Energy: Technology, Markets, Networks

Mathematical Topics

Analysis of Partial Differential Equations and Evolutionary Equations

Partial differential equations offer a powerful and versatile framework for the continuum description of phenomena in nature and technology with complex coupling and dependencies. At the Weierstrass Institute this research has three essential focuses: (a) Rigorous mathematical analysis of general evolution equations in terms of existence, uniqueness and regularity of different types of solutions, (b) Development of variational approaches using the toolbox of the calculus of variations, (c) Regularity results for solutions of elliptic and parabolic partial differential equations. [>> more]

Direct and inverse problems for the Maxwell equations

The work is focussed on models for inductive heating of steel and for light scattering by periodic surface structures. For this the quasi-stationary Maxwell equation is coupled with nonlinear partial differential equations and the timeharmonic Maxwell equation is combined with special radiation conditions, respectively. The convergence of numerical methods and several inverse promblems are analyzed. [>> more]

Free boundary problems for partial differential equations

Free boundary problems for partial differential equation describe problems such that a partial differential equation is considered on a domain depending on the solution to the equation. [>> more]

Functional analysis and operator theory

At WIAS, functional analysis and operator theory are related, in particular, to problems of partial differential equations and evolutions equations as well as to analysis of multiscale, hybrid, and rate-independent models. [>> more]

Modeling, analysis and numerics of phase field models

A diffuse phase field model is a mathematical model for describing microstructural phenomena and for predicting morphological evolution on the mesoscale. It is applied to a wide variety of material processes such as solidification, coarsening in alloys, crack propagation and martensitic transformations. [>> more]

Multi-scale modeling, asymptotic analysis, and hybrid models

To understand the interplay between different physical effects one often needs to consider models involving several length scales. The aim in this mathematical topic is the derivation of effective models for the efficient description of the processes. The understanding of the transfer between different scales relies on mathematical methods such as homogenization, asymptotic analysis, or Gamma convergence. The generated effective models are coupled partial differential equations combining volume and interfacial effects. [>> more]

Numerical Methods for PDEs with Stochastic Data

Models of real-world phenomena inevitably include uncertainties which influence the solutions in a nonlinear way. Numerical methods for PDE with stochastic data enable to quantify such uncertainties of the solution in dependence of the stochastic input data. Due to the high problem complexity, modern compression techniques are mandatory. [>> more]

Numerical methods for problems from fluid dynamics

A main research field is the development, analysis, improvement and application of numerical methods for equations coming from CFD. The spatial discretization of the equations is based on finite element and finite volume methods. A focus of research is on so-called physically consistent methods, i.e., methods where important physical properties of the continuous problem are transferred to the discrete problem. [>> more]

Optimal control of partial differential equations and nonlinear optimization

Many processes in nature and technics can only be prescribed by partial differential equations,e.g. heating- or cooling processes, the propagation of acoustic or electromagnetic waves, fluid mechanics. Additionally to challenges in modeling, in various applications the manipulation or controlling of the modeled system is also of interest in order to obtain a certain purpose... [>> more]

Statistical inverse problems

In many applications the quantities of interest can be observed only indirectly, or they must be derived from other measurements. Often the measurements are noisy and the reconstruction of the quantities of interest from noisy measurements is unstable. [>> more]

Stochastic Optimization

Stochastic Optimization in the widest sense is concerned with optimization problems influenced by random parameters in the objective or constraints. [>> more]

Systems of partial differential equations: modeling, numerical analysis and simulation

The mathematical modelling of many scientific and technological problems leads to (initial) boundary value problems with systems of partial differential equations (PDEs). [>> more]

Archive

Further mathematical topics where the institute has expertise in:

Algorithms for the generation of 3D boundary conforming Delaunay meshes

This work is motivated by the needs of the numerical solution of partial differential equations by finite element or finite volume methods. In order to apply these methods, a given domain has to be subdivided into a number of cells with simple geometry. The quality of this subdivision affects the accuracy and convergence of the method. Boundary conforming Delaunay meshes with good quality are the prerequisite for the construction of the Voronoi-box based finite volume method. This method allows to carry over important qualitative properties from the continuous problem to the discretized one. The project is devoted to the construction of boundary conforming Delaunay meshes for three-dimensional domains. [>> more]