Numerical solution algorithms for large-scale (non)linear algebraic systems and shape optimization problems are studied. Emphasis is given to efficient parallel algorithms suitable for distributed memory multiprocessor computers.
Fast direct methods for solving linear systems with separable matrices have been considered. They have many applications, for example, in the fictitious domain preconditioning. A new method, a so-called Divide & Conquer algorithm, has been developed. It can be applied for solving higher-order partial differential equations, as well. Another research topic has been the development of (approximate) partial solution techniques.
The aim is to develop and analyze efficient numerical methods based on the fictitious domain approach for solving elliptic partial differential equations with mixed boundary conditions. Similar approach is applied also in acoustic scattering problems. Special emphasis is given to new research areas such as the use of Lagrange multipliers and nonmatching discretization meshes. The methods are further applied, for example, in shape optimization problems and in convection-diffusion problems.
Numerical solution of unilateral variational inequalities (obstacle problems) has been considered. Based on finite difference or finite element discretization, various iterative methods have been proposed for the resulting algebraic system. Special emphasis has been given to the development of the methods, which can be (theoretically) treated within the unifying framework of the block relaxation methods. Also, using the theoretical results, implementation algorithms based on domain decomposition and fictitious domain approaches for large-scale algebraic obstacle problems have been studied.
Domain decomposition algorithms based on the Schwarz alternating method are developed for the numerical solution of singularly perturbed semi-linear problems. The emphasis is given to the parallel computer realization of the algorithms on distributed memory multiprocessor computers as well as investigation of their behaviour with respect to the ``critical'' parameters, such as the perturbation parameter, the overlapping size, and the number of parallel processors.
Efficient and robust methods for multidisciplinary shape optimization problems governed by state equations modelling problems in computational fluid dynamics and electromagnetics have been studied and developed. Typically, two-dimensional state equations are described by the (full) potential flow and the time-harmonic Helmholtz equations. The shape optimization problem is formulated as a nonlinear minimization problem. Each objective function evaluation requires one solution of a state equation, which is obtained by using a fictitious domain method or a multigrid method. In the minimization of the objective function, parallel implementations of genetic algorithms and gradient-based methods such as SQP have been employed.