Can We Have Superconvergent Gradient Recovery Under Adaptive Meshes?

We study adaptive finite element methods for elliptic problems with domain corner singularities. Our model problem is the two-dimensional Poisson equation. Results of this paper are twofold. First, we prove that there exists an adaptive mesh (gauged by a discrete mesh density function) under which the recovered gradient by the polynomial preserving recovery (PPR) is superconvergent. Second, we demonstrate by numerical examples that an adaptive procedure with an a posteriori error estimator based on PPR does produce adaptive meshes that asatisfy our mesh density assumption, and the recovered gradient by PPR is indeed superconvergent in the adaptive process.

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