Exponentially Accurate Approximations for Dirichlet Problems with Discontinuous data

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Abstract

A method to obtain exponentially accurate approximations to solutions for Dirichlet problems with discontinuous boundary data for Laplace's equation in two dimensions is presented and discussed. Model problems with circular or rectangular boundaries, whose solutions can be obtained by separation of variables involving Fourier series, are discussed in detail. First, the boundary data g is expressed as the sum of a singular function S˜M, which is a certain linear combination of specially constructed ‘singular basis functions’ {Sn}, and a function, namely, g − s˜M, which is much smoother than the original data. The function S˜M is constructed so that its discontinuities, and those of its first M derivatives, coincide with the corresponding discontinuities of g. The solution u to the boundary-value problem is then expressed as the sum of a linear combination of the harmonic extensions {ϕn} of {Sn}, and a function v, which satisfies the boundary condition v = g − S˜M. Since the boundary data for v has at least M continuous derivatives, the partial sum approximations for v obtained by separation of variables converge much faster than the corresponding partial sum approximations for u. Formally, by letting N, the number of terms retained in the solution for v, be proportional to M, a sequence of approximations can be constructed which converges to u exponentially in the maximum norm, as M → ∞. In particular, this implies that, when g is discontinuous, the unwanted effects of the Gibbs phenomenon can be completely overcome! The method is illustrated by several examples, and some possible applications to related problems are discussed.

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