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Weak Form of the Poisson Problem

  1. Mar 25, 2014 #1
    Hi,

    I know the weak form of the Poisson problem

    [itex]\nabla^2 \phi = -f[/itex]

    looks like

    [itex]\int \nabla \phi \cdot \nabla v = \int f v[/itex]

    for all suitable [itex]v[/itex]. Is there a similarly well-known form for the slightly more complicated poisson problem?

    [itex]\nabla (\psi \nabla \phi ) = -f[/itex]

    I am writing some finite element code and variational/weak forms are very handy.

    Thanks in advance
     
    Last edited: Mar 25, 2014
  2. jcsd
  3. Mar 25, 2014 #2

    pasmith

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    Homework Helper

    By the product rule
    [tex]
    \int_V v\nabla \cdot(\psi \nabla \phi)\,dV = \int_V\nabla\cdot(v\psi \nabla \phi) - \psi (\nabla \phi) \cdot (\nabla v)\,dV = \int_{\partial V} v\psi \nabla \phi \cdot dS - \int_V\psi (\nabla \phi) \cdot (\nabla v)\,dV[/tex]
    and hence the weak form of [itex]\nabla \cdot(\psi\nabla\phi) = - f[/itex] is
    [tex]
    \int_V \psi (\nabla \phi) \cdot (\nabla v)\,dV = \int_V fv\,dV
    [/tex]
     
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