Weitere Informationen zum ausgewählten Kolloquiumsvortrag:

Discrete Alexandroff estimate and pointwise rates of convergence for FEMs

Prof. Ricardo Nochetto, University of Maryland

Donnerstag, 2017-01-19 17:00Uhr

We derive an Alexandroff estimate for continuous piecewise linear functions which states that the max-norm of their negative part is controlled by the Lebesgue measure of the sub-differential of their convex envelope at the contact nodes. We develop a discrete Alexandroff-Bakelman-Pucci estimate which controls the Lebesgue measure of the sub-differential in terms of the discrete Laplacian via gradient jumps. We further apply these estimates in the analysis of three finite element methods (FEMs).

We first discretize the Monge-Ampere equation (MA) with a FEM based on the geometric interpretation of MA. We next discretize MA with a two-scale FEM which exploits an eigenvalue representation of the determinant of SPD matrices. We finally present a two-scale FEM for linear elliptic PDEs in non-divergence form. We prove rates of convergence in the max-norm for all three FEMs, study their optimality, and check it computationally.

This is joint work with D. Ntogkas and W. Zhang.