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PoissonSolver

-              v2
+              v3
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+= Elliptic equations background =
+Elliptic equations can be thought of as being the steady state limit of the diffusion equation,
+[[latex($u_t = (\kappa u_x)_x + (\kappa u_y)_y + \psi$)]]
+when both the 1) boundary conditions and 2) the source (or forcing term) [[latex($\psi$)]] is time-independent. Under these conditions, the time-dependent terms vanish, and we are led to the elliptic equation (here in 2D),
+[[latex($(\kappa u_x)_x + (\kappa u_y)_y = f $)]]
+where u is the dependent variable we are solving for, and here f is the forcing term.
+This equation needs to 1) be satisfied by all points within a bounding region and 2) satisfy the boundary conditions on that region. This then can be interpreted as an instantaneous constraint on the system, much different than the wave-like solutions of hyperbolic equations which travel with finite speed.
 = Equation Discretization =
 Can be thought of the steady state, static form of the diffusion equaiton - begin with equaiotn, expect relaxation to steady state (i.e. as t-> inf, d/dt -> 0), get the following Poisson equation:
+The Poisson equation can be thought of as being the steady state limit of the diffusion equation. Can be thought of the steady state, static form of the diffusion equaiton - begin with equaiotn, expect relaxation to steady state (i.e. as t-> inf, d/dt -> 0), get the following Poisson equation:
 = Matrix form, relaxation form =
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 = References =