Pointwise a posteriori error estimates for elliptic problems on highly graded meshes

Source abstract

Pointwise a posteriori error estimates are derived for linear second-order elliptic problems over general polygonal domains in 2D. The analysis carries over regardless of convexity, accounting even for slit domains, and applies to highly graded unstructured meshes as well. A key ingredient is a new asymptotic a priori estimate for regularized Green’s functions. The estimators lead always to upper bounds for the error in the maximum norm, along with lower bounds under very mild regularity and nondegeneracy assumptions. The effect of both point and line singularities is examined. Three popular local estimators for the energy norm are shown to be equivalent, when suitably interpreted, to those introduced here.