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Chrys Correa September 6, 2000 12:09

Analytical solution to the 3d Laplace equation
 
Hi,

I've written a 3-d finite volume code to solve the Laplace equation, and I would like to test it. Does anyone know of an analytical solution that I can use for this purpose?

Thanks.

Chrys

Jim Park September 6, 2000 12:56

Re: Analytical solution to the 3d Laplace equation
 
Chrys,

Check texts on heat transfer. Look for steady state conduction. With constant properties and no heat generation or phase change terms, the physics are captured by a LaPlace equation with temperature as the dependent variable.

Analytic solutions for one, two or three dimensions are usually given as infinite series. Details depend on the boundary conditions, which must fit some mathematical requirements.

Good luck.

John C. Chien September 6, 2000 19:01

Re: Analytical solution to the 3d Laplace equation
 
(1). Look for Heat Conduction book by Carslaw and Jaeger. (Series expansion solutions)

eddy September 7, 2000 02:26

Re: Analytical solution to the 3d Laplace equation
 
write a sencond order 3-d polynomial,for example,

a*x*x+b*y*y+c*z*z=d

take a laplace operation on it, you will get the source item 2d+2b+2c,

calculate boundary condtions according to above polynomial. Now, you can use you solver to get results and compare with the exact results calculated from polynomieal.

good luck

clifford bradford September 11, 2000 19:50

Re: Analytical solution to the 3d Laplace equation
 
yeah you can come up with some yourself by using separation of variables and tossing in some eigenvalues...you know, stuff you learned in PDEs. Pretty soon you'l have an analytical solution. also try solving 2d problems as well


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