Optimal adaptivity for space-time methods

Time-dependent partial differential equations arise as typical models in many scientific and engineering applications, e.g., heat conduction and diffusion, changing in time processes in social and life sciences, etc. In general, these equations can only be solved approximately by numerical methods. The goal of the proposed research is to significantly improve the performance of numerical space-time methods. In contrast to time-stepping methods, which approximate the solution at some timepoints, space- time methods aim to approximate the solution as a whole in the so-called space-time cylinder and treat time as yet another dimension. To this end, the space-time cylinder is partitioned into a four- dimensional mesh and a piecewise polynomial approximation to the solution is computed. Refinement of the underlying mesh leads to an increase of accuracy. However, in general, the solution exhibits singularities, which have to be resolved appropriately. In order to detect these singularities, one requires a-posteriori computable error estimators that locally measure the quality of the current approximation. The development and mathematical analysis of such estimators for time-dependent problems is one of the key tasks of the proposed research. In the next step, we will then use these estimators within an adaptive algorithm that automatically refines the underlying mesh at those points, where it is necessary. Our main goal is to mathematically prove that the adaptive algorithm leads to optimal convergence of the generated approximations towards the exact solution, i.e., the algorithm leads to the best possible convergence behavior. Finally, all theoretical findings will be implemented for simple model problems and provided to the academic public to underline the practical impact of the developed mathematical concepts and results. In the long run, the research might even result in specially developed software for more complicated time-dependent problems as it has been the case for new a-posteriori estimators and adaptive algorithms for time-independent problems that were developed in theoretical studies. Indeed, they found their way relatively fast to academic (e.g., iFEM, Alberta, PLTMG, Netgen/NGSolve, BEM++) and commercial (e.g., FEMLAB) software packages. This will allow to substitute costly experiments with prototypes by reliable and well-performing simulations providing approximations at an accuracy, which is yet out of reach for existing numerical schemes.

Time-dependent partial differential equations arise as typical models in many scientific and engineering applications, e.g., heat conduction and diffusion, changing-in-time processes in social and life sciences, etc. In general, these equations can only be solved approximately by numerical methods. The goal of the proposed research was to significantly improve the performance of numerical space-time methods. In contrast to time-stepping methods, which approximate the solution at some timepoints, space-time methods aim to approximate the solution as a whole in the so-called space-time cylinder and treat time as yet another dimension. To this end, the space-time cylinder is partitioned into a four-dimensional mesh and a piecewise polynomial approximation to the solution is computed. Refinement of the underlying mesh leads to an increase of accuracy. However, in general, the solution exhibits singularities, which have to be resolved appropriately. In order to detect these singularities, one requires a-posteriori computable error estimators that locally measure the quality of the current approximation. In the frame of my research, I developed and analyzed such estimators for time-dependent problems. In the next step, I used these estimators within an adaptive algorithm that automatically refines the underlying mesh at those points where it is necessary. I was able to prove mathematically that this algorithm always converges towards the exact solution, i.e., it achieves any desired given accuracy after a certain runtime. Finally, the theoretical findings were implemented for simple model problems and provided to the academic public to underline the practical impact of the developed mathematical concepts and results. In the long run, the research might even result in specially developed software for more complicated time-dependent problems as it has been the case for new a-posteriori estimators and adaptive algorithms for time-independent problems that were developed in theoretical studies. Indeed, they found their way relatively fast to academic (e.g., iFEM, Alberta, PLTMG, Netgen/NGSolve, BEM++) and commercial (e.g., FEMLAB) software packages. This will allow to substitute costly experiments with prototypes by reliable and well-performing simulations providing approximations at an accuracy, which is yet out of reach for existing numerical schemes.