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Abstract. In this paper, in order to numerically solve for multiple positive solutions to a singularly perturbed Neumann boundary value problem in mathematical biology and other applications, a local minimax method is modified with new local mesh refinement and other strategies. Algorithm convergence and other related properties are verified. Motivated by the numerical algorithm and convinced by the numerical results, a Morse index approach is used to identify the Morse index of the root solution u 1 ε = 1 at any perturbation value, its bifurcation points and then the critical perturbation value. Many interesting numerical solutions are computed for the first time and displayed with their contours and mesh profiles to illustrate the theory and method.

This paper provides a rigorous and delicate analysis for exponential decay of Jacobi polynomial expansions of analytic functions associated with the Bernstein ellipse. Using an argument that can recover the best estimate for the Chebyshev expansion, we derive various new and sharp bounds of the expansion coefficients, which are featured with explicit dependence of all related parameters and valid for degree n ≥ 1. We demonstrate the sharpness of the estimates by comparing with existing ones, in particular, the very recent results in SIAM J. Numer. Anal., 50 (2012), pp. 1240-1263. We also extend this argument to estimate the Gegenbauer-Gauss quadrature remainder of analytic functions, which leads to some new tight bounds for quadrature errors.

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Abstract. This paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the Gegenbauer-Gauss and Gegenbauer-Gauss-Lobatto points, when the underlying function is analytic on and within an ellipse. Sharp error estimates in the maximum norm are derived.

Based on an asymptotic expansion of finite element, a new extrapolation formula and extrapolation cascadic multigrid method (EXCMG) are proposed, in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid. In the case of triple grids, the error of the new initial value is analyzed in detail. A larger scale computation is completed in PC.

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