A posteriori estimators for the h – p version of the finite element method in 1D Original Research Article

We consider the h – p finite element method for elliptic problems in one dimension. The strategy for choosing an h - or p -enrichment for an element which is subject to a refinement in an adaptive method is not well understood; in particular, this is the most important open problem associated with h – p -refinement. A mathematical derivation of a posteriori estimates for the error and the error reduction corresponding to an h - or p -refinement of elements is presented. The estimation of the error reduction uses the solution of local problems, the estimate is bounded by the true reduction from below and above with constants only depending on the differential operator. Based on these a posteriori estimates an adaptive algorithm is derived. Numerical results show the efficiency of the estimators for several problem classes. For the xα model singularity the a priori known optimal h – p mesh is obtained by this algorithm.