# Finite Element Method - 3D Mesh Generator - Metfem3D

## Linear Examples

### Laplace equation

On the other hand, assuming that the time derivative in the diffusion equation equals 0 and putting D = 1 we end up with the boundary value problem of the Laplace type[1, 2]

Δu = 0 in Ω
u = g on ∂Ω.

Let us consider Ω being a cubic domain i.e. [0 π] ✗ [0 π] ✗ [0 π]. And for the g function equals 0 everything on ∂Ω apart from g(x = π, y,z) = φ0 one can approximate the exact solution by

 ∞ φ(x,y,z) = 16φ0/π2 ∑ ( sinh((n2 + m2)1/2x) sin(ny) sin(mz) )/ n, m = 1, 3, …
 /( nm sinh((n2 + m2)1/2π) )

For φ(x,y,z) = v(r) where r = (x2 + y2 + z2)1/2 the Laplace equation has the solution defined in ℜ3 for r ≠ 0

φ(x,y,z) = 1/(3α(3)) 1/(x2 + y2 + z2)1/2

where α(3) denotes volume of B(0,1) in ℜ3 and equals π3/2/Γ(3/2 + 1).

### References

1. ^ L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998
2. ^ R. Courant and D. Hilbert, Methods of mathematical physics vol. 1, Interscience Publishers Ltd., London, 1953