A Stable Affine-Approximate Finite Element Method

Abstract

The notion of the affine figure closest to a given quadrilateral can be given a precise mathematical definition. The resulting figure is referred to as the equivalent parallelogram associated with a quadrilateral. Equipped with such a concept, it is then feasible to consider finite element approximations in which the original quadrilateral elements are replaced by their equivalent parallelograms. The idea is appealing, not least because of the resulting economy arising from computations performed on an element generated by an affine map. Furthermore, numerical experiments reported recently indicate that highly efficient and accurate schemes result when such a concept is combined with the enhanced strain method or the method of incompatible modes. The purpose of this work is to analyze finite element schemes resulting from approximation of quadrilaterals by their equivalent parallelograms. The focus is on low-order (bilinear) elements, and the analysis is carried out in the context of linear elasticity for standard approximations as well as for those which use enhanced strains. The affine approximation applies only to the element map, and the primary unknown (the displacement vector in the context of elasticity) is approximated by conventional piecewise bilinear functions. The analysis confirms convergence at the optimal rate, provided that the deviations of the quadrilaterals from their equivalent parallelograms are at most O(h).