Difference formulas for the surface Laplacian on a triangulated surface

Different approximating expressions for the surface Laplacian operator on a triangulated surface are derived. They are evaluated on a triangulated spherical surface for which the analytical expression of the surface Laplacian is known. It is shown that in order to obtain accurate results, due care has to be taken of irregularities present in the triangulation grid. If this is done, the approximation will equal the performance of an expression based on least squares which can be derived. Next the different approximations obtained are used as a regularization operator in the solution of an ill-posed inverse problem in electrical volume conduction. It is shown that in this application a crude approximation to the surface Laplacian suffices.