Dual Reciprocity Method with Multigrid Technique
by C. Gaspar
Szechenyi Istvan College, Department of Mathematics
P.O.Box 701, H--9007 Gyor, Hungary
e-mail: csgaspar@d7.szif.hu
Abstract
The Dual Reciprocity Method is a well known and widely used tool to
generalize the Boundary Integral Equation Method to more complex problems
(even to nonlinear problems). Its essential advantage is that both the
discretization of the domain and the evaluation of domain integrals can be
avoided by converting these integrals to a sum of boundary integrals. Thus,
the main advantages of the Boundary Integral Equation Methods are preserved.
However, the Dual Reciprocity Method is based on a special interpolation of
irregularly spaced data, which has serious drawbacks from a numerical point
of view (it produces large, dense and ill-conditioned system of algebraic
equations) and the usual fast methods such as multigrid tools seem to be
inapplicable. We present a method based on a biharmonic interpolation,
which allows the use of highly efficient multigrid techniques. Though this
interpolation, strictly speaking, leads to a domain type solution procedure,
this subproblem can be solved completely independently of the original
boundary integral equations, without using and discretizing its domain.
The resulting method can be regarded as a combination of the dual and the
multiple reciprocity methods. However, the solution of dense and ill-
conditioned systems of equations is thus avoided, and also the memory
requirement is significantly reduced.