Thoughts on Fast Multipole II

In last post, I sort of believe that KIFMM was limited to a small portion of PDE, especially for second order elliptic equations, while the deeper idea is not the same thing at all.

Equivalence of information

I realized that there is something called Equivalence of information here, for example, Laplacian $\Delta u=0$ made solution depending only information on boundary, as we called boundary value problems. Then we can define the equivalence class of equation $Pu=0$ that associated with pseudo-differential operator $P$.

In theory, even higher ordered elliptic equation

$P(x,D)u={\sum }_{|\alpha |\le 2m}{c}_{\alpha }{D}^{\alpha }u=0$

we can see that for this equation, classical results could apply easily, and we consider the weak formulation, if vector-function $\tilde{g}=(g_{\alpha }{)}_{|\alpha |\le m-1}$, which satisfies $\Vert \tilde{g}\Vert ={\sum }_{\alpha }\Vert g_{\alpha }\Vert$ is finite, we say $\tilde{g}\in W^{m-1}(\partial \Omega )$. We are looking for $u\in W^{2m}(\Omega )\cap W^{m-1}(\overline{\Omega })$ such that $Pu=0$ and $D^{\alpha }u{|}_{\partial \Omega }=g_{\alpha }$.

For any bounded domain $\Omega$, there is unique weak solution to the Dirichlet problem $Pu=0$. When boundary is smooth enough, the weak solution is also classical solution.

On the other hand, the exterior problem also has unique solution for the same Dirichlet boundary conditions.

If $Pu=0$, for any given measure ${\nu }^{\alpha }\in \Omega$, there exists ${\mu }^{\alpha }\in \partial \Omega$ that ${\sum }_{|\alpha |\le m-1}\int (D^{\alpha }u)d{\mu }^{\alpha }={\sum }_{|\alpha |\le m-1}\int (D^{\alpha }u)d{\nu }^{\alpha }$

The proof should be found somewhere in ADN (Agmon, Douglis, Nirenberg), uniqueness could be concluded from something similar of Lax-Milgram. That also reveals the fact that this operator $P$ has equivalence class $\Omega \sim (\partial \Omega {)}^m$. Since we can use finite difference method to approximate $D^{\alpha }u$ on $\partial \Omega$, by giving $u$ at $\partial \Omega +t_i\mathbf{n}$ surfaces, where $t_0,\cdots ,t_{m-1}\in (0,ϵ)$. Now our problem is how to generate those surfaces/or small domains for equivalence potential.

Equivalent cluster of surfaces

Define cluster of surfaces $C(t_i),i=0,\cdots ,m-1$, where $C(t_i)=\{x+t_i\mathbf{n},x\in \partial \Omega \}$. We also consider the solutions on each surfaces $C_i=C(t_i)$ as $u_i$.

Then $D^{\alpha }u$ can be computed through finite difference. Especially, on square surface, this is quite straightforward to compute each derivatives.

Exterior to Interior problem

Consider free space Green function for internal source $f$, the whole space solution is $u(x)={\int }_{\Omega }K(x,y)f(y)dy$ and here we are looking for ${\Lambda }_{\alpha }$ such that $x\in {\Omega }^c$, $u(x)={\sum }_{|\alpha |\le m-1}{\int }_{\partial \Omega }{\Lambda }_{\alpha }(x,y)\underline{D^{\alpha }u(y)}dy$

Interior to Exterior problem

The exterior problem will be the same, except the source $f$ lives outside of the domain $\Omega$.