Adaptive Mesh Refinement for High-Resolution Finite Element Schemes

New a posteriori error indicators based on edgewise slope-limiting are presented. The L2-norm is employed to measure the error of the solution gradient in both global and element sense. A second order Newton-Cotes formula is utilized in order to decompose the local gradient error from a P1-finite element solution into a sum of edge contributions. The gradient values at edge midpoints are interpolated from the two adjacent vertices. Traditional techniques to recover a (superconvergent) nodal gradient from the consistent finite element gradients are reviewed. The deficiencies of standard smoothing procedures – global L2-projection and the Zienkiewicz-Zhu patch recovery – as applied to non-smooth solutions are illustrated for simple academic configurations. The recovered gradient values are corrected by applying a slope limiter edge-by-edge so as to satisfy geometric constraints. The direct computation of slopes at edge midpoints by means of limited averaging of adjacent gradient values is proposed as an inexpensive alternative. Numerical tests for various solution profiles in one and two space dimensions are presented to demonstrate the potential of this postprocessing procedure as an error indicator. Finally, it is used to perform adaptive mesh refinement for compressible inviscid flow simulations.