Research

collaborators

L. Banjai (MPI Leipzig)
C. Carstensen (HU Berlin)
A. Målqvist (U Uppsala)
H. Rabus (HU Berlin)
S. A. Sauter (U Zurich)
M. Schedensack (HU Berlin)
A. Schröder (HU Berlin)

Computational Homogenization

The numerical solution of partial differential equations with rough data is a challenging part in material simulations and geophysical applications such as ground water flow, oil recovery modeling, and CO2 sequestration. Standard finite element methods are unable to capture the solution - neither its microscopic nor the macroscopic behavior - unless the meshwidth is chosen fine enough (that is smaller than the smallest scale).
We are interested in the design of new methods that model macroscopic quantities sufficiently accurate without solving the global problem on the microscopic scale.

related publications

[19, 9](rough coefficient), [16] (rough interfaces), [13, 11] (rough domain)

Composite Material Modeling

Composite materials (composites) are engineered multiphase materials with physical properties better than what their constituents have on their own. Manipulation of the characteristics and relative volumes of the constituents allow one to equip these materials with certain targeted physical properties. Hence, these materials are of crucial importance for many application areas in high technology.
By the development of efficient and reliable numerical methods for the PDE models of certain material responses, this research aims to understand how certain material properties (conductivity, permeability, etc.) depend on controllable variables.

further information

MATHEON project C33 "Modeling and Simulation of Composite Materials"
[project webpage]

Adaptive Algorithms

Nowadays, a posteriori error estimation is commonly used for the control of adaptive mesh refinement or, in general, the control of the adaptive enrichment of finite element spaces. We develop estimators and refinement/enrichment strategies with the aim to minimize computational complexity. We emphasize, however, that for many singularly perturbed or parameter dependent problems the condition "the mesh width has to be sufficiently small" still arises. The generation of optimal initial meshes is, hence, of utmost importance and we develop new concepts for this purpose.

related publications

[21], [20], [18], [15], [10], [3]

Parallel Algorithms

The success of an algorithm strongly depends on its ability to advantage of the available computational resources, that is, its ability for parallelism. We develop new methodologies for parallelism in the context of computational partial differential equations. Our decoupling approaches (in space & time) allow for distributed computing with minimal communication.

related publications

[12], [19], [9]