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