運用多重網格法(Multigrid method)延伸出的演算法, 作為平行 Newton-Krylov-Schwarz 演算法的預處理, 降低迭代次數與計算時間, 加速求得大型鬆弛非線性方程式組的解, 此非線性系統是介由有限元素法, 作離散化在三維的 Poisson-Boltzmann 方程式; 於膠質科學的應用中, 做帶電膠質微粒在電解液中的三維數值模擬, 並進一步探討對稱與非對稱電解質溶液對於電場與電位能的影響. Poisson-Boltzmann 方程式, 為描述帶電膠體粒子於電解液中, 其電位能分佈狀況的方程式. 並進行關於平行效能的研究, 使用多層次法 (Multilevel) 優化迭代次數及時間, 和比較多層次法使用不同聚集方法的效益。

The use of multi-grid (Multigrid method) extending algorithm as preconditioner parallel Newton-Krylov-Schwarz algorithms to reduce the number of iterations and calculation time determined to accelerate the solution of nonlinear equations large relaxation. The group, this nonlinear system is mediated by the finite element method, as in the
three-dimensional discrete Poisson-Boltzmann equation; in glial scientific applications, do the three-dimensional numerical simulation of charged colloidal particles in the electrolyte, and to further explore symmetric and asymmetric electrolyte solution for electric field and the potential energy of the impact. Poisson-Boltzmann equation for the description of charged colloidal particles in the electrolyte, the potential energy distribution formula. And conduct research on parallel performance, optimization iterations, and time, and compare the effectiveness of different aggregation methods.

Tables......................................... vi
Figures ......................................... ix
Symbols ........................................ x
1 Introduction . .................................. 1
2 Mathematical model . .............................. 3
3 Solution Algorithm . ............................... 6
3.1 Inexact Newton-Krylov........................... 6
3.2 Calculation of Jacobian matrices...................... 7
3.3 A parallel one-level overlapping Schwarz preconditioner......... 8
3.4 Two-level methods with a parallel coarse preconditioner......... 9
3.5 Multilevel Method: A Parallel Smoothed Aggregation Multigrid Method. 11
3.5.1 Uncoupled Parallel Aggregation Scheme............. 13
3.5.2 MIS-based Parallel Aggregation Scheme............. 14
3.5.3 METIS-based Parallel Aggregation Scheme............ 15
4 Numerical Results . ............................... 16
4.1 Simulation domains............................. 16
4.2 Two Isolated Charged Particles....................... 18
4.3 Parameters study and parallel performance................ 22
5 Conclusions . ................................... 27
5.1 The derivation of Poisson-Boltzmann model in a Asymmetric electrolyte. 30
5.2 Particles Interaction Force......................... 34
5.3 The computation of the non-dimensionalized electrostatic force acting on the charged particles............................ 36
5.4 Richardson’s Extrapolation......................... 40

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