laplace

# laplace - Poisson's & Laplace's Equations Poisson equation...

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Poisson’s & Laplace’s Equations Poisson equation 2 u = 2 u ∂x 2 + 2 u ∂y 2 = - ρ ( x, y ) Laplace equation 2 u = 2 u 2 + 2 u 2 =0 Discretization of Laplace equation (set Δ x y = h ) ( u i +1 ,j + u i - 1 ,j + u i,j +1 + u i,j - 1 - 4 u ij ) /h 2 Figure 1: Numerical solution to the model Laplace problem on a 40 × 40 grid.

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u i +1 ,j + u i - 1 ,j + u i,j +1 + u i,j - 1 - 4 u ij =0= Au = b iterative method u ij 1 4 ( u i +1 ,j + u i - 1 ,j + u i,j +1 + u i,j - 1 ) b 11 b 10 b 9 b 3 b 4 b 5 b 0 b 1 b 2 b 8 b 7 b 6 0 1 2 3 4 5 6 7 8 Au = - 410100000 1 - 41010000 01 - 4001000 100 - 410100 0101 - 41010 00101 - 4001 000100 - 410 0000101 - 41 00000101 - 4 u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 = b = - b 0 - b 11 - b 1 - b 2 - b 3 - b 10 0 - b 4 - b 8 - b 9 - b 7 - b 6 - b 5
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## This note was uploaded on 03/11/2012 for the course MAT 421 taught by Professor Staff during the Fall '11 term at ASU.

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laplace - Poisson's & Laplace's Equations Poisson equation...

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