Helmholtz equation and the Maximum Principle

[Tammo]

hi everyone,

I am dealing with a numerical method to solve the Helmholtz equation.
As far as I know it is a second order elliptic PDE. I would like to
know, if the Maximum Principle (aka Boundary Maximum Principle) holds
for the Helmholtz eq., or where can I find explicit literature about
it.
In Collatz's 'Numerical Treatment of Differential Equations' it is written, that it is
generally applicable to elliptic PDE-s in case of certain conditions.
In other literature I read that it does not hold for the Helmholtz
eq., because of it's special structure.
The relevance of the question for me is the following: if one uses the
solutions of the homogenous PDE as approximation functions for the
numerical solution, the solution will satisfy the PDE, but not the
boundary conditions. But in this case the error function will also
satisfy the PDE, which if the Max. Principle is applicable, takes its
maximum on the boundary, which is then good to have an upper bound on
the error within the domain.

thx,
t.

 

[jvicens]

Dear t.,

Thank you for bringing up this interesting question about the Maximum Principle and its applicability to the Helmholtz equation.

The Maximum Principle is a powerful tool in the analysis of elliptic PDEs and can provide important insights into the behavior of solutions. However, its applicability to the Helmholtz equation is not straightforward and has been a topic of debate in the literature.

Some studies have shown that the Maximum Principle does hold for certain types of boundary conditions and specific domains. However, in general, it does not hold for the Helmholtz equation due to its special structure and the presence of complex-valued solutions. Therefore, it is important to carefully consider the conditions under which the Maximum Principle can be applied to the Helmholtz equation.

I recommend looking into more recent literature on the topic, as well as consulting with experts in the field, for a more comprehensive understanding of the current understanding of the Maximum Principle for the Helmholtz equation.

I also want to address your concern about the boundary conditions and the error function in the numerical solution. While it is true that using the solutions of the homogeneous PDE as approximation functions can lead to a solution that satisfies the PDE but not the boundary conditions, this issue can be addressed by using appropriate boundary conditions or incorporating them into the approximation functions. Additionally, the error function can still be useful in estimating the error within the domain, even without the application of the Maximum Principle.

I hope this helps clarify the applicability of the Maximum Principle to the Helmholtz equation. Thank you for your question and good luck with your research.