On the Accuracy of Multigrid Truncation Error Estimates

Scott R. Fulton

Department of Mathematics and Computer Science
Clarkson University, Potsdam, NY 13699-5815


In solving boundary-value problems, multigrid methods can provide computable estimates of the truncation error, which can then be used in adaptive grid refinement algorithms or in extrapolation to higher-order accuracy (tau-extrapolation). To be useful, these estimates must be accurate, i.e., if the truncation error itself is order p, then the computed estimate must differ from it by a term of order p+m for some positive m.

This paper analyzes the accuracy of multigrid truncation error estimates, examining how m depends on the grid transfers employed. In particular, we compare two definitions of the relative local truncation error (a computable estimate of the truncation error difference between two grids) found in the literature. One definition requires a careful choice of high-order grid transfers to achieve accurate estimates (e.g., Bernert, 1997), while the other (e.g., Schaffer, 1984) can utilize simpler grid transfers (and is itself simpler to compute). Our analytical results are illustrated with numerical calculations for several model problems.