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]

Interacting stochastic particle systems

In the mathematical modeling of many processes and phenomena in the Sciences and Technology one employs systems with many random particles and interactions. In this context, we define the term "particle systems" very broadly and also include point processes with percolation properties and random graph structures as well as Gibbs interactions. It also includes random movements of these particles, such as those that occur in spatial models for communication. At WIAS, many macroscopic properties of these particle systems are investigated that arise from the microscopic rules, such as phase transitions (condensation, percolation, crystallization) and critical properties such as rescaling limits. [>> 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 Inference

The term statistical inference summerizes methods to extract information from observed data in order to characterize properties in populations. This involves statistical modeling, estimation and uncertainty assessment of parameters, testing of hypothesis. [>> 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]

Variational methods

Many physical phenomena can be described by suitable functionals, whose critical points play the role of equilibrium solutions. Of particular interest are local and global minimizers: a soap bubble minimizes the surface area subject to a given volume and an elastic body minimizes the stored elastic energy subject to given boundary conditions. At WIAS, methids from the calcuus of variations are applied and further developed to solve problems in physics and technology such as continuum mechanics, quantum mechanics, and optimal control. [>> 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]

Magnetohydrodynamics

For the production of semiconductor crystals, electromagnetic fields are often used to produce heat by induction. Moreover, Lorentz forces can improve the melt motion during crystal growth processes. Their modeling leads to a system of coupled partial differential equations. [>> more]

Nonlinear kinetic equations

Kinetic equations describe the rate at which a system or mixture changes its chemical properties. Such equations are often non-linear, because interactions in the material are complex and the speed of change is dependent on the system size as well as the strength of the external influences. [>> more]