Solving PDE with non-trivial null space


I am solving the following problem (solvability condition satisfied)

"a(u,v)- lam* b(u,v) = L(v),

where lam (non-zero) is the eigenvalue and the non-trivial null space is spanned by gfu, i.e. a(gfu,v) = lam* b(gfu,v)."

I assume that one cannot simply assemble the bilinear form "a(u,v)- lam* b(u,v) ", since lam is an eigenvalue. Is there a way to solve this PDE in netgen?

Any help is greatly appreciated.


Could this be realized through defining a FE space incorporating that the solution u is orthogonal to gfu? If yes, does netgen provides such modifications?

I don’t think you can do this purely in NGSolve without some C++ hacking, but if efficiency is not the key point you can use NGSolve to assemble the matrices, then convert them to scipy matrices and do the orthogonalization and solving in scipy…

Thanks for the reply. Efficiency is not an issue here :slight_smile:


I think we need more information for this problem.

Have you computed the Eigenvalue/Eigenvector in advance, or should it be part of the same equation ?
Are your matrices symmetric and positive definite ?

If you know the Eigensystem, you can pose your equation orthogonal to the Eigenvector using a scalar Lagrange parameter (from NumberSpace).


Hi Joachim:

Thank you very much for the reply. I have computed the eigenvalues/eigenfunctions in advance using Arnoldi solver; the matrices are symmetric and non-negative definite.

I am not sure yet how to use a scalar Lagrange parameter (from NumberSpace), is there a short illustrative example on this?


Ok I’m sorry I was wrong… With Joachims comment I think it’s actually quite easy…
I think this should do what you want:

from netgen.geom2d import unit_square
from ngsolve import *

mesh = Mesh(unit_square.GenerateMesh(maxh=0.3))


u,v = h1.TnT()

a = BilinearForm(h1)
a += SymbolicBFI(grad(u)*grad(v))
b = BilinearForm(h1)
b += SymbolicBFI(u*v)


evecs = GridFunction(h1,multidim=50)
vlam = ArnoldiSolver(a.mat,b.mat,h1.FreeDofs(),evecs.vecs,1)

# get a non multidim GF out of the multidim one
evec0 = GridFunction(h1) = evecs.vecs[0]

number = NumberSpace(mesh,complex=True)
fes = FESpace([h1,number])

(u,unum), (v,vnum) = fes.TnT()

# The orthogonality condition is satisfied because vnum = 1 means that evec0 * u = 0
# unum incorporates the evec0 part of the right hand side, if the argument of f is orthogonal to
# evec0, then unum should be 0
alam = BilinearForm(fes)
alam += SymbolicBFI(grad(u) * grad(v) - vlam[0] * u * v + unum * v * evec0)
alam += SymbolicBFI(evec0 * u * vnum)

f = LinearForm(fes)
f += SymbolicLFI(1*v)


sol = GridFunction(fes,"solution") = alam.mat.Inverse(fes.FreeDofs()) * f.vec


This is great! I appreciate this short example and thank you very much!