Crouzeix-Raviart and Raviart-Thomas finite-element error analysis on anisotropic meshes violating the maximum-angle condition

We investigate the piecewise linear nonconforming Crouzeix-Raviar and the lowest order Raviart-Thomas finite-element methods for the Poisson problem on three-dimensional anisotropic meshes. We first give error estimates of the Crouzeix-Raviart and the Raviart-Thomas finite-element approximate problems. We next present the equivalence between the Raviart-Thomas finite-element method and the enriched Crouzeix-Raviart finite-element method. We emphasise that we do not impose either shape-regular or maximum-angle condition during mesh partitioning. Numerical results confirm the results that we obtained.