By Wolfgang Hackbusch, Ulrich Trottenberg

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Extra info for Multigrid Methods II

Example text

We observe that β1,2 = − 12 , as required. As α decreases from the value of 2 and approaches the threshold α = 1, the coeﬃcient β1,α diverges. However, values of β1,α can be computed beyond this threshold toward α = 0 where β1,α tends to zero. 1 Dependence of the coeﬃcient β1,α , associated with the Green’s function of the fractional Laplacian in one dimension, on the fractional order, α. As α tends to 1 or 3, the coeﬃcient β1,α diverges. 2 for several values of α. 2(a). 2(b). 2(b). The discontinuity amounts to an inﬁnite local curvature mandated by the Dirac delta function.

9) The coeﬃcient β1,α can be evaluated from these expressions for any value of α inside or outside the parameter space of interest, [0, 2). 1) by β1,α = −c1,−α . 11) and thus β1,α = 1 c1,2−α . 1. We observe that β1,2 = − 12 , as required. As α decreases from the value of 2 and approaches the threshold α = 1, the coeﬃcient β1,α diverges. However, values of β1,α can be computed beyond this threshold toward α = 0 where β1,α tends to zero. 1 Dependence of the coeﬃcient β1,α , associated with the Green’s function of the fractional Laplacian in one dimension, on the fractional order, α.

The diﬀusivity is given by the generalized Einstein relation κ= 1 2 s2d . 1 for symmetric random walkers, q = 12 , where sd = Δx2 . 2 Anomalous diﬀusion It is possible that the random particle jump lengths in each step do not possess a ﬁnite standard deviation, sd , due to the slow decay of the probability πk for large |k|, endowing the discrete probability distribution with a heavy tail. Physically, a signiﬁcant fraction of particles are able to perform long excursions. In that case, the classical central limit theorem does not apply and the collective particle motion describes irregular (anomalous) diﬀusion.