Abstract. In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of the second kind with a weakly singular kernel. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval [−1, 1], so that the solution of the new equation possesses better regularity and the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high-order accuracy for the approximation, the integral term in the resulting equation is approximated by using Jacobi spectral quadrature rules. The convergence analysis of this novel method is based on the Lebesgue constants corresponding to the Lagrange interpolation polynomials, polynomial approximation theory for orthogonal polynomials and operator theory. The spectral rate of convergence for the proposed method is established in the L ∞ -norm and the weighted L 2 -norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.

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Thermal transport properties of isotopic-superlattice graphene nanoribbons with zigzag edge (IS-ZGNRs) are investigated. We find that by isotopic superlattice modulation the thermal conductivity of a graphene nanoribbon can be reduced significantly. The thermal transport property of the IS-ZGNRs strongly depends on the superlattice period length and the isotopic mass. As the superlattice period length decreases, the thermal conductivity undergoes a transition from decreasing to increasing. This unique phenomenon is explained by analyzing the phonon transmission coefficient. While the effect of isotopic mass on the conductivity is monotone. Larger mass difference induces smaller thermal conductivity. In addition, the influence of the geometry size is also discussed. The results indicate that isotopic superlattice modulation offers an available way for improving the thermoelectric performance of graphene nanoribbons.

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In this paper, we study the Legendre-Galerkin spectral approximation for a constrained optimal control problem. We first derive a priori error estimates for the spectral approximation of optimal control problems. Then a posteriori error estimates are obtained for both the state and the control approximation. A preconditioning projection algorithm is proposed with some numerical tests. Introduction.The spectral method employs global polynomials as the trial functions for the discretization of PDEs. It provides very accurate approximations with a relatively small number of unknowns when the solutions are smooth. Recently the spectral method has been extended to approximate unconstrained optimal control problems; see, for example, [10].However spectral accuracy generally cannot be achieved when the approximated solutions have lower regularities, and this is typically the case when, for example, there exist control constraints in an optimal control problem (the so-called constrained optimal control problem). Thus the spectral method is not widely used in solving constrained distributed optimal control problems where the solutions often have singularities at the boundary of constraints even though all the initial data are smooth. Thus although there has been much work on the finite element method for numerically solving constrained optimal control problems, it seems that there has not been much work on the spectral method. Furthermore, the optimality conditions, which are normally the starting point of spectral approximation, are just PDE systems for unconstrained control problems, while those for constrained control problems contain variational inequalities, as shown later on. This also raises new issues in analyzing and solving the systems discretized using the spectral method.In this work we study a constrained optimal control problem. It in fact represents a class of useful optimal control problems. We show that the solution of this particular control problem can be infinitely smooth if the initial data are so. Then we propose to use the Galerkin spectral method to approximate the solution. As there

Abstract. In this paper, we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods. The state and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique. We derive L 2 superconvergence properties for the flux functions along the Gauss lines and for the scalar functions at the Gauss points via mixed projections. Moreover, global L 2 superconvergence results are obtained by virtue of an interpolation postprocessing technique. Thus, based on these superconvergence estimates, some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods. Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence.

SUMMARYThe aim of this work is to investigate the discretization of a quadratic convex optimal control problem using the mixed finite element method. The state and co-state are approximated by the order k 1 RaviartThomas mixed finite element spaces, and the control is approximated by piecewise constant functions. We construct an interpolation of the exact control and a projection of the discrete scalar co-state to be the approximated solution of the control variable for the continuous optimal control problem. As a result, it can be proved that the difference between the interpolation and the piecewise constant approximation has superconvergence property for the control of order h 3/2 for k = 0 and of order h 2 for k = 1. Moreover, only for the order k = 1 Raviart-Thomas mixed finite element approximation does the postprocessing technique possess the superconvergence property of order h 2 . Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence.

SUMMARYWe present two efficient methods of two-grid scheme for the approximation of two-dimensional semilinear reaction-diffusion equations using an expanded mixed finite element method. To linearize the discretized equations, we use two Newton iterations on the fine grid in our methods. Firstly, we solve an original non-linear problem on the coarse grid. Then we use twice Newton iterations on the fine grid in our first method, and while in second method we make a correction on the coarse grid between two Newton iterations on the fine grid. These two-grid ideas are from Xu's work (SIAM J. Sci. Comput. 1994; 15:231-237; SIAM J. Numer. Anal. 1996; 33:1759-1777) on standard finite element method. We extend the ideas to the mixed finite element method. Moreover, we obtain the error estimates for two algorithms of two-grid method. It is showed that coarse space can be extremely coarse and we achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O(h 1/4 ) in the first algorithm and H = O(h 1/6 ) in second algorithm.