Asymptotic Error Analysis

When computing numerical approximations to problems with smooth solutions using regular grids, the error can have additional structure. The historical example of the Euler-McLaurin formula for the approximation of integrals with the trapezoidal rule is shown. This expansion can be used to justify Richardson extrapolation of the approximations leading to the Romberg integration formula. For approximation of differential equations, similar error expansions can be derived. These expansions justify the higher order discrete regularity that is sometimes called super-covergence in the finite element literature. The expansion can also be used as a analytical tool in the convergence analysis of methods used to compute nonlinear problems. For standard methods, an expansion for the error can be constructed that is regular in the grid spacing. For some other methods, numerical artifacts (boundary layers and errors that alternate in sign between adjacent grid points) can also be present. Identifying the types of errors that are generated by a given scheme and the order at which they occur is called Asymptotic Error Analysis. Several examples are shown, including the error analysis of cubic spline interpolation which is shown to have numerical boundary layers. A new result of a numerical artifact from an idealized adaptive grid with hanging nodes used to approximate a simple elliptic problem is presented.