Non-Iterative Parallel Schwarz Domain Decomposition Algorithms for a Two-Domain Parabolic Problem

Based on overlapping domain decomposition, two parallel Schwarz finite element algorithms are presented for solving a two-domain parabolic problem. Our algorithms are based on two ideas. One is the observation that the influence of the non-homogeneous boundary data diminishes exponentially towards the interior of the subdomains. Another is to apply proper partitions of unity to distribute the residuals of the discretized systems reasonably into many subdomains so as to avoid repeated correction in the overlapping parts. The resulting algorithms are fully parallel in each subdomain. At each time step, only one iteration step is required to reach the optimal order of accuracy. The convergence rate depending on the mesh parameters are analyzed. Finally, numerical results are reported, which verify the theoretical analysis.