Nonseparable geometry

The implementation of the Capacitance Matrix method for Dirichlet or Neumann boundary conditions is straightforward. We consider an example that is nonstandard and could apply to the problem of nonreflecting boundary conditions. At the boundary of domain, $D$, the noslip boundary condition for Navier-Stokes flow requires that $\psi=0$ and $\frac{\partial\psi}{\partial n}=0.$ We use iteration to solve the implicit scheme at each time step. At each step of the iteration we are required to solve for the stream function

$$\Delta \psi = -c$$

subject to the no-slip boundary conditions. This is a nonstandard boundary value problem in the sense that we have two data for the stream function, rather than one.

We have found it convenient to adopt a new technique for solving this problem. That is, we solve the system $\Delta \psi = f + \nabla\cdot \hat\rho$ in a periodic domain containing $D$ where $\hat \rho$ is supported on the boundary of $D$, $\partial D$. The boundary conditions on $\partial D$ are $\nabla \psi = 0.$ We specify the vector $\hat \rho$ by the capacitance matrix for the system

$$\nabla\psi - \Delta^{-1}\left(\nabla f\right) = \nabla\Delta^{-1}\nabla\cdot \hat\rho$$

restricted to the boundary The operator $\nabla\Delta^{-1}\nabla\cdot$ projects a vector onto its gradient component (is a projection in $L^2(R^2)$. The capacitance matrix for this problem is symmetric. In spite of the fact that the capacitance matrix is twice as large, the Navier-Stokes algorithm is only a third slower than the Euler algorithm. The algorithm, subject to further testing, appears to be accurate and fast.

This algorithm can be adapted to define nonreflecting boundary conditions.