Review of Discretization Error Estimators in Scientific Computing

Discretization error occurs during the approximate numerical solution of differential equations. Of the various sources of numerical error, discretization error is generally the largest and usually the most difficult to estimate. The goal of this paper is to review the different approaches for estimating discretization error and to present a general framework for their classification. The first category of discretization error estimator is based on estimates of the exact solution to the differential equation which are higher-order accurate than the underlying numerical solution(s) and includes approaches such as Richardson extrapolation, order refinement, and recovery methods from finite elements. The second category of error estimator is based on the residual (i.e., the truncation error) and includes discretization error transport equations, finite element residual methods, and adjoint method extensions. Special attention is given to Richardson extrapolation which can be applied as a post-processing step to the solution from any discretization method (e.g., finite different, finite volume, and finite element). Regardless of the approach chosen, the discretization error estimates are only reliable when the numerical solution, or solutions, are in the asymptotic range, the demonstration of which requires at least three systematically refined meshes. For complex scientific computing applications, the asymptotic range is often difficult to achieve. In these cases, it is appropriate to treat the numerical error estimates as an uncertainty. Issues related to mesh refinement are addressed including systematic refinement, the grid refinement factor, fractional refinement, and unidirectional refinement. Future challenges in discretization error estimation are also discussed.