×

zbMATH — the first resource for mathematics

A domain splitting method for heat conduction problems in composite materials. (English) Zbl 0952.65070
The linear evolutionary heat equation is considered on a medium containing fibres placed in mutually parallel planes. These fibres are assumed dense, which approves their homogenization considered in the paper. The resulting problem then involves subdomains divided by these plates. A two-dimensional situation is then considered and discretized by (bi-)linear finite elements on triangles or quadrilaterals and the backward Euler scheme, and a noniterative overlapping domain splitting method is adapted. Convergence analysis and even rate-of-error estimates taking into account special regularity properties are presented as well as numerical experiments. The method is claimed suitable for nonlinear problems, too.

MSC:
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
80A22 Stefan problems, phase changes, etc.
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI Link EuDML
References:
[1] H. Blum, S. Lisky and R. Rannacher, A domain splitting algorithm for parabolic problems. Computing49 (1992) 11-23. Zbl0767.65073 · Zbl 0767.65073 · doi:10.1007/BF02238647
[2] D. Braess, W. Dahmen and Chr. Wieners, A multigrid algorithm for the mortar finite element method. SIAM J. Numer. Anal.37 (1999) 48-69. · Zbl 0942.65139 · doi:10.1137/S0036142998335431
[3] H. Chen and R.D. Lazarov, Domain splitting algorithm for mixed finite element approximations to parabolic problems. East-West J. Numer. Math.4 (1996) 121-135. Zbl0878.65081 · Zbl 0878.65081
[4] Z. Chen and J. Zou, Finite element methods and their convergence analysis for elliptic and parabolic interface problems. Numer. Math.79 (1998) 175-202. Zbl0909.65085 · Zbl 0909.65085 · doi:10.1007/s002110050336
[5] W. Hackbusch, Theorie und Numerik elliptischer Differentialgleichungen. Teubner, Stuttgart (1986). Zbl0609.65065 · Zbl 0609.65065
[6] W. Hackbusch and S. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated microstructures. Numer. Math.75 (1997) 447-472. · Zbl 0874.65086 · doi:10.1007/s002110050248
[7] H. Haller, Composite materials of shape-memory alloys: micromechanical modelling and homogenization (in German). Ph.D. thesis, Technische Universitt Mnchen (1997). Zbl0885.73041 · Zbl 0885.73041
[8] F.H. Hebeker, An a posteriori error estimator for elliptic boundary and interface problems. Preprint 97-46 (SFB 359), Universitt Heidelberg (1997); submitted.
[9] F.K. Hebeker, Multigrid convergence analysis for elliptic problems arising in composite materials (in preparation).
[10] F.K. Hebeker and Yu.A. Kuznetsov, Unsteady convection and convection-diffusion problems via direct overlapping domain decomposition methods. Preprint 93-54 (SFB 359), Universitt Heidelberg, 1993; Numer. Methods Partial Differential Equations14 (1998) 387-406. Zbl0919.65058 · Zbl 0919.65058 · doi:10.1002/(SICI)1098-2426(199805)14:3<387::AID-NUM7>3.0.CO;2-I
[11] K.H. Hoffmann and J. Zou, Finite element analysis on the Lawrence-Doniach model for layered superconductors. Numer. Funct. Anal. Optim.18 (1997) 567-589. Zbl1114.65349 · Zbl 1114.65349 · doi:10.1080/01630569708816779
[12] J. Jäger, An overlapping domain decomposition method to parallelize the solution of parabolic differential equations (in German). Ph.D. thesis, Universitt Heidelberg (1994). Zbl0801.65094 · Zbl 0801.65094
[13] C. Kober, Composite materials of shape-memory alloys: modelling as layers and numerical simulation (in German). Ph.D. thesis, Technische Universitt Mnchen (1997). Zbl0909.73053 · Zbl 0909.73053
[14] Yu.A. Kuznetsov, New algorithms for approximate realization of implicit difference schemes. Sov. J. Numer. Anal. Modell.3 (1988) 99-114. · Zbl 0825.65066
[15] Yu.A. Kuznetsov, Domain decomposition methods for unsteady convection diffusion problems. Comput. Methods Appl. Sci. Engin. (Proceedings of the Ninth International Conference, Paris 1990) SIAM, Philadelphia (1990) 211-227. · Zbl 0744.65063
[16] Yu.A. Kuznetsov, Overlapping domain decomposition methods for finite element problems with singular perturbed operators. in Domain decomposition Methods for Partial differential equations, R. Glowinski et al. Eds., SIAM, Philadelphia. Proc. of the 4th Intl. Symp. (1991) 223-241 Zbl0766.65089 · Zbl 0766.65089
[17] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer, Berlin etc. (1994). · Zbl 0803.65088
[18] R. Rannacher and J. Zhou, Analysis of a domain splitting method for nonstationary convection-diffusion problems. East-West J. Numer. Math.2 (1994) 151-172. · Zbl 0836.65100
[19] J. Wloka, Partielle Differentialgleichungen. Teubner, Stuttgart (1982).
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.