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The main focus of our group is on the developement of efficient numerical methods for partial differential equations (   more).

In particular we are concerned with:

  A Posteriori Error Estimation and Adaptivity

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.
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  Model Reduction with Reduced Basis Methods

Reduced basis (RB) methods have been developed during the last decades with the aim to make known discretization methods, like Finite Element methods, usable for parametrized applications. These are applications for which not only a single simulation has to be performed, but solutions for a range of different configurations of the same problem are desired.  [...]

  Numerical Multiscale Algorithms

In general, Multiscale Algorithms or Multiscale Methods partially decouple finescale problems into macroscale portions and into microscale portions. In the macroscale parts, only the macroscopic behaviour of the solution is regarded. For instance small holes in a porous medium are not taken into account. On this scale, it is pretended they were not there. We can say that no microscopic oscillations will occur in macroscale approximations.  [...]

  Model Reduction for Bayesian Inverse Problems

Bayesian inversion describes the process of inverting a model with prior knowledge about the associated parameters to aquire a distribution for each parameter. Usually realistic models of complicated dynamic systems, in example brain connectivity, require large amounts of parameters, for what often some previous knowledge due to experiments or a hypothesis is given. The model reduction for bayesian inverse problems focusses on reducing the dimensionality of the parameter (and state) spaces of these models, to minimize the computational expense of the inversion procedure, given some experimental data.

 
 

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  DUNE

DUNE, the Distributed and Unified Numerics Environment is a modular toolbox for solving partial differential equations with grid-based methods. It supports easy discretization using methods like Finite Elements, Finite Volumes, and also Finite Differences.  DUNE Homepage

  DUNE-FEM

DUNE-FEM is a DUNE module which defines interfaces for implementing discretization methods like Finite Element Methods (FEM) and Finite Volume Methods (FV) and Discontinuous Galerkin Methods (DG). The module is developed in cooperation with the group of Dietmar Kröner Freiburg.  DUNE-FEM Homepage

  ALUGrid

For numerical computation in the field of fluid dynamics a flexible implementation of a discretization grid is needed. The ALUGrid Library provides both hexahedral and tetrahedral grids which can be locally adapted and when used for parallel computations the decomposition of the domain can be recomputed.  ALUGrid Homepage

  GRAPE

GRAPE is a package for mathematical visualization. It has been particularly effective in the fields of differential geometry and continuum mechanics. But it will probably help to understand any other problem involving the numerics of partial differential equations or the need of advanced three-dimensional computer graphics.  GRAPE Homepage

  GFLOW

Gflow is a numerical toolbox for the approximation of density driven and two phase flow problems in porous media. It provides adaptive triangular meshes in two space dimensions. 
Developer M. Ohlberger

  RBmatlab

RBmatlab is a MATLAB library containing all our reduced simulation approaches for linear and nonlinear, affine or arbitrarily parameter dependent evolution problems with finite element, finite volume or local discontinuous Galerkin discretizations.   RBmatlab Homepage

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