A lowest-order weak Galerkin method for linear elasticity

The lowest-order weak Galerkin (WG) method is considered for linear elasticity based on the displacement formulation. The new method approximates the displacement using piecewise constant vector functions both inside and on the boundary of each mesh element and its bilinear form does not require a stabilization term for the existence and uniqueness of the solution. A-priori error estimates of optimal order in the discrete H-1- and L-2-norms for the displacement are proved when the solution is smooth. The error estimates are independent of the Lame constant A, thus the performance of the new method does not deteriorate as the elastic material becomes incompressible. Further, a simple post processing technique to obtain a numerical approximation of the stress is presented. A careful error analysis reveals that the L-2-norm error in the stress is also optimal and independent of A. Several numerical experiments confirm the locking-free property and optimal convergence rates of the new method. (C) 2018 Elsevier B.V. All rights reserved.