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On the Accuracy of Multigrid Truncation Error Estimates

Scott R. Fulton

Department of Mathematics and Computer Science

Clarkson University, Potsdam, NY 13699-5815

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Abstract

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.