On the nested refinement of quadrilateral and hexahedral finite elements and the affine approximation

Summary. It is shown in this paper that the optimal approximation property of bilinear or trilinear finite elements would be retained when the affine mapping is used to replace the Q ₁ mapping on each element, if the grids are refined nestedly. The new method truncates the quadratic and cubic terms in reference mappings and produces constant Jacobians and Jacobian matrices. This would avoid a shortcoming of the quadrilateral and hexahedral elements where the integrals of rational functions have to be computed or approximated. Numerical tests verify the analysis.