lec21 (1)

# lec21 (1) - Introduction to Simulation - Lecture 21...

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Introduction to Simulation - Lecture 21 Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy Boundary Value Problems - Solving 3-D Finite-Difference problems Jacob White

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Outline • Reminder about FEM and F-D –1 - D E x a m p l e • Finite Difference Matrices in 1, 2 and 3D – Gaussian Elimination Costs • Krylov Method – Communication Lower bound – Preconditioners based on improving communication
1-D Example Normalized 1-D Equation Normalized Poisson Equation 2 2 () s Tx u x hf x xx x κ ∂∂ =− ⇒− = Heat Flow ux f x −=

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SMA-HPC ©2002 MIT FD Matrix properties Finite Differences 1-D Poisson 0 x 1 x 2 x n x 1 n x + 1 2 ˆˆ ˆ 2 () jj j j uu u fx x + + −= xx u f = 11 2 ˆ 21 0 0 12 1 1 00 1 1 2 ˆ nn uf x x x   −− =  MM L OM OOO M L A
SMA-HPC ©2002 MIT Residual Equation Using Basis Functions 2 2 u f x −= Partial Differential Equation form (0) 0 (1) 0 uu = = Basis Function Representation Plug Basis Function Representation into the Equation () ( ) 2 2 1 n i i i dx Rx f x dx ϕ ω = =+ { 1 Basis Functions n hi i i ux u x x ωϕ = ≈=

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SMA-HPC ©2002 MIT Basis Weights Using Basis functions Galerkin Scheme Generates n equations in n unknowns () ( ) 2 1 2 1 0 0 n i li i dx xf x d x dx ϕ ϕω =  + =   Force the residual to be “orthogonal” to the basis functions { } 1,. .., ln 1 0 0 l xRxd t =
SMA-HPC ©2002 MIT Basis Weights Using Basis Functions Galerkin with integration by parts Only first derivatives of basis functions () 1 11 00 0 n ii i l i x d dx dx x f x dx dx dx ωϕ ϕ = = ∫∫ { } 1,. .., ln

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