# Helmholtz equation and the Maximum Principle

## Main Question or Discussion Point

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.