Interpolation error-based a posteriori error estimation for \(hp\)-refinement using first and second derivative jumps. (Q1419856)

This demanding paper presents relevant new results concerning the a posteriori error estimation for the \(hp\)-adaptive finite-element method. These results can be used to optimise the \(hp\)-adaptive refinement process. After introducing the reader into the treated subject, the needed function-analytical framework and assumptions are settled. In the following the Lobatto polynomials together with some facts concerning their interpolational properties are given. Using a Taylor like expansion of the interpolation error, relations between jumps of the first and second derivative of the interpolant and the finite-element-solution, obtained with a \(p\)-order increased by one respectively increased by two, are derived. The strong assumption for these results, namely uniform meshes, are in the following alleviated to the assumption of uniform \(p\)-order. The paper includes also computational results comparing the newly derived estimates with those obtained by earlier approaches together with a discussion of the impact of the presented results on the \(hp\)-refinement strategy.