Asymptotic Error Analysis of Finite Difference Methods

When computing approximations to PDE problems with smooth solutions using regular grids, the error has additional structure. For second order methods applied to elliptic or parabolic problems, an expansion for the error can be constructed that is regular in the grid spacing. This expansion can be used to justify convergence for nonlinear problems, and is an easy way to see why convergence with higher regularity is observed (a phenomena sometimes called superconvergence in the FE community). When artificial boundary conditions are introduced for higher order finite difference methods, numerical boundary layers result. Identifying the types of errors that are generated by a given scheme and the order at which they occur can be called Asymptotic Error Analysis. Several examples of the technique and its uses will be given. This will be an overview talk also with some material useful to anyone trying to implement "unusual" boundary conditions for PDE problems.