Numerical Analysis, Eighth Edition

by Richard L. Burden and J. Douglas Faires

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Poisson Equation Finite-Difference Method

To approximate the solution of the Poisson equation

  D(D(u)) = F(x,y), a<= x <= b, c <= y <= d, Subject to the boundary conditions: u(x,y) = G(x,y)

  if x = a or x = b for c <= y <= d, if y = c or y = d for a <= x <= b.