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]

Analysis of ordinary and partial stochastic differential equations

An ordinary differential equation is often used to model the movement of a particle. Similarly, partial differential equation can be used to describe the evolution of a total of trajectories of particles. It is natural to add randomness to such models: sometimes because this is a more realistic description which takes into account random noise, sometimes because this randomness is fundamental to the model itself as is the case for financial markets. [>> 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]

Hysteresis operators and rate-independent systems

Time-dependent processes in physics, biology, and economics often exhibit a rate-independent input-output behavior. Quite often, such processes are accompanied by the occurrence of hysteresis phenomena induced by inherent memory effects. There are two methods to describe such processes at WIAS: rate independent systems and. hysteresis operators . [>> more]

Large deviations

The theory of large deviations, a branch of probability theory, provides tools for the description of the asymptotic decay rate of a small probability, as a certain parameter diverges or shrinks to zero. Examples are large times, low temperatures, large numbers of stochastic quantities, or an approximation parameter. This probabilistic theory is also indispensable in the treatment of a number of models in statistical physics, as it makes them accessible for analysis using variational techniques. Both theory and sophisticated applications in physics and chemistry are being investigated at WIAS. [>> 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]

Random geometric systems

Systems with many random components distributed in space (points, edges, graphs, trajectories, etc.) with many short- or long-range interactions are examined at the WIAS for their macroscopic properties. Particular attention is paid to the formation of particularly large structures in the system or other phase transitions. [>> 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]

Spectral theory of random operators

Spectra of random operators are used to describe physical processes and phenomena. Our research in this area focuses on the mathematical investigation of concentration phenomena of spectral states at the edge of the spectrum using probabilistic methods (Feynman-Kac formula, large deviations) and their effects on electrical conduction properties of a metal alloy described by the model. These investigations are combined with the analysis of the analogous probabilistic models (random walks in random potential) and thus lead to further results of independent interest. [>> more]