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Verwertbare Ergebnisse
The major outcome of the project was solving stiff elliptic equations arising in elasticity theory, fluid flow (convection diffusion), scattering theory and certain inverse problems for both linear and nonlinear problems.
New strategies based on defect correction methods for constructing adaptively refined meshes to capture layers for convection-diffursion equations were developed.
A multilevel approach for second order nonlinear problems solved via least squares methods for first order systems and multi-level Newton-like methods for nonlinear boundary value problems were developed. A least squares method for Stokes problem was also analysed.
Acoustic and electromagnetic stationary problems in homogeneous and heterogeneous media were studied. New results were obtained with fictitious domain methods and domain decomposition methods using mortar finite elements, in particular for application in case of strongly varying coefficients. Boundary integral methods were also studied. For the latter methods a splitting technique was introduced to avoid the storage of dense matrices and the multiplication with such matrices.
Certain aspects of inverse problem solvers modelled by 2D and 3D wave equations were considered related to the behaviour and the properties of the so-called « minimum point of the data misfit functional », which plays an important role when solving seismic and impedance tomography inverse problems.
Multilevel solvers for elliptic problems which converge uniformly in the discretisation and reaction coefficient parameter were developed. A robust method for linear elasticity problems based on a bordering method to handle anisotropy was developed.
A new approach for constructing robust methods for the generalised Stokes equations was developed.
Linear equation solvers were studied from a mainly algebraic point of view with new results for the construction of factorized sparse approximate inverses, Krylov subspace solvers and Jacobi-Davidson method for eigenvalue problems regularizing properties of the CG algorithm, and the influence of orderings on the incomplete factorization type of preconditioners.
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