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I'm new to this weak formulation theories, and I'm having a hard time deriving the weak form to following PDE:

$$E(u)=-\nabla^2u -k^2 u=0$$

using the fact that for the minimizer $\displaystyle\lim_{\epsilon\to 0}\frac{d}{d\epsilon}E(u+\epsilon v) = 0$

  1. Is this method general for all kind of PDE equations (integral form and ...)? see link below Derivation of weak form for variational problem
  2. If not, could you suggest any other method? Thanks.
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Ghartal
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  • Quick comment: for derivation of weak PDE form $E(u)$ must be a mapping from functions space to $\mathbb{R}$. In your case $E(u)$ is an operator. – Evgeny Aug 07 '13 at 18:12

1 Answers1

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Is this method general for all kind of PDE equations (integral form and ...)? see link below:Derivation of weak form for variational problem

This method actually originates from calculus of variations when looking for the Euler-Lagrange equation for a functional. This functional usually describes certain potential and kinetic energy (this is why people usually call this functional energy functional). For example, from electrostatic potential energy functional to Poisson equation. The expression $$ \lim_{\epsilon\to 0} \frac{d}{d\epsilon}E(u+\epsilon v) = 0 \;\text{ for any }v\in V \iff u = \operatorname{argmin}\limits_{v\in H} E(v). $$

$E(u)=-\nabla^2u -k^2 u=0.$

This is not accurate. $E(u)$ should be identified with the energy functional I mentioned above, not the underlying PDE you gave, which is the Helmholtz equation (or say the eigenvalue problem for $-\Delta = -\nabla^2$) $$ -\Delta u = k^2u.\tag{1} $$ The energy functional you wanna minimize is actually the Rayleigh quotient: $$ E(v) = \frac{\int_{\Omega} |\nabla v|^2}{\int_{\Omega} v^2}, $$ given the Rayleigh quotient is $k^2$. The weak form you will obtain will be (left for you as an exercise, this is called the Minimum Principle of the Eigenvalue) $$ \lim_{\epsilon\to 0} \frac{d}{d\epsilon} E(u+\epsilon v) = 0 \implies \int_{\Omega} \nabla u\cdot \nabla v - k^2 \int_{\Omega} uv = 0.\tag{2} $$

If not, could you suggest any other method? Thanks.

A simpler, but containing less physics based heuristics, method is that multiplying both sides of (1) by a test function $v$, then integration by parts using Green's formula (or divergence theorem, Stokes, etc):

$$ \int_{\Omega }-\Delta u v =k^2\int_{\Omega } uv \implies \int_{\Omega} \nabla u\cdot \nabla v -\color{blue}{\int_{\partial \Omega} v\frac{\partial u}{\partial \nu} \,dS}= k^2 \int_{\Omega} uv. $$ Then we set the test function $v=0$ on the boundary to make blue term vanish. Hence you have (2). The reason to do this for Dirichlet eigenvalue problem (equation (1) with Dirichlet boundary value) is that: The test functions should not change the boundary value of the solution to the underlying PDE.

Some remark: in the question you link, the OP's notation was a bit inaccurate, the bilinear form should NOT be identified with the same notation with the energy functional.

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Shuhao Cao
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  • +1 for this extensive answer, very nice. I was wondering, i had never learnt or even heard of this variational or weak form approach to PDE's, i definitely intend to discover more about it. Do you happen to know of any suited textbook for beginners that covers this? (Or to see solved examples of famous equation, in order to get a grip on the idea) many thanks in any case. For example i got completely lost trying to understand the min. principle of the eigenvalue method you used for helmholtz. – user929304 Sep 30 '16 at 14:00
  • @user929304 Any linear algebra book covering eigenvalue should have the derivation using matrices for the minimization of the Rayleigh quotient. For the variational approach of PDEs, the book Convex Analysis and Variational Problems by Temam is a good read. It has a lot example using PDEs. – Shuhao Cao Sep 30 '16 at 16:22