| High order methods, such as spectral methods and the p-version of the finite element method, are based on approximation by polynomial interpolation, where the degree of the polynomial may become large. We analyze the L('(INFIN)) error. If the L('(INFIN)) error is large, then the numerical implementation dictates that L('p) errors will probably also be large.; Derivatives are approximated by derivatives of interpolants, and convergence of this approximation is determined by the choice of node spacing and by the smoothness of the function. The optimal mesh for approximating derivatives is given by the points, cos j(pi)/N, j = 0, 1, ..., N. Using results known from the literature, we easily obtain an error estimate for approximating derivatives on arbitrary meshes, but this estimate is not sharp. For the optimal mesh, we prove an optimal estimate. Extensions to arbitrary meshes and higher dimensions are discussed. Finally, we apply our results to the solution of a partial differential equation with special attention to the control of round-off error. |
