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Helmholtz equation and the Maximum Principle

  1. Jan 14, 2010 #1
    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.
     
  2. jcsd
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