A posteriori error estimates for the generalized overlapping domain decomposition methods. (English) Zbl 1264.65202

Benlarbi, Hakima; Chibi, Ahmed-Salah

J. Appl. Math. 2012, Article ID 947085, 15 p. (2012).

Summary: A posteriori error estimates for the generalized overlapping domain decomposition method (GODDM) (i.e., with Robin boundary conditions on the interfaces), for second order boundary value problems, are derived. We show that the error estimate in the continuous case depends on the differences of the traces of the subdomain solutions on the interfaces. After discretization of the domain by finite elements we use the techniques of the residual a posteriori error analysis to get an a posteriori error estimate for the discrete solutions on subdomains. The results of some numerical experiments are presented to support the theory.

MSC:

65N55

Multigrid methods; domain decomposition for boundary value problems involving PDEs

References:

[1]	J. Douglas, Jr. and C.-S. Huang, “An accelerated domain decomposition procedure based on Robin transmission conditions,” BIT Numerical Mathematics, vol. 37, no. 3, pp. 678-686, 1997. · Zbl 0886.65114
[2]	B. Engquist and H.-K. Zhao, “Absorbing boundary conditions for domain decomposition,” Applied Numerical Mathematics, vol. 27, no. 4, pp. 341-365, 1998. · Zbl 0952.65098
[3]	P.-L. Lions, “On the Schwarz alternating method. I,” in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J. Périaux, Eds., pp. 1-42, SIAM, Philadelphia, Pa, USA, 1988. · Zbl 0658.65090
[4]	T. F. Chan, T. Y. Hou, and P.-L. Lions, “Geometry related convergence results for domain decomposition algorithms,” SIAM Journal on Numerical Analysis, vol. 28, no. 2, pp. 378-391, 1991. · Zbl 0724.65109
[5]	A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, The Clarendon Press, Oxford, UK, 1999. · Zbl 0931.65118
[6]	A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory, vol. 34 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 2005. · Zbl 1069.65138
[7]	Y. Maday and F. Magoulès, “Improved ad hoc interface conditions for Schwarz solution procedure tuned to highly heterogeneous media,” Applied Mathematical Modelling, vol. 30, no. 8, pp. 731-743, 2006. · Zbl 1102.65126
[8]	Y. Maday and F. Magoulès, “A survey of various absorbing interface conditions for the Schwarz algorithm tuned to highly heterogeneous media,” in Domain Decomposition Methods: Theory and Applications, vol. 25 of Gakuto International Series. Mathematical Sciences Applications, pp. 65-93, Gakk\Botosho, Tokyo, Japan, 2006. · Zbl 1103.65124
[9]	C. Farhat and P. Le Tallec, “Vista in Domain Decomposition Methods,” Computer Methods in Applied Mechanics and Engineering, vol. 184, no. 2-4, pp. 143-520, 2000.
[10]	F. Magoulès and D. Rixen, “Domain decomposition methods: recent advances and new challenges in engineering,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 8, pp. 1345-1346, 2007.
[11]	F. Nataf, “Recent developments on optimized Schwarz methods,” in Domain Decomposition Methods in Science and Engineering XVI, vol. 55 of Lecture Notes in Computational Science and Engineering, pp. 115-125, Springer, Berlin, Germany, 2007.
[12]	M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley-Interscience, New York, NY, USA, 2000. · Zbl 1049.65135
[13]	A. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley Teubner, Stuttgart, Germany, 1996. · Zbl 0853.65108
[14]	F.-C. Otto and G. Lube, “A posteriori estimates for a non-overlapping domain decomposition method,” Computing, vol. 62, no. 1, pp. 27-43, 1999. · Zbl 0936.65116
[15]	C. Bernardi, T. Chacón Rebollo, E. Chacón Vera, and D. Franco Coronil, “A posteriori error analysis for two-overlapping domain decomposition techniques,” Applied Numerical Mathematics, vol. 59, no. 6, pp. 1214-1236, 2009. · Zbl 1166.65053