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Nonconforming finite elements of higher order satisfying a new compatibility condition

Abstract For a general diffusion-convection-reaction equation, we analyze families of nonconforming finite elements of arbitrary order on a sequence of multilevel grids consisting of quadrilaterals or hexahedra. We prove existence and uniqueness of the discrete solution and optimal order of convergence in the broken H 1-seminorm and the L 2-norm. The novelty of our approach is that a new integral compatibility condition of the discrete functions across the element faces is introduced such that it can be solely treated on the reference element once for all faces of the grid. A numerical comparison between conforming and nonconforming discretizations will be given in the three-dimensional case.