H-Matrix approximation for the operator exponential with applications

110 I. P. Gavrilyuk et al.
Using Lemma 2.3 and setting in (2.14) α = t a k
, we get
(6.22) ‖η N (F, h)‖ ≤
Mc √ [
π
2 √ k exp(−2πd/h)
√ + k exp[−(N +1)2 h 2 a k t]
at(1 − exp(−2πd/h)) ath(N +1)
Equalising
√
the exponents by setting −2πd/h = −(N +1) 2 h 2 a/k, weget
h = 3 2πdk
a
(N +1)−2/3 . Substituting this value into (6.22) leads to the
estimate
‖η N (F, h)‖ ≤
Mc √ [
(6.23) π
2 √ ke −s(N+1)2/3
√
at(1 − e −s(N+1) 2/3 ) +
]
.
]
ke −ts(N+1)2/3
t(N +1) 1/3 3√ ,
2πdka 2
which completes our proof.
1
Note that our estimate implies ‖η N (F, h)‖ = O( ) as t → 0,
t(N+1) 1/3
1
but numerical tests even indicate an error order O( ) as t → 0.
(N+1) 1/3
Acknowledgements. The authors would like to thank Prof. C. Lubich (University Tübingen)
for valuable comments und useful suggestions. We appreciate Lars Grasedyck (University
of Kiel) for providing the numerical computations.
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