Multipoint Stress Mixed Finite Element Methods for Elasticity on Distorted Quadrilateral Grids

Abstract We develop multipoint stress mixed finite element methods for linear elasticity with weakly enforced stress symmetry on distorted quadrilateral grids, which can be reduced to positive definite cell-centered systems. The methods utilize the lowest-order Brezzi–Douglas–Marini finite element spaces for the stress and employ vertex quadrature rules to localize the interaction of degrees of freedom. This approach allows for 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 the rotation. The second method employs a continuous piecewise bilinear rotation, enabling further elimination of the rotation and resulting in a cell-centered system for the displacement only. The methods utilize a non-symmetric vertex quadrature rule for the stress bilinear form and both non-symmetric and symmetric vertex quadrature rules for the asymmetry bilinear forms. Stability and error analysis are performed for both methods. First-order convergence is established for all variables in the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {L^{2}} -norm. Numerical results are presented that verify the theoretical results.