Reliable Residual-Based Error Estimation for the Finite Cell Method

In this work, the reliability of a residual-based error estimator for the Finite Cell method is established. The error estimator is suitable for the application of hp-adaptive finite elements and allows for Neumann boundary conditions on curved boundaries. The reliability proof of the error estimator relies on standard arguments of residual-based a posteriori error control, but includes several modifications with respect to the error contributions associated with the volume residuals as well as the jumps across inner edges and Neumann boundary parts. Important ingredients of the proof are Stein's extension theorem and a modified trace theorem which estimates the norm of the trace on (curved) boundary parts in terms of the local mesh size and polynomial degree. The efficiency of the error estimator is also considered by discussing an artificial example which yields an efficiency index depending on the mesh-family parameter h. Numerical experiments on more realistic domains, however, suggest global efficiency with the occurrence of a large overestimation on only few cut elements. In the experiments the reliability of the error estimator is demonstrated for h- and p-uniform as well as for hp-geometric and h-adaptive refinements driven by the error estimator. The practical applicability of the error estimator is also studied for a 3D problem with a non-smooth solution.