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Strang’s formula for holomorphic semi-groups. (English) Zbl 1030.35095
In the finite-dimensional case G. Strang proved the following approximation formula \[ \|e^{-t(A+B)}-e^{-tA/2}e^{-tB}e^{-tA/2} \|= O(t^3). \] The authors prove generalizations of this formula to the case of infinite dimensional Banach and unbounded generators \(A,B\). Detailed analysis of Schrödinger operators (\(A=-\Delta, B=V(x)\)) in \(L^p({\mathbb R}^d)\), matrix Schrödinger operator and the pair of elliptic second-order operators (\(A={d\over{dx}}a{d\over{dx}}, A={d\over{dx}}b{d\over{dx}}\)) in \(L^2({\mathbb R}^d)\) is realized.

MSC:
35K15 Initial value problems for second-order parabolic equations
47D06 One-parameter semigroups and linear evolution equations
35A35 Theoretical approximation in context of PDEs
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[1] S. Descombes, Systèmes semilinéaires diffusifs ou dispersifs. Etude théorique. Schémas précis d’intégration en temps par décomposition d’opérateurs, PhD thesis, Université Claude-Bernard Lyon 1, France, June 1998
[2] S. Descombes, Convergence of a splitting method of high order for reaction-diffusion systems, Math. Comp. (2001), to appear · Zbl 0981.65107
[3] Descombes, S.; Dia, B.O., An operator-theoretic proof of an estimate on the transfer operator, J. funct. anal., 165, 2, 240-257, (1999) · Zbl 0946.47026
[4] Dia, B.O.; Schatzman, M., Estimations sur la formule de strang, C. R. acad. sci. Paris Sér. I math., 320, 7, 775-779, (1995) · Zbl 0827.47034
[5] Dia, B.O.; Schatzman, M., Commutateurs de certains semi-groupes holomorphes et applications aux directions alternées, RAIRO modél. math. anal. numér., 30, 3, 343-383, (1996) · Zbl 0853.47024
[6] Dia, B.O.; Schatzman, M., An estimate of the Kac transfer operator, J. funct. anal., 145, 1, 108-135, (1997) · Zbl 0919.47031
[7] Helffer, B., Around the transfer operator and the trotter – kato formula, (), 161-174 · Zbl 0835.47050
[8] Henry, D., Geometric theory of semilinear parabolic equations, (1981), Springer-Verlag Berlin · Zbl 0456.35001
[9] Ichinose, T.; Takanobu, S., Estimate of the difference between the Kac operator and the Schrödinger semigroup, Comm. math. phys., 186, 1, 167-197, (1997) · Zbl 0912.47025
[10] Ichinose, T.; Takanobu, S., The norm estimate of the difference between the Kac operator and Schrödinger semigroup. II. the general case including the relativistic case, Electron. J. probab., 5, 4, (2000), 47 pp. (electronic) · Zbl 0987.47032
[11] Kato, T., Perturbation theory for linear operators, Grundlehren math. wiss., 132, (1966), Springer-Verlag New York
[12] R. Kozlov, B. Owren, Order reduction in operator splitting methods, Technical Report 6/1999, Numerics, The Norwegian University of Science, Trondheim, Norway, 1999. http://www.math.ntnu.no/num/synode/papers/ps/kozlov99ori.ps
[13] Lanser, D.; Verwer, J.G., Analysis of operator splitting for advection-diffusion-reaction problems from air pollution modelling, J. comput. appl. math., 111, 1-2, 201-216, (1999), Numerical methods for differential equations (Coimbra, 1998) · Zbl 0949.65090
[14] C. Lubich, T. Jahnke, Errors bounds for exponential operator splitting, Technical Report, Universität Tübingen, Germany, 1999 · Zbl 0972.65061
[15] Neidhardt, H.; Zagrebnov, V.A., Trotter – kato product formula and operator-norm convergence, Comm. math. phys., 205, 1, 129-159, (1999) · Zbl 0949.47019
[16] Neidhardt, H.; Zagrebnov, V.A., Fractional powers of self-adjoint operators and trotter – kato product formula, Integral equations operator theory, 35, 2, 209-231, (1999) · Zbl 0945.47030
[17] Schatzman, M., Stability of the peaceman-Rachford formula, J. funct. anal., 162, 219-255, (1999) · Zbl 0920.47021
[18] Sheng, Q.; Agarwal, R.P., A note on asymptotic splitting and its applications, Math. comput. modelling, 20, 12, 45-58, (1994) · Zbl 0822.65046
[19] Sheng, Q., Global error estimates for exponential splitting, IMA J. numer. anal., 14, 1, 27-56, (1994) · Zbl 0792.65067
[20] Stein, E.M., Singular integrals and differentiability properties of functions, Princeton mathematical series, 30, (1970), Princeton University Press Princeton, NJ
[21] Strang, G., On the construction and comparison of difference schemes, SIAM J. numer. anal., 5, 506-517, (1968) · Zbl 0184.38503
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