``FAST'' POISSON SOLVER

These are fast, in the sense that if you use an FFT to do the Fourier computations, it is faster that the Discrete Fourier Transform (DFT). Competitive in speed with ADI and SOR optimized. It proves useful, in any event, to know how to use this technology because its applicable to other boundary value problems.

Motivations There are a number of reasons for visiting this problem:

• You get an idea of what is involved in solving problems in two-space dimensions and of the importance of boundary data in fixing a solution.
• You see how advantageous it is, in many cases, to use analytical means as much as possible to pose a problem for solution BEFORE actually coding it up. In general, one should explore all possible means to advance a calculation by analytical means before resorting to numerics...of course, this is not a theorem, but merely a rule of thumb.
• Following the lead in the last item, we could find that the problem posed below can be solved exactly via analytical means (as are many of the problems covered in this course). Nevertheless, we want to remind you that in many instances, problems in several space dimensions can be solved most easily numerically and analytically if you happen to choose the right reference frame and/or coordinate system. In this instance we'll emphasize the issue of choosing the coordinate system, and in this case, the choice is based on symmetries in the boundary geometry and the type of PDE (consult an elementary PDE book, particularly, one geared towards engineers).
• The reason for wanting to solve the disk problem is that we'll get come practice in solving PDE in coordinates other than Cartesian, and we'll show how the boundary conditions must be payed special attentions to.
• We will also use this problem to introduce, albeit in an elementary way, how spectral methods can be used numerically. Also, many books do not emphasize the fact that numerical methods can often be combined, a fact that is obvious later on, but sometimes not so obvious as a beginning computational scientist.

We will solve the Helmholtz Equation

$$\big(\nabla^2+\kappa^2\big)U=g \left\{ \begin{array}{l} g, \qquad... ...ox {real} k, \qquad \mbox {real} u, \qquad \mbox {real} \end{array} \right.$$ (172)

on a unit disk with domain $$D=\left\{(x,y)\in R^2:x^2+y^2<1\right\}$$

with boundary conditions $\left\{\begin{array}{ll} U(\cos \theta, \sin \theta)=\phi (\theta) & 0\le \the... ...)=\phi (2\pi)\quad ,& \mbox {Dirichlet boundary conditions} \end{array}\right.$

Remarks Note that (173), with $k^2=0$, we get the Poisson Equation. The Helmholtz Equation originates from the linear 2-way wave equation, where the time dependence of the solution is assured time harmonic:

To see this, take the Wave Equation (with no forcing term, for simplicity)

$$\frac{1}{c^2}\psi_{tt}-\nabla^2 \psi=0\quad c^2>0 \mbox { constant,}$$ (173)

where $c$ is the wave speed.

Substitute

$$U(x,y)= \psi e^{-i\omega t}$$ (174)

where $\omega$ is the frequency of the wave, assumed constant. Let $$\frac{\omega}{c} = \kappa$$ be the wavenumber, then the (175) solution to (174) is found by solving

$$(\kappa^2+\nabla^2)U=0$$ (175)

When the problem is defined on a disk and a forcing term $g$ is added to the equation (176), we get (173).