Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter e which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.

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The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series

Kowledge of a truncated Fourier series expansion for a 2π-periodic function of finite regularity, which is assumed to be piecewise smooth in each period, is used to accurately reconstruct the corresponding function. An algebraic equation of degree M is constructed for the M singularity locations in each period for the function in question. The M coefficients in this algebraic equation are obtained by solving an algebraic system of M equations determined by the coefficients in the known truncated expansion. If discontinuities in the derivatives of the function are considered, in addition to discontinuities in the function itself, that algebraic system will be nonlinear with respect to the M unknown coefficients. The degree of the algebraic system will depend on the desired order of accuracy for the reconstruction, i.e., a higher degree will normally lead to a more accurate determination of the singularity locations. By solving an additional linear algebraic system for the jumps of the function and its derivatives up to the arbitrarily specified order at the calculated singularity locations, we are able to reconstruct the 2π-periodic function of finite regularity as the sum of a piecewise polynomial function and a function which is continuously differentiab1e up to the specified order

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Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions

Neville extrapolation is generalized to expansions for which the extrapolation coefficients, used in the calculation of the elements in the related extrapolation table, are not known. It is proved that the extrapolation coefficients are the multipliers of a special Gaussian elimination process. The stability of the generalized extrapolation process is discussed. Application of generalized Neville extrapolation to integrals with weight functions/singularities in one or more dimensions is indicated.

Generalized neville type extrapolation schemes

Introduction. Solving boundary value problems for partial differential operators by integral equation methods is not a new idea. However, the classical way to do it consists in representing the unknown solution as a potential of the type that will lead to an integral equation of the second kind. Then, Fredholm's theorems can be used. Thus, the Dirichlet problem is usually solved with the help of a double layer potential, and the Neumann problem with the use of a single layer potential. We shall have a different point of view. Our aim will be to obtain a variational formulation of the problem in order to obtain the existence and unicity of a solution and error estimates. This philosophy leads to opposite choices for the representation of the solution. Thus, J. C. Nedelec and J. Planchard, for the three-dimensional case, and M. N. Leroux for the two-dimensional case, have solved the Dirichlet problem by using a single layer potential. We propose here the solution of a Neumann problem by using a double layer potential. Let 2 be a bounded open set of R3. Let r be the boundary of 2 and Qc denote the complementary set of Q. We assume that r is sufficiently smooth, and we put the coordinates' origin in 2. We shall write nt, for the exterior normal to r, r, for the distance to the origin, [V] = vlint vie,xt, for the jump through r, of the function v defined in Ri3.

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Numerical solution of an exterior Neumann problem using a double layer potential

For an integrand with a 1/r vertex singularity, the Duffy transformation from a triangle (pyramid) to a square (cube) provides an accurate and efficient technique to evalu- ate the integral. In this paper, we generalize the Duffy trans- formation to power singularities of the form p(x)/r α , where p is a trivariate polynomial and α> 0 is the strength of the singularity. We use the map (u ,v , w)→ (x, y, z) : x = u β , y = xv, z = xw, and judiciously choose β to accurately estimate the integral. For α = 1, the Duffy transformation (β = 1) is optimal, whereas if α �= 1, we show that there are other values of β that prove to be substantially better. Numerical tests in two and three dimensions are presented that reveal the improved accuracy of the new transforma- tion. Higher-order partition of unity finite element solutions for the Laplace equation with a derivative singularity at a re-entrant corner are presented to demonstrate the benefits of using the generalized Duffy transformation.

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J.N. Lyness

Papers

Applications of extrapolation techniques to multidimensional quadrature of some integrand functions with a singularity

The Calculation of Trigonometric Fourier Coefficients