Reformulating a Boundary-Integral Equation in Three Dimensions as as Integral-Operator Problem in a Plane Region

Dallas, A.G.
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Department of Mathematical Sciences
Motivated by a desire to simplify the design of numerically stable and efficient approximation schemes for boundary-operator problems, we develop a framework in which an integral equation on the boundary of a domain ${I\kern-.30em R}^3$ can be systematically reformulated as an integral-operator problem set in a region in the plane; some geometric restrictions are imposed on the shape of the (smooth) boundary. When the plane region is chosen to be a rectangle, the necessary Sobolev-space structures can be handled numerically rather easily in the new simpler geometry, in contrast to the situation on the original boundary. Moreover, familiar trial- and test- functions can then be employed in the construction of approximate solutions of the reformulated problems. We show for two examples how a well-posed problem can be transferred from the domain-boundary setting to the plane-region setting. We describe a numerical implementation of these ideas to a lower-dimensional example involving the approximate solution of a first-kind integral equation associated with the Helmholtz equation that is originally set on the boundary of a domain in ${I\kern-.30em R}^2$.