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Spurious oscillations in computing microstructures

The purpose of this paper is to discuss some numerical difficulties in a two-dimensional mathematical model for displacive phase transformations. The usual approach to discretizing this model uses finite elements. Some previous work has computed certain microstructures by using a uniform square mesh discretization together with a form of derivative averaging. The overall scheme is equivalent to using bilinear elements with a one point quadrature approximation to discretize the energy functional E below. The resulting functional will be denoted by Eh. Although the square mesh respects a key symmetry property of the energy functional, when using this technique we have found a computational 'skewing' problem. This problem has a deleterious effect on the numerical solutions since it generates a grid scale spurious oscillation. In this paper, we will study this skewing phenomenon and attempt to show its effect on the computed microstructure.