## Mathematical Topics

Algorithms for the generation of 3D boundary conforming Delaunay meshesThis 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]

Analysis of Partial Differential Equations and Evolutionary EquationsPartial differential equations form an adequate and powerful instrument to provide a mathematical model for nature. At the Weierstrass Institute this research has two essential focuses: (a) Regularity for the solutions of linear elliptic equations and (b) Existence, uniqueness and regularity for evolution equations. [>> more]

Direct and inverse problems for the Maxwell equationsThe 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 equationsFree boundary problems are investigated in connection with energy technology and coating of surfaces. [>> more]

Functional analysis and operator theoryAt WIAS, functional analysis and operator theory are related, in particular, to problems of partial differential equations and evolutions equations, to analysis of multiscale, hybrid and rate-independent models and, finally, to mathematical problems of semiconductor models. [>> more]

Modeling, analysis and numerics of phase field modelsA 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 and hybrid modelsBecause of the ongoing miniaturization, modern devices in mechanics, electronics, or optics become smaller and smaller. Their working principles depend more and more on effects on different spatial scales. The aim is to optimize the efficiency by adapting the arrangement of interfaces or periodic microstructures. 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]

Multiscale Modeling and Asymptotic AnalysisThis work is focussed on plates, beams, shells and arches; sharp interface limits of generalized Navier-Stokes-Korteweg systems; sharp limits of regularized diffusion equations with mechanical coupling for applications in energy technology as well as thin film equations. [>> more]

Nonlinear kinetic equationsKinetic 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]

Numerical Methods for PDEs with Stochastic DataModels 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 inputdata. [>> more]

Numerical methods for coupled systems in computational fluid dynamicsThe main field of research are schemes for convection-diffusion equations, transport equations with exponential nonlinearities and Navier-Stokes equations (turbulent flows). The methods are based on FEM and FVM spatial discretizations and on implicit temporal discretizations. Considered applications involve population balance systems and Roosbroeck systems. [>> more]

Optimal control of partial differential equations and nonlinear optimizationMany 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 problemsIn 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 OptimizationStochastic 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 simulationThe mathematical description of many scientific and technological problems leads to systems of partial differential equations (PDEs). [>> more]

Variational methodsMany 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. [>> more]