HW8. Poisson equations
Two-dimensional Poisson equation is written as
∇2u(x,y) = f(x,y) for (x,y) ∈ Ω
and on the boundary ∂Ω = ∂ΩD∪ ∂ΩN
u(x,y) = g(x,y) on ∂ΩD and ∂u/∂n = h(x,y) on ∂ΩN
Note that n is the normal to the boundary, ∂ΩD is the Dirichlet boundary, and ∂ΩN is the Neumann boundary.
1. (Iterative Poisson solver) Let’s consider the Poisson equation in the square domain [0,1] × [0,1] with homogenous boundary conditions u = 0 at all boundaries and f(x,y) = sin(πx)sin(πy).
2. (Linearity) Let’s consider a following Poisson equation
∇2u(x,y) = f1(x,y) + f2(x,y) for (x,y) ∈ Ω
and u(x,y) = 0 on the boundary ∂Ω of the square domain Ω ≡ [0,1] × [0,1].
(1) Find u(x,y), the solution of Poisson equation, ∇2u(x,y) = f1(x,y) + f2(x,y) using Gauss-Seidel SOR. The forcing functions are defined as
f1(x,y) = sin(πx)sin(πy)
f2(x,y) = exp(−100.0((x − 0.5)2 + (y − 0.5)2))
(2) Find u2(x,y) the solution of Poisson equation, ∇2u2(x,y) = f2(x,y) using Gauss-Seidel SOR.
(3) Discuss the solution u(x,y) by comparing with the solutions u2(x,y) and u1(x,y) that is obtained in Problem 1.
