Enriched two dimensional mixed finite element models for linear elasticity with weak stress symmetry

Abstract

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).

NewH(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes

“…Proof. By means of Theorem 4.2, we derive the error estimates for the equivalent MFEM-WS(E γ ) formulation, for which the following estimates in terms of interpolation errors hold (see [19] or the references therein):…”

“…, where b K = λ 1 λ 2 λ 3 are bubble functions defined by the barycentric coordinates λ i of K, and Q CR k (K) = P k−1 (K). [19]:…”

“…The stability of some particular scenarios were recently analyzed in [19] for single-scale triangular or quadrilateral meshes (i.e. h in = h sk = h), and internal polynomial degrees k in = k sk + 1 or k in = k sk + 2 (see Table 1).…”

“…Divergenceconstraint is obtained by forming rows of tensors and displacements with Poisson-compatible FE pairs widely used for flux and potential approximations in mixed methods. Concerning the enforcement of the Stokesconstraint, we extend the methodology proposed in [19] to construct new stable Stokes-compatible pairs: the pair used for stability analysis at the coarsest single-scale space setting is incremented with extra refined composite bubble terms for the velocity in order to restore stability when using enlarged pressure spaces, in the spirit of the methodology suggested in [10]. Classical tools are applied to the equivalent two-scale MFEM-WS framework, guiding the error analysis of the MHM-WS solutions.…”

“…Since k in ≥ 2, the property U rm ⊂ U γin , required by the MHM-WS(E BDMγ ) scheme, holds. The particular cases based on the conformal coarse partitions T Ωi h sk (h in = h sk ), and polynomial increment k in = k sk + n, for n = 1, 2, correspond to the FE spaces denoted by E BDM + γ sk and E BDM ++ γ sk considered in [19]. For these methods, the composite rotation space of piecewise polynomials Q BDMγ in (Ω i ) := P kin−1 (T Ωi h sk ) is stable.…”