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A scheme for rapidly and accurately computing solutions to boundary integralequations (BIEs) on rotationally symmetric surfaces in three dimensions ispresented. The scheme uses the Fourier transform to reduce the original BIEdefined on a surface to a sequence of BIEs defined on a generating curve forthe surface. It can handle loads that are not necessarily rotationallysymmetric. Nystrom discretization is used to discretize the BIEs on thegenerating curve. The quadrature used is a high-order Gaussian rule that ismodified near the diagonal to retain high-order accuracy for singular kernels.The reduction in dimensionality, along with the use of high-order accuratequadratures, leads to small linear systems that can be inverted directly via,e.g., Gaussian elimination. This makes the scheme particularly fast inenvironments involving multiple right hand sides. It is demonstrated that forBIEs associated with Laplace's equation, the kernel in the reduced equationscan be evaluated very rapidly by exploiting recursion relations for Legendrefunctions. Numerical examples illustrate the performance of the scheme; inparticular, it is demonstrated that for a BIE associated with Laplace'sequation on a surface discretized using 320 000 points, the set-up phase of thealgorithm takes 2 minutes on a standard desktop, and then solves can beexecuted in 0.5 seconds.

A Direct Solver for the Rapid Solution of Boundary Integral Equations on Axisymmetric Surfaces in Three Dimensions