ADAPTIVE REFINEMENT FOR ARBITRARY FINITE-ELEMENT SPACES WITH HIERARCHICAL BASES

An adaptive refinement algorithm is presented and interpreted as the selective enrichment of a finite-element space through the hierarchical basis. Each element division corresponds exactly to the inclusion of a small number of new basis functions, while existing basis functions remain unchanged. Which bases to add, i.e., which divisions to perform, are determined so that those that make the largest contribution to the function are included first. The space of C0pth-degree piecewise polynomials over triangles are used for illustration, but the techniques apply to any family of function spaces that can be represented with a hierarchical basis. Numerical examples are presented to show that the technique can regain the optimal (smooth) order of convergence for the solution of partial differential equations with nonsmooth solutions.