Research On Algebraic Multigrid Method And Application To Signal Integrity Analysis System

In this thesis the algebraic multigrid method is considered for solving system equations which arise from a finite element discretization of partial differential equations. In particular we consider system equations that originate from an edge finite element discretization of Maxwell's equation in time and frequency domain, which is widely used in signal integrity analysis systems. The general algebraic method is modified to fit the special mesh structure we encounter in practice, and then is used as a preconditioner in several preconditioned Krylov subspace methods to solve the system equations in signal integrity analysis system efficiently.In the modified algebraic multigrid method, as the 3D finite element mesh is composed of several 2D finite element meshes which have the same structure, only one 2D mesh need to be coarsened. The construction of auxiliary matrices and their prolongation operators is the same as in the general algebraic multigrid method. But after the system matrix prolongation operators being defined via the method proposed in general multigrid method, they have to be extended to the 3D mesh. Auxiliary matrices and system matrices on the coarsen level are computed by the Galerkin method as usual.In this thesis, several classical iterative methods are introduced, such as Gauss-Seidel method, conjugate gradient method, conjugate gradient method for complex matrices, BiCG method and BiCGSTAB method. Preconditioned conjugate gradient method and preconditioned BiCGSTAB method are derived from the conjugate gradient method and BiCGSTAB method.Both the AMG preconditioned conjugate gradient method and AMG preconditioned BiCGSTAB method can efficiently solve system equations in the time domain. The former may fail in the frequency domain, but the later is also an efficient solver.Finally, numerical studies are given for problems in signal integrity analysis system. The coarsened meshes are given in figures. Several solvers are applied and their convergence behaviors are compared. AMG preconditioned methods have much higher convergence than other methods.