A posteriori error estimators

One claim of the project is to develop a posteriori error estimators for high order elements. For convenience, let us consider the Poisson model problem $ -\Delta u = f$. Our approach is to solve the primal problem on a mesh with the polynomial orders prescribed for all elements, faces, and edges. Then, solve also the dual problem $ -\operatorname{div}p = f$, $ p = \nabla u$ by a mixed method (using high order $ H(\operatorname{div})$ and $ L_2$ elements). The difference

$\displaystyle \Vert u_{hp} - p_{hp} \Vert _{L_2} $

is an upper bound for the error. Instead of computing the flux by solving a global problem, it could be computed by a local postprocessing method involving $ H(\operatorname{div})$ quasi-interpolation operators. First results are available for 2D problems. Figure 6 shows a model consiting of a small square (of conductivity 100) embedded into a larger one (of conductivity 1). The left picture draws the absolut value of the flux showing singularities at the vertices. We have chosen a mesh with a priori strong refinement towards the singularities. The right picture shows the polynomial distribution obtained from an adaptive process driven by the primal-dual error estimator. More work in this direction is in progress.

Figure 6: Flux singularity and polynomial distribution
Image fluxsing Image poldist