15_pde - Domain Decomposition Computation with Grids...

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Unformatted text preview: Domain Decomposition Computation with Grids Scalability and Fault Tolerance Parallel Numerical Algorithms Chapter 15 Partial Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign CSE 512 / CS 554 Michael T. Heath Parallel Numerical Algorithms 1 / 66 Domain Decomposition Computation with Grids Scalability and Fault Tolerance Outline 1 Domain Decomposition Overlapping Subdomains Non-Overlapping Subdomains 2 Computation with Grids Parallel Computation with Grids Ghost Points Multigrid 3 Scalability and Fault Tolerance Scalability Fault Detection and Fault Tolerance Michael T. Heath Parallel Numerical Algorithms 2 / 66 Domain Decomposition Computation with Grids Scalability and Fault Tolerance Numerical Methods for Partial Differential Equations Partial differential equations are typically solved numerically by finite difference, finite element, finite volume, or spectral discretization Such discretization yields system of linear or nonlinear algebraic equations whose solution gives approximate solution to PDE Solving linear or nonlinear algebraic system is one major source of parallelism in solving PDEs numerically We will consider domain decomposition methods that exploit natural parallelism in PDE and its discretization Michael T. Heath Parallel Numerical Algorithms 3 / 66 Domain Decomposition Computation with Grids Scalability and Fault Tolerance Overlapping Subdomains Non-Overlapping Subdomains Alternating Schwarz Method Consider elliptic PDE Lu = f on domain = 1 2 , with boundary condition u = g on 1 2 1 2 Michael T. Heath Parallel Numerical Algorithms 4 / 66 Domain Decomposition Computation with Grids Scalability and Fault Tolerance Overlapping Subdomains Non-Overlapping Subdomains Alternating Schwarz Method Given initial guess u (0) 2 on 2 , for k = 0 , 1 ,... On 1 , solve Lu ( k +1) 1 = f with boundary conditions u ( k +1) 1 = g on 1 \ 1 u ( k +1) 1 = u ( k ) 2 on 1 On 2 , solve Lu ( k +1) 2 = f with boundary conditions u ( k +1) 2 = g on 2 \ 2 u ( k +1) 2 = u ( k +1) 1 on 2 Michael T. Heath Parallel Numerical Algorithms 5 / 66 Domain Decomposition Computation with Grids Scalability and Fault Tolerance Overlapping Subdomains Non-Overlapping Subdomains Alternating Schwarz Method Alternating iterations continue until convergence to solution u on entire domain Schwarz proposed this method in 1870 to deal with regions for which analytical solutions are not known Today it is of interest, in discretized form, for suggesting one of two major paradigms for solving PDEs numerically by domain decomposition Overlapping subdomains (Schwarz) Non-overlapping subdomains (Schur) Michael T. Heath Parallel Numerical Algorithms 6 / 66 Domain Decomposition Computation with Grids Scalability and Fault Tolerance Overlapping Subdomains Non-Overlapping Subdomains Discretized Schwarz Method Discretization yields n n symmetric positive definite...
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15_pde - Domain Decomposition Computation with Grids...

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