A multipoint stress mixed finite element method for elasticity on quadrilateral grids. Issue 3 (10th November 2020)

• Record Type: Journal Article
• Authors: Ambartsumyan, Ilona; Khattatov, Eldar; Nordbotten, Jan Martin; Yotov, Ivan
• Is Part Of: Numerical methods for partial differential equations. Volume 37:Issue 3(2021)
• Page Start: 1886
• Page End: 1915
• Publication Date: 2020-11-10
• Subjects: elasticity -- mixed finite element -- multipoint stress; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22624
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract: We develop a multipoint stress mixed finite element method for linear elasticity with weak stress symmetry on quadrilateral grids, which can be reduced to a symmetric and positive definite cell centered system. The method utilizes the lowest order Brezzi–Douglas–Marini finite element spaces for the stress and the trapezoidal quadrature rule in order to localize the interaction of degrees of freedom, which allows for local stress elimination around each vertex. We develop two variants of the method. The first uses a piecewise constant rotation and results in a cell‐centered system for displacement and rotation. The second uses a continuous piecewise bilinear rotation and trapezoidal quadrature rule for the asymmetry bilinear form. This allows for further elimination of the rotation, resulting in a cell‐centered system for the displacement only. Stability and error analysis is performed for both methods. First‐order convergence is established for all variables in their natural norms. A duality argument is employed to prove second order superconvergence of the displacement at the cell centers. Numerical results are presented in confirmation of the theory.