A Novel Multipoint Stress Control Volume Method for Linear Elasticity on Quadrilateral Grids (Communications in Computational Physics, 2025)
In this paper, we develop a novel control volume method that is locallyconservative and locking-free for linear elasticity problem on quadrilateral grids. Thesymmetry of stress is weakly imposed through the introduction of a Lagrange multiplier. As such, the method involves three unknowns: stress, displacement and rotation.To ensure the well-posedness of the scheme, a pair of carefully defined finite elementspaces is used for the stress, displacement and rotation such that the inf-sup condition holds. An appealing feature of the method is that piecewise constant functionsare used for the approximations of stress, displacement and rotation, which greatlysimplifies the implementation. In particular, the stress space is defined delicately suchthat the stress bilinear form is localized around each vertex, which allows for the local elimination of the stress, resulting in a cell-centered system. By choosing differentdefinitions of the space for rotation, we develop two variants of the method. In particular, the first method uses a constant function for rotation over the interaction region,which allows for further elimination and results in a cell-centered system involvingdisplacement only. A rigorous error analysis is performed for the proposed scheme.We show the optimal convergence for $L^2$-error of the stress and rotation. Moreover,we can also prove the superconvergence for $L^2$-error of displacement. Extensive numerical simulations indicate that our method is efficient and accurate, and can handleproblems with discontinuous coefficients.