A hierarchical matrix adaptation on a family of iterative method for solving poisson equation

This thesis deals with an adaptation of hierarchical matrix (H -matrix) techniques in iterative methods for solving the Poisson equation, which is a representative of partial differential equations. The research examines different iterative techniques and ordering strategies in Gauss-Seidel method which are easy to implement on a computer. The H -matrix techniques allows an efficient treatment of a dense matrix. This treatment will lead to less memory utilizations. Three types of finite-difference approximations in the form of the full-sweep (FS), half-sweep (HS) and quarter-sweep (QS) approaches are considered in this research. An extension of this approach where a faster convergence rate can be achieved is by grouping the iteration points into a single iteration unit. Implemented with the finite-difference schemes mentioned above, this approach produces Explicit Group (EG), Explicit Decoupled Group (EDG) and Modified Explicit Group (MEG) methods. All of these iterative methods are yet to be implemented with H -matrix. The construction of an H -matrix relies on a hierarchical partitioning of the dense matrix. To set up this partitioning, a so-called admissibility condition must be satisfied. Two types of admissibility conditions namely the weak admissibility and standard admissibility will be considered in this research. This will produce two different H -matrix structures, HW- and HS-matrices, which consists of different memory utilizations. The main objective of this thesis is to develop an adaptation of the H - matrix structures with the iterative method. Both of these structures will be compared with each other. The HW-matrix should produce a more accurate solution with a faster execution time and utilizes less memory when compared to the HS-matrix.

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