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A new semidiscrete finite element method for the solution of second order nonlinear parabolic boundary value problems is formulated and analyzed. The test and trial spaces consist of discontinuous piecewise polynomial functions over quite general meshes with interelement continuity enforced approximately by means of penalties. Optimal order error estimates in energy and L²-norms are stated in terms of locally expressed quantities. They are proved first for a model problem and then in general.
Douglas N. Arnold (1982) studied this question.