A Posteriori Error Estimates and Adaptivity
Motivation
Many physical, chemical or biological processes can be described by means of partial differential equations. If the underlying processes exhibit a nonlinear dynamic, an analysis and prediction of the complex behaviour is often only possible by solving the partial differential equations numerically. That is why the design of efficient numerical schemes is central to our research. In spite of the increasing computational capacities many problems are still only solvable with severe simplifications. This is due to the high complexity of the problems. One may think of e.g. the weather forecast and the huge amount of processes which have to be considered in this context. This example also makes clear, why it is of great importance to design efficient numerical schemes, i.e. that the numerical methods lead with least possible effort to the best possible solution: The weather forecast for yesterday is not very interesting at all.
Adaptive modelling and model reduction, adaptive grid refinement, multiscale methods and parallelisation are important methods to increase the efficiency of numerical schemes. But the quality of the numerical approximation is essential, too, meaning that we want to know how good the computed solution is in comparison to the exact one. In this context a posterior error estimates play a crucial role. They allow to estimate the approximation error without using the exact solution of the partial differential equation, which is generally unknown. In short we aim at developing numerical methods, which are as exact as possible and as fast as possible at the same time.
Furthermore, a posteriori error estimates can be used to design efficient schemes. One example is adaptive grid refinement. Here, we refine a computational grid only in regions where the local a posteriori error estimate is high. If the estimated error is small, the grid might even be coarsened. In this way, we can save computational costs and get more efficient numerical schemes.
Focus of our Work
The development of effizient adaptive numerical methods for convection dominated flow problems is one of the main focuses of the group. Efficiency is obtained by using modern higher order Finite Volume and Discontinuous Galerkin approximations on locally adaptive meshes with dynamic load balancing on parallel computers.
The local mesh adaption algorithms are derived from rigorous a posteriori error estimates. Significant theoretical contributions have been obtained for hyperbolic and convection dominated parabolic equations.
The resulting numerical schemes were successfully applied to flow problems in porous media, such as miscible and immiscible displacement, single or two phase flow with species transport as a model for biodegradation or gas and water flow with species transport and reaction as a model for PEM fuel cells.
Selected Publications
