Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

Preprint 2014-15

C. Carstensen, D. Gallistl, M. Schedensack


L2 Best-Approximation of the Elastic Stress in the Arnold-Winter FEM


Abstract: The first part of this paper enfolds a medius analysis for mixed finite
element methods (FEMs) and proves a best-approximation result in L2
for the stress variable independent of the error of the Lagrange multiplier
under the abstract conditions (LBB), condition (C) and efficiency (E).
The second part applies the general result to the FEM of Arnold and
Winther for linear elasticity: The stress error in L2 is controlled by the
L2 best-approximation error of the true stress by any discrete function
plus data oscillations. The analysis is valid without any extra regularity
assumptions on the exact solution and also covers coarse meshes and
Neumann boundary conditions. Further applications include Raviart-
Thomas finite elements for the Poisson and the Stokes problem. The
result has consequences for nonlinear approximation classes related to
adaptive mixed finite element methods.


Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), P-2014-15


35 pp.