Date & Time: Wednesday, February 01, 2012, 15:30-16:30.
Venue: Ramanujan Hall
Title:On the lumped mass finite element method for parabolic problems
Speaker: Prof. Vidar Thomee, Department of Mathematics, Chalmers Univ. of Technology, Sweden
Abstract: We study the lumped mass method for the model homogeneous heat equation with homogeneous Dirichlet boundary conditions. We first recall that the maximum principle for the heat equation does not carry over to the the spatially semidiscrete standard Galerkin finite element method, using continuous, piecewise linear approximating functions. However, for the lumped mass variant the situation is more advantageous. We present necessary and sufficient conditions on the triangulation, expressed in termsof properties of the stiffness matrix, for the semidiscrete lumped mass solution operator to be a positive operator or a contraction in the maximum-norm.
We then turn to error estimates in the $L_2$-norm. Improving earlier results we show that known optimal order smooth initial data error estimates for the standard Galerkin method carry over to the lumped mass method, whereas nonsmooth initial data estimates require special assumptions on the triangulations.
We also discuss the application to time discretization by the backward Euler and Crank-Nicolson methods.
About the Speaker: Professor Vidar is renowned Mathematician working on Computational PDE. He is one of the few researchers who worked on mathematical foundation of Finite Element Methods specially for parabolic problems.