×

Maximal \(L_{p }\) regularity for parabolic difference equations. (English) Zbl 1112.65046

Summary: Maximal regularity for a family of finite difference operators \(\{A_h\}_{h>0}\) on discrete \(L_q\)-spaces asigned to an elliptic differential operator \(A:D (A)\to L_2(\mathbb{R}^n)\) with constant coefficients of order \(m\), is investigated. It is shown that the family \(\{A_p\}_{h>0}\) have discrete maximal \(L_p\) regularity on the discrete \(L_q\)-space, what means that the numerical scheme is stable.

MSC:

65J10 Numerical solutions to equations with linear operators
34G10 Linear differential equations in abstract spaces
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory (Birkhäuser, Basel, 1995).
[2] and , Well-posedness of Parabolic Difference Equations (Birkhäuser, Basel, 1994).
[3] Blunck, Studia Math. 146 pp 157– (2001)
[4] Clément, Adv. Math. Sci. Appl. 3 pp 17– (1993/94)
[5] , and , -boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type, Memoirs of the American Mathematical Society Vol. 166, No. 788 (Amer. Math. Soc., Providence, RI, 2003).
[6] Escher, J. Reine Angew. Math. 563 pp 1– (2003) · Zbl 1242.35220 · doi:10.1515/crll.2003.082
[7] Guidetti, Numer. Funct. Anal. Optim. 20 pp 251– (1999)
[8] Hieber, J. Fourier Anal. Appl. 6 pp 467– (2000)
[9] Hieber, Comm. Partial Differential Equations 22 pp 1647– (1997)
[10] Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals (Princeton University Press, Princeton, NJ, 1993).
[11] Weis, Math. Ann. 319 pp 735– (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.