Plenary talk: On the lumped mass finite element method for<br/><br/>parabolic problems

Friday, November 5, 2010 - 5:45pm - 6:30pm
Keller 3-180
Vidar Thomée (Chalmers University of Technology)
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 terms of 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 L2-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.