Thoughts on Fast Multipole III

Continue with last post. Consider the pseudo-differential elliptic operator $P={\sum }_{\Vert \alpha \Vert \le 2m}{c}_{\alpha }(x)D^{\alpha }$, if we have $Pu(x)=f(x)$, with Dirichlet boundary condition $u^{\alpha }=0$ for $\Vert \alpha \Vert \le m-1$.

${\int }_{\Omega }(P^{\ast }v(x,y))u(x)-(Pu(x))v(x,y)dx={\sum }_{\beta }{\int }_{\partial \Omega }{\psi }_{\beta }(x)v^{\beta }(x,y)dx$

Here consider Green function $P^{\ast }v(x,y)={\delta }_y(x)$, which is $Pv(y,x)={\delta }_y(x)$. If we say $K(x,y)$ is Green function of $P$, then we have to replace $x$ and $y$ in $v$, thus $K(x,y)=v(y,x)$.

$u(y)-{\int }_{\Omega }f(x)K(y,x)dx={\sum }_{\beta }{\int }_{\partial \Omega }{\psi }_{\beta }(x)K^{\beta }(y,x)dx$

Now come back to our previous exterior problem, if $f=0$ in ${\Omega }^c$, then for $y\in \Omega$, here $\Omega$ could be internal domain or external domain.

$u(y)={\sum }_{|\beta |\le m-1}{\int }_{\partial \Omega }{\psi }_{\beta }(x)K^{\beta }(y,x)dx$

In 2D, there are $C_{m+1}^2$ terms in this summation, with discretization, and merge weights into $\psi$.

$u(y_j)={\sum }_{\beta }{\sum }_{k=1}^p{\psi }_{\beta }(x_k)K^{\beta }(x_k,y_j)$

There are $p{C}_{m+1}^2$ unknowns for this problem. Just one surface is enough. This can be seen as an extension of KIFMM method.

Numerical concerns

The summation over $\partial \Omega$ requires too many points (unknowns) to find out. Here we can simulate $K^{\beta }(x_k,y_j)$ by taking finite difference to approximate the derivatives. Then we only need $m$ surfaces to find out all $K^{\beta }(x,\cdot )$, each surfaces we can place equally spaced $p$ points.

Infinite order case

It seems to me that given all points inside a annulus will be enough for this.

Failure for numerics

Just for a update. In numerical world, finite difference is rather a really bad idea to approximate derivatives, esp. those high order terms. Thus KIFMM is an effective fast low order PDE solver.