A locking-free <i>P</i>0 finite element method for linear elasticity equations on polytopal partitions
Yujie Liu, Junping Wang
Abstract
Abstract This article presents a $P_0$ finite element method for boundary value problems for linear elasticity equations. The new method makes use of piecewise constant approximating functions on the boundary of each polytopal element and is devised by simplifying and modifying the weak Galerkin finite element method based on $P_1/P_0$ approximations for the displacement. This new scheme includes a tangential stability term on top of the simplified weak Galerkin to ensure the necessary stability due to the rigid motion. The new method involves a small number of unknowns on each element, it is user friendly in computer implementation and the element stiffness matrix can be easily computed for general polytopal elements. The numerical method is of second-order accurate, locking-free in the nearly incompressible limit, and ease polytopal partitions in practical computation. Error estimates in $H^1$, $L^2$ and some negative norms are established for the corresponding numerical displacement. Numerical results are reported for several two-dimensional and three-dimensional test problems, including the classical benchmark Cook’s membrane problem in two dimensions as well as some three-dimensional problems involving shear-loaded phenomena. The numerical results show clearly the simplicity, stability, accuracy and efficiency of the new method.