DFG Research Center "Mathematics for key technologies - MATHEON": Adaptive solution of parametric eigenvalue problems for partial differential equations (Project C 22)

Parameter dependent eigenvalue problems for partial differential equations (PDEs) arise in a large number of current technological applications. To mention only one example, parameter dependent eigenvalue problems arise in the computation of the acoustic field inside vehicles, such as cars or trains, or in analyzing the noise compensation in highly efficient motors and turbines. It is well understood that numerical methods for PDEs, such as finite element methods (FEM) with very fine meshes give good approximation but lead to a very high computational effort. Therefore it is important to use adaptively refined meshes to reduce the computational complexity while retaining good accuracy. This can be done by the adaptive finite element method (AFEM). But to do so, reliable and efficient error estimators especially for parameter dependent eigenvalue problem are needed. Since the general algebraic eigenvalue problem requires a lot of computational effort, especially for large sparse nonsymmetric systems, it is also an important task to develop fast iterative and multilevel eigenvalue solvers. In combining these two research fields together, the goal is to equilibrate the errors and computational work between the discretization and the approximation errors and the errors in the solution of the resulting finite dimensional linear and nonlinear eigenvalue problems.