The purpose of this article is to derive and analyze new discrete mixed approximations for linear elasticity problems with weak stress symmetry. These approximations are based on the application of enriched versions of classic Poisson-compatible spaces, for stress and displacement variables, and/or on enriched Stokes-compatible space configurations, for the choice of rotation spaces used to weakly enforce stress symmetry. Accordingly, the stress space has to be adapted to restore stability. Such enrichment procedures are done via space increments with extra bubble functions, which have their support on a single element (in the case of H 1-conforming approximations) or with vanishing normal components over element edges (in the case of H(div)-conforming spaces). The advantage of using bubbles as stabilization corrections relies on the fact that all extra degrees of freedom can be condensed, in a way that the number of equations to be solved and the matrix structure are not affected. Enhanced approximations are observed when using the resulting enriched space configurations, which may have different orders of accuracy for the different variables. A general error analysis is derived in order to identify the contribution of each kind of bubble increment on the accuracy of the variables, individually. The use of enriched Poisson spaces improves the rates of convergence of stress divergence and displacement variables. Stokes enhancement by bubbles contributes to equilibrate the accuracy of weak stress symmetry enforcement with the stress approximation order, reaching the maximum rate given by the normal traces (which are not affected).

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In this paper, we propose and analyze the extrapolation methods and asymptotically exact a posterior error estimates for eigenvalues of the Morley element. We establish an asymptotic expansion of eigenvalues, and prove an optimal result for this expansion and the corresponding extrapolation method. We also design an asymptotically exact a posterior error estimate and propose new approximate eigenvalues with higher accuracy by utilizing this a posteriori error estimate. Finally, several numerical experiments are considered to confirm the theoretical results and compare the performance of the proposed methods.

In this article, a family of H 2 -nonconforming finite elements on tetrahedral grids is constructed for solving the biharmonic equation in 3D. In the family, the P ℓ polynomial space is enriched by some high order polynomials for all ℓ ≥ 3 and the corresponding finite element solution converges at the optimal order ℓ − 1 in H 2 norm. Moreover, the result is improved for two low order cases by using P6 and P7 polynomials to enrich P4 and P5 polynomial spaces, respectively. The optimal order error estimate is proved. The numerical results are provided to confirm the theoretical findings.2000 Mathematics Subject Classification. 65N15, 65N30S. Key words and phrases. nonconforming H 2 element, finite element method, biharmonic problem, tetrahedral grid.