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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.

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# Helmholtz equation and the Maximum Principle

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