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We investigate the piecewise linear nonconforming Crouzeix–Raviart 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 emphasize that we do not impose either shape-regular or maximum-angle condition during mesh partitioning. Numerical results confirm the results that we obtained.

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