Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-24T19:27:23.208Z Has data issue: false hasContentIssue false
  • English
  • Français

A Full Space-Time Convergence Order Analysis of Operator Splittings for Linear Dissipative Evolution Equations

Part of: Numerical analysis in abstract spaces Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  17 May 2016

Eskil Hansen  and
Erik Henningsson
Eskil Hansen*
Affiliation:
Centre for Mathematical Sciences, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden
Erik Henningsson*
Affiliation:
Centre for Mathematical Sciences, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden
*
*Corresponding author. Email addresses:eskil@maths.lth.se (E. Hansen), erikh@maths.lth.se (E. Henningsson)
*Corresponding author. Email addresses:eskil@maths.lth.se (E. Hansen), erikh@maths.lth.se (E. Henningsson)
Get access

Abstract

The Douglas-Rachford and Peaceman-Rachford splitting methods are common choices for temporal discretizations of evolution equations. In this paper we combine these methods with spatial discretizations fulfilling some easily verifiable criteria. In the setting of linear dissipative evolution equations we prove optimal convergence orders, simultaneously in time and space. We apply our abstract results to dimension splitting of a 2D diffusion problem, where a finite element method is used for spatial discretization. To conclude, the convergence results are illustrated with numerical experiments.

Keywords

Douglas/Peaceman-Rachford schemesfull space-time discretizationdimension splittingconvergence orderevolution equationsfinite element methods

MSC classification

Secondary: 65J08: Abstract evolution equations 65M12: Stability and convergence of numerical methods 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 5 , May 2016 , pp. 1302 - 1316
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Akrivis, G. and Crouzeix, M.. Linearly implicit methods for nonlinear parabolic equations. Math. Comp., 73(246):613–635, 2004.Google Scholar
[2]Bátkai, A., Csomós, P., Farkas, B., and Nickel, G.. Operator splitting with spatial-temporal discretization. In Arendt, W., Ball, J. A., Behrndt, J., Förster, K.-H., Mehrmann, V., and Trunk, C., editors, Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, volume 221 of Operator Theory: Advances and Applications, pages 161–171. Springer, Basel, 2012.Google Scholar
[3]Ciarlet, P. G.. The Finite Element Method for Elliptic Problems, volume 4 of Studies in Mathematics and its Applications. North-Holland, Amsterdam, 1978.Google Scholar
[4]Ciarlet, P. G. and Raviart, P.-A.. The combined effect of curved boundaries and numerical integration in isoparametric finite element method. In Aziz, A. K., editor, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, pages 409–474, New York and London, 1972. Academic Press.CrossRefGoogle Scholar
[5]Crouzeix, M.. Parabolic evolution problems. Lecture notes, Université de Rennes 1, http://perso.univ-rennes1.fr/michel.crouzeix/, accessed 2014-03-04.Google Scholar
[6]Descombes, S. and Ribot, M.. Convergence of the Peaceman-Rachford approximation for reaction-diffusion systems. Numer. Math., 95(3):503–525, 2003.Google Scholar
[7]Grisvard, P.. Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics. Pitman, 1985.Google Scholar
[8]Hackbusch, W.. Elliptic Differential Equations: Theory and Numerical Treatment, volume 18 of Springer Series in Computational Mathematics. Springer, Berlin, 1992.Google Scholar
[9]Hairer, E., Lubich, C., and Wanner, G.. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, volume 31 of Springer Series in Computational Mathematics. Springer, Berlin, 2006.Google Scholar
[10]Hansen, E. and Henningsson., E.A convergence analysis of the Peaceman-Rachford scheme for semilinear evolution equations. SIAM J. Numer. Anal., 51(4):1900–1910, 2013.Google Scholar
[11]Hansen, E. and Ostermann, A.. Dimension splitting for evolution equations. Numer. Math., 108(4):557–570, 2008.Google Scholar
[12]Hansen, E., Ostermann, A., and Schratz, K.. The error structure of the Douglas–Rachford splitting method for stiff linear problems. Preprint, 2014, Lund University, http://www.maths.lu.se/staff/eskil-hansen/publications/, accessed 2015-02-25.Google Scholar
[13]Hundsdorfer, W. and Verwer, J.. Stability and convergence of the Peaceman–Rachford ADI method for initial-boundary value problems. Math. Comp., 53(187):81–101, 1989.CrossRefGoogle Scholar
[14]Hundsdorfer, W. and Verwer, J.. Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, volume 33 of Springer Series in Computational Mathematics. Springer, 2003.Google Scholar
[15]Kadlec, J.. O reguljarnosti rešenija zadači Puassona na oblasti s granicej, lokol'no podobno granice vypukloj oblasti (On the regularity of the solution of the Poisson equation on a domain with boundary locally similar to the boundary of a convex domain). Czech. Math. J., 14:386–393, 1964.Google Scholar
[16]Larsson, S.. Nonsmooth data error estimates with applications to the study of the longtime behavior of finite element solutions of semilinear parabolic problems. Preprint, 1992, Chalmers University of Technology, http://www.math.chalmers.se/~stig/papers/preprints.html, accessed 2015-02-25.Google Scholar
[17]Larsson, S. and Thomée, V.. Partial Differential Equations with Numerical Methods, volume 45 of Texts in Applied Mathematics. Springer, 2003.Google Scholar
[18]McLachlan, R. I. and Quispel, G. R. W.. Splitting methods. Acta Numer., 11:341–434, 2002.Google Scholar
[19]Nie, Y.-Y. and Thomée, V.. A lumped mass finite-element method with quadrature for a nonlinear parabolic problem. IMA J. Numer. Anal., 5(4):371–396, 1985.Google Scholar
[20]Pazy, A.. Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences. Springer, New York, 1983.Google Scholar
[21]Peaceman, D. W. and Rachford, H. H.. The numerical solution of parabolic and elliptic differential equations. J. Soc. Indust. Appl. Math., 3(1):28–41, 1955.Google Scholar
[22]Raviart, P.-A.. The use of numerical integration in finite element methods for solving parabolic equations. In Miller, J., editor, Topics in Numerical Analysis, pages 263–264, New York and London, 1973. Academic Press.Google Scholar
[23]Schatzman, M.. Stability of the Peaceman-Rachford approximation. J. Funct. Anal. 162(1):219–255, 1999.Google Scholar
[24]Strang, G. and Fix, G.. An Analysis of the Finite Element Method. Series in Automatic Computation. Prentice-Hall, 1973.Google Scholar
[25]Thomée, V.. Galerkin Finite Element Methods for Parabolic Problems, volume 25 of Computational Mathematics Series. Springer, Berlin, 1997.Google Scholar