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 best-approximation error estimate for the stress error in |$L^2$| 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 those of the conforming first-order FEM. The explicit role of the parameter |$\lambda$| in those equivalence constants leads to an advertisement of the robust and quasi-optimal Kouhia–Stenberg FEM, in particular for nonconvex 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 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 discretization in linear elasticity in the presence of Neumann boundary conditions.