L^2 best-approximation of the elastic stress in the Arnold-Winther FEM

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 stability, compatibility and efficiency conditions. 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 problems. The result has consequences for nonlinear approximation classes related to adaptive mixed FEMs.