Parallel space-time hp adaptive discretization scheme for parabolic problems

We introduce an integration scheme for parabolic problems. Our parallelizable method uses adaptive hp-finite elements in space, and finite differences in time. The strategy can also be combined with a classical finite element method parallelization technique based on domain decomposition. We verified the performance of our method against two different benchmarks, in both two-dimensional (model problem on an L-shaped domain) and three-dimensional (Pennes bioheat equation) settings. Results show a significant speedup in computational time when compared with the sequential version of the algorithm. Moreover, we develop a mathematical framework to analyze similar schemes which include hp spatial adaptivity. Our framework describes error propagation rigorously, and as such allows to analyze convergence properties of these mixed methods.