A locally conservative stress recovery technique for continuous Galerkin FEM in linear elasticity

The standard continuous Galerkin finite element method (FEM) is a versatile and well understood method for solving partial differential equations. However, one shortcoming of the method is lack of continuity of derivatives of the approximate solution at element boundaries. This leads to undesirable consequences for a variety of problems such as a lack of local conservation. A two-step postprocessing technique is developed in order to obtain a local conservation from the standard continuous Galerkin FEM on a vertex centered dual mesh relative to the finite element mesh when applied to displacement based linear elasticity. The postprocessing requires an auxiliary fully Neumann problem to be solved on each finite element where local problems are independent of each other and involve solving two small linear algebra systems whose sizes are 3-by-3 when using linear finite elements on a triangular mesh for displacement based linear elasticity. The postprocessed stresses then satisfy local conservation on the dual mesh. An a priori error analysis and numerical simulations are provided.