The Approximation of Solutions of Elliptic Boundary-Value Problems via the p-Version of the Finite Element Method

The approximation theory developed in [11] is used to determine the piecewise polynomial approximability of solutions of elliptic problems on polygonal domains in R2 and polyhedra in R3. From these estimates, convergence orders for the p-version of the finite element method applied to such problems are readily obtained. The critical issue is the approximation of the singularities which occur at the nonsmooth parts of the domain boundaries. It is seen that the estimates of [11] involving the weighted Sobolev spaces Zsl are well suited for treating such singular functions, yielding directly the optimal approximation degree, up to an arbitrarily small ε. Numerical results for two problems from two-dimensional linear elasticity are also presented. The computations show that the predicted order of convergence is achieved even for low values of p. Moreover, in contrast to the usual h-version of the finite element method, the point at which the p-version enters the asymptotic range does not depend on problem parameters such as the Poisson ratio. Some practical implications of the p-version convergence orders for the solvability of elliptic problems with strong singularities are also discussed.