Adventures in adaptivity

In this work, we compare and contrast a few finite element h-adaptive and hp-adaptive algorithms. We test these schemes on three example PDE problems and we utilize and evaluate an a posteriori error estimate. In the process, we introduce a new framework to study adaptive algorithms and a posteriori error estimators. Our innovative environment begins with a solution u and then uses interpolation to simulate solving a corresponding PDE. As a result, we always know the exact error and we avoid the noise associated with solving. Using an effort indicator, we evaluate the relationship between accuracy and computational work. We report the order of convergence of different approaches. And we evaluate the accuracy and effectiveness of an a posteriori error estimator.