Galerkin method reaches a given accuracy with lower computational costs than the collocation methods starting from a few hundreds of unknowns. With larger meshes the difference for the Galerkin method increases significantly.