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From: Tammo on 12 Jan 2010 17:35
hi,

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.

thx,
t.
From: eric gisse on 12 Jan 2010 20:30
Tammo wrote:

> hi,
>
> 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.

If you are referring to how the function has its' maximum on the boundary of
the region in question, that is only true for Laplace's equation as I
recall. Expecting the extremes to be on the boundary in an elliptic PDE
isn't the best plan.

As for literature on the subject, I'd say any comprehensive book on partial
differential equations will have what you need to learn how to solve it
numerically. Or at least guide you in the right direction.

>
> thx,
> t.

From: Tammo on 13 Jan 2010 03:39
On jan. 13, 02:30, eric gisse <jowr.pi.nos...(a)gmail.com> wrote:
> Tammo wrote:
> > hi,
>
> > 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.
>
> If you are referring to how the function has its' maximum on the boundary of
> the region in question, that is only true for Laplace's equation as I
> recall. Expecting the extremes to be on the boundary in an elliptic PDE
> isn't the best plan.
>
> As for literature on the subject, I'd say any comprehensive book on partial
> differential equations will have what you need to learn how to solve it
> numerically. Or at least guide you in the right direction.
>
>
>
> > thx,
> > t.

In case of the Laplace eq. it is for sure true.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.
From: Willi =??B?TcO2aHJpbmc=?= on 15 Jan 2010 06:06
Tammo wrote:

> hi,
>
> 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

In 3-d sin(k|x|)/|x| is a solution of the Helmholtz equation \Delta f +k^2 f = 0. It has a local maximum, therefore a maximum principle cannot be valid.

On the other hand \Delta f has to be non-positive for a local maximum. Therefore, because of \Delta f = -k^2 f there cannot be a maximum of f, where f is negative.

Willi
From: Tammo on 19 Jan 2010 16:30

hi,

I found your comment to be very valuable.
Somehow, as a prejudgement, I expected that the maximum principle
wouldn't be true for the Helmholtz equation,
as the solutions are wave functions which do not necessarily take
their extremities on the boundary.

However from your second statement "because of \Delta f = -k^2 f
there cannot be a maximum of f, where f is negative."
there could still be a local minimum (extremum) where f is negative
and \Delta f is then positive.

Could you maybe suggest some literature references regarding the
topic ?

On jan. 15, 12:06, Willi =??B?TcO2aHJpbmc=?= <wmoe...(a)gwdg.de> wrote:
> Tammo wrote:
> > hi,
>
> > I am dealing with a numerical method to solve theHelmholtzequation.
> > 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 theHelmholtzeq., or where can I find explicit literature about
>
> In 3-d sin(k|x|)/|x| is a solution of theHelmholtzequation \Delta f +k^2 f = 0. It has a local maximum, therefore a maximum principle cannot be valid.
>
> On the other hand \Delta f has to be non-positive for a local maximum. Therefore, because of \Delta f = -k^2 f  there cannot be a maximum of  f, where f is negative.
>
> Willi

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