Multipoint stress mixed finite element methods for elasticity on cuboid grids

TL;DR

This paper introduces two novel multipoint stress mixed finite element methods for linear elasticity on cuboid grids, achieving stable, efficient, and accurate solutions with superconvergence properties.

Contribution

The paper develops two new multipoint stress mixed finite element methods with local stress elimination and superconvergence analysis for elasticity on cuboid grids.

Findings

Methods are stable and have first-order convergence.

Displacement exhibits second-order superconvergence at cell centers.

Numerical results confirm theoretical error estimates.

Abstract

We develop multipoint stress mixed finite element methods for linear elasticity with weak stress symmetry on cuboid grids, which can be reduced to a symmetric and positive definite cell-centered system. The methods employ the lowest-order enhanced Raviart-Thomas finite element space for the stress and piecewise constant displacement. The vertex quadrature rule is employed to localize the interaction of stress degrees of freedom, enabling local stress elimination around each vertex. We introduce two methods. The first method uses a piecewise constant rotation, resulting in a cell-centered system for the displacement and rotation. The second method employs a continuous piecewise trilinear rotation and the vertex quadrature rule for the asymmetry bilinear forms, allowing for further elimination of the rotation and resulting in a cell-centered system for the displacement only. Stability and error analysis is performed for both methods. For the stability analysis of the second method, a new auxiliary H-curl conforming matrix-valued space is constructed, which forms an exact sequence with the stress space. A matrix-matrix inf-sup condition is shown for the curl of this auxiliary space and the trilinear rotation space. First-order convergence is established for all variables in their natural norms, as well as second-order superconvergence of the displacement at the cell centers. Numerical results are presented to verify the theory.