# eigenvalues of inhomogeneous Helmholtz equation violate superposition using FEM

I am trying to solve the in-homogeneous Helmholtz equation with damping and forcing using finite element method (FEM) with FEniCSx. The equation is;

$$c^2\nabla^2p - c^2\omega i d \nabla^2p + \omega^2 p = f$$

where the $$d$$ is the damping parameter, the $$i$$ is complex number, the f is the forcing, the $$\omega$$ is the eigenvalue and the $$p$$ is the eigenvector. In matrix form;

$$(\mathbf{A} + \omega \mathbf{B} + \omega^2 \mathbf{C})\mathbf{p} = \mathbf{D}\mathbf{p}$$

where $$\mathbf{B}$$ is the damping matrix and $$\mathbf{D}$$ is the forcing matrix. For 1D case, the boundary conditions are Neumann boundary at both ends.

I am aiming to damp the system with $$\mathbf{B}$$ and did try 4 different cases;

• $$\mathbf{A} + \omega^2 \mathbf{C} = 0$$ (no damping and no forcing)
• $$\mathbf{A} + \omega \mathbf{B} + \omega^2 \mathbf{C} = 0$$ (damping and no forcing)
• $$(\mathbf{A}-\mathbf{D}) + \omega^2 \mathbf{C} = 0$$ (no damping and forcing)
• $$(\mathbf{A}-\mathbf{D}) + \omega \mathbf{B} + \omega^2 \mathbf{C} = 0$$ (damping and forcing)

When I run this cases, I get very weird behaviour. Here is the list of the eigenvalues for the cases;

• $$\mathbf{A} + \omega^2 \mathbf{C} = 0$$ gives $$\omega=1069.2399-0j$$
• $$\mathbf{A} + \omega \mathbf{B} + \omega^2 \mathbf{C} = 0$$ gives $$\omega=1139.6967-0.2463j$$
• $$(\mathbf{A}-\mathbf{D}) + \omega^2 \mathbf{C} = 0$$ gives $$\omega=1074.748+3.2431j$$
• $$(\mathbf{A}-\mathbf{D}) + \omega \mathbf{B} + \omega^2 \mathbf{C} = 0$$ gives $$\omega=1146.0758+3.5129j$$

The complex part of the eigenvalue is representing the growth rate of the system. As I impose a damping, the last case's complex part (+3.5129j) should be lower than the 3. case's complex part (3.2431j) according to the law of superposition but it is not.

Would anyone can enlight me about what I am doing wrong, I am struggling with this for more than a week.

Thank you!

• Usually, the forced Helmholtz equation is not an eigenvalue problem. Could you elaborate? Nov 30, 2023 at 0:25
• Sure, the forcing is a function of an eigenvalue $\omega$. Hence I find the matrix D iteratively and I use a method called fixed point iteration. The fundamental paper about this problem stated in Section IV.B in hal.science/file/index/docid/908192/filename/paper.pdf Nov 30, 2023 at 10:29
• Which eigenvalues are being shown here? The one with the largest magnitude? Largest real part? Nov 30, 2023 at 13:26
• These eigenvalues belong to 1st axial mode of the 1m tube. Nov 30, 2023 at 14:32
• Even though you are approximating the solution via a sequence of standard eigenvalue problems, the problem you are attempting to solve is a quadrativ eigenvalue problem and you are trying to udnerstand the effect of perturbing one of the matrices in the problem via a damping matrix. I don't see any easy way of analyzing the effect of this perturbation. Nov 30, 2023 at 20:37