| Author | Yong Qu |
|---|---|
| Title | MULTILEVEL METHODS FOR INDEFINITE SYSTEMS |
| Year | 2000 |
| School | RPI |
| Abstract | The purpose of this thesis is to develop a general purpose multilevel solver for indefinite systems which arise in many areas of scientific computing. A family of local-basis two-level solvers for indefinite systems is first presented. Local approximations of the lowfrequency modes of the source system are used to construct the prolongation operator. The methods utilize normal equations for highly indefinite problems and the source system for weakly indefinite problems. The concept of the optimal prolongation operator for indefinite systems is then introduced and studied. It is shown that the optimal prolongation operator is spanned by the spectrum of the highest eigenmodes of the smoothing iteration matrix. Convergence studies conducted on a model prolongation operator reveal pathological sensitivity to any deviation from the optimal prolongation operator. Based on these notions, a global-basis two-level solver for highly indefinite systems is developed. The algorithm includes effi- cient construction of the global basis prolongator using the Lanczos vectors, predictorcorrector smoothing procedures, and a heuristic two-level feedback loop aimed at ensuring convergence. Finally, two important issues concerning the construction of preconditioners, namely improving the memory-system performance and adaptive selection of algorithmic parameters, are addressed and discussed. A new class of preconditioners based on the adaptive threshold incomplete multifrontal factorization for indefinite and complex symmetric systems is presented. Numerical experiments consisting of the 3D Helmholtz equations, the fluidstructure interaction problem and the shear banding problems demonstrate the excellent performance of the proposed methods. |
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