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

Preprint 2014-20

C. Carstensen, M. Schedensack


Medius Analysis and Comparison Results for First-Order Finite Element Methods in Linear Elasticity.


Abstract: This paper enfolds a medius analysis for first-order nonconforming finite element methods (FEMs) in linear elasticity named after Crouzeix-Raviart and Kouhia-Stenberg, which are robust with respect to the incompressible limit as the Lamé parameter lambda tends to infinity. The new result is a bestapproximation error estimate for the stress error in L2 up to data-oscillation terms. Even for very coarse shape-regular triangulations, two comparison results assert that the errors of the nonconforming FEM are equivalent to that of the conforming first-order FEM. The explicit role of the parameter lambda in those equivalence constants leads to an advertisment of the robust and quasi-optimal Kouhia-Stenberg FEM in particular for non-convex polygons. The proofs are based on conforming companions, a new discrete Helmholtz decomposition, and a new discrete-plus-continuous Korn inequality for Kouhia-Stenberg finite element functions. Numerical evidence strongly supports the robustness of the nonconforming FEMs with respect to the incompressibility locking and with respect to singularities and underlines that the dependence of the equivalence constants on lambda in the comparison of conforming and nonconforming FEMs cannot be improved. This work therefore advertises the Kouhia-Stenberg FEM as a first-order robust discretisation in linear elasticity in the presence of Neumann boundary conditions.


Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), P-2014-20