Abstract
Iterative methods are widely used to solve sparse linear systems due to the improvements which can be achieved in reducing the solution time and increasing the size of the problem which can be solved on a given computer compared to traditional direct solvers. The theory behind the convergence rate relationship and storage requirements for the preconditioned conjugate gradient methods using the diagonal scaling, incomplete Cholesky decomposition and SSOR preconditioners is explained in detail in this study. Sparse matrix storage techniques, such as profile, element-by-element, and compact row storage, are described along with the redefined matrix operations for each storage technique which must be used to eliminate the operations on zero elements. A procedure to directly assemble the global stiffness in compact row storage format from element stiffness matrices is introduced. Numerical studies have been performed to compare the storage requirements, the convergence rate, and the solution time for the direct and PCG methods using various storage formats. Effects of different material properties and external loading on the convergence rate and solution time are also analyzed. The test problems for this study are based on the three-dimensional linear elasticity finite element equations. The physical memory of 64 MB of RAM of the IBM RISC/6000 Model 355 workstation was the limiting factor for the size of the sparse linear system that could be solved in this study. The diagonal preconditioned conjugate gradient method with the compact row storage has solved a three-dimensional finite element problem up to a maximum of 50,000 equations on an IBM RISC/6000 Model 355 workstation with 64 MB of RAM. To apply adaptive mesh refinement on certain regions of a coarse mesh, the modeling error over a coarse mesh must be estimated. This thesis will show that the modeling error from an intermediate unconverged coarse mesh solution will closely match the modeling error from the converged solution. This result may lead to quicker solution times for a highly accurate mesh based on adaptive mesh refinement iterative methods.
Wang, Hongbing (1995). Iterative solutions to large sparse finite element equations. Master's thesis, Texas A&M University. Available electronically from
https : / /hdl .handle .net /1969 .1 /ETD -TAMU -1995 -THESIS -W364.