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Zbl 0853.47024
Dia, Boun Oumar; Schatzman, Michelle
Commutators of certain holomorphic semigroups and applications to alternate directions. (Commutateurs de certains semi-groupes holomorphes et applications aux directions alternées.) (French)
[J] RAIRO, Modélisation Math. Anal. Numér. 30, No.3, 343-383 (1996). ISSN 0764-583X

Summary: Let $A$ and $B$ be two non necessarily commuting operators. Two high-order alternate direction formulae are defined by $$M_1(t)= \textstyle{{4\over 3}} e^{tA/4} e^{tB/2} e^{tA/2} e^{tB/2} e^{tA/4}- \textstyle{{1\over 3}} e^{tA/2} e^{tB} e^{tA/2},$$ $$M_2(t)= \textstyle{{2\over 3}} (e^{tA/2} e^{tB} e^{tA/2}+ e^{tB/2} e^{tA} e^{tB/2})- \textstyle{{1\over 6}} (e^{tA} e^{tB}+ e^{tB} e^{tA})$$ and in the sense of formal series $$\cases e^{t(A+ B)}- M_1(t/n)^n= O(n^{- 4})\\ e^{t(A+ B)}- M_2(t/n)^n= O(n^{- 3})\endcases\tag{$*$}$$ If $a$, $a_0$, $b$ and $b_0$ are strictly positive and infinitely differentiable from $\bbfT^2= (\bbfR/\bbfZ)^2$ to $\bbfR$, operators $A$ and $B$ are defined by $$A= {\partial\over \partial x_1} \Biggl(a(x_1, x_2) {\partial\over \partial x_1}\Biggr)- a_0(x_1, x_2);\ B= {\partial\over \partial x_2} \Biggl(b(x_1, x_2) {\partial\over \partial x_2}\Biggr)- b_0(x_1, x_2).$$ These two operators generate holomorphic semigroups in $L^2(\bbfT^2)$ and the following estimates hold in operator norm in $L^2(\bbfT^2)$, $$|M_1(t)|= O(t)\quad \text{and} \quad |M_2(t)|= O(t).$$ Here, there exists a constant $c$ such that $$|M_1(t/n)^n|\le e^{ct}\quad \text{and} \quad |M_2(t/n)^n|\le e^{ct},$$ which implies that formulae $(*)$ are stable.

MSC 2000:: *47D06 One-parameter semigroups and linear evolution equations
47F05 Partial differential operators

Keywords: high-order alternate direction formulae; holomorphic semigroups

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