http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
Optimal order preconditioners for mixed and nonconforming finite element approximations of elliptic problems with anisotropy
The general area of this thesis is preconditioning techniques for mixed and nonconforming finite element approximations of elliptic boundary value problems. A special emphasis is placed on problems in three dimensions with possibly large anisotropy in the coefficients of the PDE's along with large jumps in the coefficients across the interfaces separating subregions. The optimal preconditioners developed exploit the techniques of domain decomposition methods, algebraic substructuring, and multigrid methods. As a result, the proposed iterative processes converge with rates independent of the mesh size, the jumps of the coefficients, and the ratio of anisotropy. Using an equivalence between nonconforming finite element methods and hybridmixed methods the iterative methods constructed for algebraic systems with symmetric positive definite matrices are extended to saddle-point problems which arise from mixed finite element approximations. A new construction of iterative methods for nonconforming approximations of elliptic PDE's on nonmatching grids is proposed. The computational domain is considered as a union of nonintersecting subdomains. In each subdomain the grid is constructed in accordance with its own coordinate system using the main directions of anisotropy. The original elliptic problem is posed as a problem with Lagrange multipliers at the interface between the subdomains, which ensure the continuity conditions of the solution. A mortar finite element subspace is constructed in the space of Lagrange multipliers, which results in algebraic systems of a saddle-point type. Based on the technique of domain decomposition and fictitious components methods conditioners for the algebraic systems arising in the a construction of block diagonal pre mortar finite element method is developed. The fictitious components method is used to precondition subdomain problems, whil interface problems are preconditioned by an innere the Chebyshev iterative procedure. It is shown that the developed preconditioner is spectrally equivalent to the original saddle-point matrix. Applications of the newly developed iterative methods and preconditioning techniques are considered. In particularly, these methods are applied in the simulator of fluid flow in porous media.
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