• 作者： 程宣仁
• 作者服務機構： 國立清華大學數學系
• 中文摘要： 我們針對一些來自於定義在複合網格上的非對稱及非正定的橢圓型有限元問題所產生大而稀疏的線性系統考慮一求解方法－不同步快速可調整的複合網格法；此方法被設計用來解決一些其解具有奇異點的橢圓型邊界值問題；應用此方法好處為在求解的過程中我們僅需求解一些在均勻網格上的子問題，而不需求解原來複雜的大問題：我們證明在一般的假設下，此方法之收斂速度與網格分割加細的次數及所有各層次網格的大小等參數均無關；關於收斂速度此方法與標準的多重網格法的主要差別，在於為了得到均勻的收斂速度界值，我們不需像在多重網格法中加入一些關於網格大小的限制；此工作推廣了我們先前關於對稱及正定橢圓型問題的工作。
• 英文摘要： We consider an AFAC (Asynchronous fast adaptive composite) algorithm for large sparse linearsystems of equations which arise from second-order nonsymmetric and indefinite elliptic finite elementproblems defined on composite meshes. The AFAC method was designed for elliptic boundary valueproblems with some solution singularities. The benefit of applying AFAC is that we only need to solvesome subproblems on uniform meshes rather than solve the original complicated problem. We prove thatthe rate of convergence is independent of all mesh size parameters and the number of refinement levelsunder some general assumptions. The main difference between the AFAC method and the standard multigridmethod with regard to the convergence rate is that unlike multigrid methods no restrictions are placedon mesh sizes for the AFAC in order to get a uniform convergence bound. This work generalizes ourprevious work on symmetric and positive definite elliptic problems.
• 中文關鍵字： Schwarz methods; GMRES method; elliptic regularity; extension theorem; nonsymmetric and idenfinite elliptic problems; finite elements
• 英文關鍵字： --