Title: Method for Imposing Boundary Conditions at Irregular Boundaries
Using a Regular Grid
Speaker: Martin Z. Bazant,
Dept. of Mathematics
University of Arizona, Tucson, AZ
Date: April 16, 1993
Place: SIAM Conference on Nonlinear Mathematics and Computation
SUNY - Stony Brook, NY
Abstract:
In the interest of accurately imposing boundary
conditions, the numerical solution of a partial differential
equation on an irregular domain is often computed with
an irregular grid that somehow conforms to the domain. The
computational simplicity offered by a regular grid, however, is
so overwhelming in many applications that it is worth
considering how to optimize its use for an irregular domain to
which it cannot conform. A finite difference method is presented
for imposing conditions at irregular boundaries represented by
points that do not conform in any way to a regular grid on which
continuum fields are computed. The fundamental idea is to use
certain grid points near the boundary to minimize the mean
square error in satisfying the boundary condition. The boundary
points are coupled to the grid via a discrete approximation to
the Dirac $\delta$-function. It is shown that the method
smoothes high frequency oscillations in boundary data and shape
in a natural way. The method also extends the range of
applicability multigrid methods because a square grid of
arbitrary spacing can be used to obtain reasonable solutions
regardless of the complexity of the computational domain.
Finally, the method allows the use of a fixed, regular grid for
free boundary problems and can be generalized to implement
implicit boundary movement that is stable and second order in
time.