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**Description.** Optimal transportation of substances, goods, or information is a classical mathematical problem with applications in economics, meteorology, and computer science. It can be formulated as a high-dimensional linear program whose direct solution is difficult. More efficient approaches are based on equivalent continuous formulations via partial differential equations or variational problems. These are nonlinear and nondifferentiable mathematical problems that require a suitable discretization and iterative solution. The project focuses on the developement and numerical analysis of finite elemet discretizations, the automatic efficient mesh refinement based on rigorous a posteriori error estimates, and the fast iterative solution. Special emphasis is on the avoidance of regularizations and use of unjustified smoothness assumptions on solutions.

**Description.** The aim of this project is to establish reliable and efficient numerical methods for models of solids with spatial discontinuities caused by the evolution of dissipative processes such as plasticification, damage or fracture. In particular, the project focuses on such prototypical models that use the class of BV-functions to mathematically describe the discontinuities, that are guaranteed to converge to a solution of the infnite-dimensional model and for which iterative solution methods can be constructed. Emphasis is on unregularized numerical approaches that lead to sharp approximations of discontinuities on coarse grids and rigorous convergence proofs. The main objectives of the research project are the development, analysis, and implementation of finite element methods for model problems describing discontinuities in BV. This includes the derivation of a priori and a posteriori error estimates as well as the construction of adaptive and extended approximation methods for BV-prototype models such as the Rudin-Osher-Fatemi and the Mumford-Shah model. The techniques will be transferred to analytically justified and closely related models for the description of rate-independent inelastic processes, in particular perfect plasticity, damage and fracture. The methods and results will be applied and transferred to particular model scenarios and benchmark problems in mechanics.

**Description.** The project investigates the development and analysis of numerical methods for the
approximation of geometric partial differential equations. Focus is on the reliability of algorithms
for weak solutions which may exhibit singularities and blowup phenomena in finite time. Applications
include the simulation of switching processes in micromagnetics, surface registration in image processing,
and the prediction of patterns of lipids and shapes of biological membranes.

Simulation of phase field models and geometric evolution problems Research Center Matheon Mathematics in Key Technologies Funding period: 2005 - 2008 |

**Description.** Allen-Cahn equations, which are the simplest case of Phase field models, occur in the mathematical modeling of various physical processes and applications in several key technologies such as phase transitions in binary alloys and superconducting materials. They are often employed for the approximation of nonlinear geometric evolution problems which arise in liquid crystal theory, micromagnetics, and color image denoising. Solutions depend sensitively on a small parameter. Owing to the limited practical regularity of these solutions, numerical schemes have to be carefully chosen in order to yield reliable approximations. In fact, classical error estimates for the numerical approximation depend exponentially on the small parameter and are therefore of limited practical use. It has only recently been shown that a robust error estimation is possible in certain situations. The proposed schemes employ or compute spectral estimates for the linearized partial differential equation about the exact or approximate solution and thereby measure the stability of a solution. In the Project C16, adaptive numerical schemes will be designed for Cahn-Hilliard and Cahn-Larche equations which allow robust error estimation. In particular, a posteriori error estimates will be derived which are the basis for local mesh-refinement and coarsening and thereby yield efficient approximation schemes. Convergence of adaptive approximation schemes will be investigated and numerical simulations will be carried out for a mathematical model in the production of lead-free electric joints in microeletronic devices. A rigorous mathematical analysis of projection schemes and fully implicit discretization methods for the approximation of geometric evolution problems will be carried out. Finite time blow-up of solutions for Landau-Lifshitz-Gilbert equations will be studied numerically and theoretically.