Parallel Numerical Methods for Partial Differential Equations

It has been rightly predicted that parallel computing is inevitable. This thesis at-tempts to study and implement the so-called geometry splitting solution paradigm as a parallel computational framework for solving elliptic partial differential equations (PDEs) on distributed memory machines. First, we formulate, analyze and implement the Schwarz Alternating Method (SAM) for elliptic PDEs defined in one and multi-dimensional domains. Specifically, we analyze SAM methods whose convergence is controlled by a different parameter in each interface condition or overlapped domain. We derive both analytical and experimental results. Furthermore, we introduce a sym-metric version of SAM and make useful observations about its convergence. Second, we implement four non-overlapping geometry splitting approaches based on finite element and difference techniques. One of them is formulated on the extended rectangular domain that encapsulates the given PDE domain and its corresponding grid. This encapsulation method assumes an extension of the PDE problem outside the specified domain of definition. This approach has reduced significantly the grid partitionin