An elliptic partial differential equation Lu=f with a zero Dirichlet boundary condition is converted to an equivalent elliptic equation on the unit ball. A spectral Galerkin method is applied to the reformulated problem, using multivariate polynomials as the approximants. For a smooth boundary and smooth problem parameter functions, the method is proven to converge faster than any power of 1/n with n the degree of the approximate Galerkin solution. Examples in two and three variables are given as numerical illustrations. Empirically, the condition number of the associated linear system increases like O(N), with N the order of the linear system.Comment: This is latex with the standard article style, produced using Scientific Workplace in a portable format. The paper is 22 pages in length with 8 figure

Let be an open, simply connected, and bounded region in R d , d ≥ 2, and assume its boundary ∂ is smooth. Consider solving the elliptic partial differential equation − u + γ u = f over with a Neumann boundary condition. The problem is converted to an equivalent elliptic problem over the unit ball B, and then a spectral method is given that uses a special polynomial basis. In the case the Neumann problem is uniquely solvable, and with sufficiently smooth problem parameters, the method is shown to have very rapid convergence. Numerical examples illustrate exponential convergence.

This paper considers the numerical solution of boundary integral equations of the second kind for Laplace's equation Au 0 on connected regions D in R with boundary S. The bounda S is allowed to be smooth or piecewisc smooth, and we let {AK _< K _< N} be a triangulation of S. The numerical method is collocation with approximations which arc pieccwise quadratic in the parametrization variables, leading to a numerical solution UN. Superconvergence results for UN are given for S a smooth surface and for a special type of refinement strategy for the triangulation. We show that u us is 0(84 log 8) at the collocation node points, with 8 being the mesh size for A K }. Error analyses are given are given for other quantities, and an important error analysis is given for the approximation of S by piecewise quadratic interpolation on each triangular element, with S either smooth or piecewise smooth. The convergence result we prove is only 0(82) but the numerical experiments suggest the result is 0(84) for the error at the collocation points, especially when S is a smooth surface. The numerical integration of the collocation integrals is discussed, and extended numerical examples are given for problems involving both smooth and piecewise smooth surfaces.

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We consider solving integral equations on a piecewise smooth surface S in R 3 with a smooth kernel function, using piecewise polynomial collocation interpolation of the surface. The theoretical analysis includes the effects of the numerical integration of the collocation coefficients and the use of the approximating surface. The resulting order of convergence is higher than had previously been expected in the literature. Introduction. Consider the integral equation( 1)with k(P, Q) continuous for P, Q ∈ S, and with S a piecewise smooth surface in R 3 . We write the equation ( 1) assymbolically. We assume λ is nonzero and is not an eigenvalue of the integral operator K defined implicitly in (1). Thus, (1) has a unique solution f ∈ C(S) for each g ∈ C(S). In this paper we use collocation with piecewise quadratic interpolation for both the surface S and the unknown function f , as proposed in Atkinson [3].In practice, most of the 3-D boundary integral equations that arise do not have a smooth kernel. The major motivation of this paper is to develop the tools needed for handling boundary integral equations. Also, this paper is the first paper of a sequence of two papers. The second paper, Atkinson and Chien [6], will discuss a nonsmooth kernel case.