A mixed finite element method for nearly incompressible elasticity and Stokes equations using primal and dual meshes with quadrilateral and hexahedral grids

•Optimal scheme is given for nearly incompressible elasticity and Stokes equations.•Primal mesh for the displacement and dual mesh for the pressure.•The earlier results are extended to quadrilateral and hexahedral meshes.•Optimal error estimates are proved.•Displacement-based formulation is derived.

We consider a mixed finite element method for approximating the solution of nearly incompressible elasticity and Stokes equations. The finite element method is based on quadrilateral and hexahedral triangulation using primal and dual meshes. We use the standard bilinear and trilinear finite element space enriched with element-wise defined bubble functions with respect to the primal mesh for the displacement or velocity, whereas the pressure space is discretized by using a piecewise constant finite element space with respect to the dual mesh.