A low-order finite element method for three dimensional linear elasticity problems with general meshes

The paper is concerned with a low-order finite element method, namely the staggered cell-centered finite element method, which has been proposed and analyzed in Ong et al. (2015) for two-dimensional compressible and nearly incompressible linear elasticity problems. In this work, we extend the results to the three-dimensional case and focus on the creating of the meshes. In particular, from a general primal mesh M, we construct a polygonal dual mesh M∗ and its submesh M∗∗ in a way such that each dual control volume of M∗ corresponds to a primal vertex and is a union (macro-element) of some fixed number of adjacent tetrahedral elements of M∗∗. The displacement is approximated by piecewise trilinear functions on the subdual mesh M∗∗ and the pressure by piecewise constant functions on the dual mesh M∗. As for two-dimensional case, such construction of the meshes and approximation spaces satisfies the macroelement condition, which implies stability and convergence of the scheme. Numerical experiments are carried out to investigate the performance of the proposed method on various mesh types.