Elliptic equations background

Elliptic equations can be thought of as being the steady state limit of the diffusion equation,

u_t = (\kappa u_x)_x + (\kappa u_y)_y + \psi

when both the boundary conditions and the source (or "forcing") term are time in-dependent. Under these conditions, it can be expected that the time dependent terms vanish as the system relaxes to steady state (i.e. in the limit t \rightarrow \infty, \partial t \rightarrow 0), and we are led to the elliptic equation (here in 2D),

(\kappa u_x)_x + (\kappa u_y)_y = f

where u is the dependent variable we are solving for, and f is the forcing term.

The solution to this equation needs to simultaneously 1) satisfy this equation at all points within a bounding region and 2) satisfy the boundary conditions on that region. This then lends an "instantaneous" feel to the system, much different than the wave-like solutions of hyperbolic equations which travel with finite speed.

In fact, apart from their classification as either a boundary value problem for the elliptic equation, or initial value problem for the hyperbolic, they can be considered of either a "time-evolution" or "static" nature. This is a more helpful classification in terms of designing numerical methods to solve these different types of equations (see Fortran Numerical Recipes, Press et al, Vol. 2, Chapt. 19 - Partial Differential Equations). The following figure from that book illustrates this concept:

Some special cases for elliptic equations, of relevance here, occur when \kappa = 1; when f is non-zero, we have the Poisson equation,

\triangledown ^2 u = f

when f is zero, we have Laplace's equation,

\triangledown ^2 u = 0

The solutions to both Poisson's and Laplace's equations satisfy the Uniqueness theorem (see Griffith's E&M textbook). That is, if the solution satisfies the aforementioned 2 conditions, it is the unique solution to the equation.

As is, Poisson's equation is set up to solve for f (be it charge density, or matter density, etc.) given some potential (e.g. electrical or gravitational, etc.). However, in most situations we have the opposite information about a system — given the density, we seek the potential. This requires some numerical techniques for solving this 2nd order differential equation, especially for those systems that do not admit closed-form solutions.

Equation Discretization

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

• Discretizatiion leads to a syustenm of eqns. There are 2 ways to solve the equations. Matrix direct methods and relaxtiuon iteration mrethods.

Matrix form (not followed)

I did not write a code that uses a direct matrix method for solving the system of equations, but include this section here for completeness.

Relaxation form (followed)

Error, accuracy, convergence, solution is guaranteed to converge

Laplace Equation (source term = 0) Physical Meaning

Note that the discretization equation with f=0 describes each point as being the average of its neighbors. This is in fact one of the key properties of the harmonic functions, which are the class of solution to the Laplace Equation. Since this is such an ubiquitious equation in physics, and a subset of the Poisson equation, I felt it was important to review some of the properties behind it.

• Boundary conditions
• Minimizing distance b/w boundaries (in 1d this is a line, in 2d this is soap bubble)
• Physical interprestaion - no charge within the domain, but elsewhere.

Poisson Equation Physical Meaning

While the interpreation of the solution propertoies are not as clear cut and intuitive as Laplace, the solution can be thoguht of as composing green's functions (cite references).

The code

Explain what my thing does, post it for download

Tests and Results

Test 1 - Laplace's Equation

Test 2 - Poisson's Equation with simple forcing

Test 3 - Poisson's Equation with complicated forcing

Properties - aka only an approximate solution now . Verify 2nd order (how?)

Cite References

References