On the order of prolongations and restrictions in multigrid procedures

摘要

It is well known in the world of multigrid that the order of the prolongation and the order of the restriction in a multigrid method should satisfy certain conditions. A rule of thumb is that the sum of the orders of the prolongation and of the restriction should at least be equal to the order of the differential equation solved. In this note we show the correctness of this rule. We notice that we have to distinguish between low frequency and high frequency orders for the transfer operators. For the restriction, the low frequency order is related with its accuracy, whereas for the interpolation operator both orders are related with the accuracy of the result of the interpolation. If an interpolation rule leaves all polynomials of degree k − 1 invariant, then both the low and the high frequency order are equal to k. It is the high frequency order that plays a role in the above-mentioned rule of thumb.