The lumped mass finite element method for surface parabolic problems: Error estimates and maximum principle

The lumped mass method is extended to the surface finite element method for solving the surface parabolic equations. The main purpose of the proposed method is to overcome the difficulty that the surface finite element method does not guarantee the maximal principle of the surface heat equation. Optimal error estimates are given for the semi-discrete and fully-discrete schemes of the proposed method respectively. The maximum principle is shown for surface heat equations and its preservation by the lumped mass surface finite element under the Delaunay type triangulation. Moreover, some results of positivity and monotonicity are derived for nonlinear parabolic equations. Finally some numerical experiments are displayed to illustrate the validity and numerical performance of the proposed method.