Description:
Description of the project: Unlike classical time-stepping methods, which discretize a time-dependent partial differential equation (PDE) first in space and then in time or vice versa, space-time methods discretize the PDE as a whole, treating time as yet another dimension. Due to their ability for massive parallelization and their potential for adaptive mesh refinement in both space and time, space-time methods have become popular. Flexible adaptive mesh refinement is indeed crucial to efficiently approximate PDE solutions, as they often exhibit geometry- or data-induced local singularities. Nevertheless, adaptive space-time methods are still in their infancy.
In this project, we will investigate such methods for parabolic PDEs. In particular, we will develop computable a-posteriori error estimators which assess the accuracy of the numerical approximation. These estimators will be used within adaptive algorithms which automatically detect singularities and locally refine the space-time mesh to efficiently enhance accuracy. The project’s overall aim and breakthrough contribution is to prove, for the first time for time-dependent PDEs, optimal convergence of such adaptive mesh refinement algorithms with respect to computational cost. Together with suitable parallelization, this mathematically guarantees accuracies, at minimal cost, beyond the reach of current approaches.
To develop the estimators and prove optimal convergence of the resulting adaptive algorithms, we can draw on my combined expertise in the well-established a-posteriori and optimality analysis for stationary PDEs and in space-time methods in general. We will focus on linear model problems related to the heat equation on bounded and unbounded domains, using the finite element method (FEM), the boundary element method (BEM), and couplings of these two methods as discretization schemes.