# help with Wentzell eigenvalue problem.

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13 months ago by
I am working with the Wentzell eigenvalue problem in a 3-D domain. I am still learning FeniCs and I want to know how to use FeniCs to compute some of the first eigenvalues for the Wentzell problem.

the Wentzell eigenvalue problem is the following.

-Δu = 0 in Ω
-βΔτu + ∂nu = λu on ∂Ω

Where Δτ is the Laplace-Beltrami operator and ∂n  is the normal derivative.

The week formulation is

$\int_{\Omega}\nabla u \cdot \nabla \phi + \beta\int_{\partial\Omega}\nabla_{\tau}u\cdot\nabla_{\tau}\phi = \int_{\partial\Omega}\lambda u \phi$

where $\nabla_{\tau}u = \nabla u - (\nabla u \cdot \textbf{n})\textbf{n}$ and $\textbf{n}$ is the normal to $\partial\Omega$

This is my attempt to deal with the weak formulation. I don´t know how to handle the boundary integral and the eigenvalue.

from fenics import *
from mshr import *
import matplotlib.pyplot as plt

beta=0
#Define 3D geometry
sphere1 = Sphere(Point(0, 0, 0), 1.0)
sphere2 = Sphere(Point(0, 0, 0), 0.5)
g3d = sphere1 - sphere2

#Define mesh and function space
mesh = generate_mesh(g3d, 32)

plot(mesh)
plt.show()

V = FunctionSpace(mesh,"Lagrange", 1)
n = FacetNormal(mesh)

# Define basis and bilinear form
u = TrialFunction(V)
v = TestFunction(V)

L = u*v*ds

I really don´t know what would be the next step I tried reading some examples but I don´t understand quite well the use of some commands as PETScMatrix(),PETScMatrix(). Any suggestion or guidance would be greatly appreciated.

Thank you very much.
1
did you see the question about Eigenvalues posted just recently?

written 13 months ago by pf4d
Thank you very much!
written 13 months ago by Leoncio