Hughes, Thomas J. R.; Belytschko, Ted; Liu, Wing Kam Convergence of an element-partitioned subcycling algorithm for the semi- discrete heat equation. (English) Zbl 0653.65054 Numer. Methods Partial Differ. Equations 3, No. 2, 131-137 (1987). Large scale systems of type \(Md'(t)+Kd(t)=F(t)\) which arise in the numerical solution of two- or three-dimensional heat equations by finite element methods are considered. Here M is a symmetric positive definite matrix and K is positive semidefinite. Solving these equations with a conditionally stable integrator requires to restrict the time-step according to the highest frequency component of the system and therefore the computation may be highly inefficient. To circumvent this drawback in the subcycling algorithms the spatial domain is partitioned into different spatial subdomains such that in some of them larger time steps can be used. Here the stability of a subcycling algorithm proposed by the authors is studied. The domain is partitioned into two subdomains which are advanced with steps \(\Delta\) t and \(m\Delta\) t and under some restrictions on \(\Delta\) t first order convergence is established. Reviewer: M.Calvo Cited in 11 Documents MSC: 65L05 Numerical methods for initial value problems involving ordinary differential equations 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 35K05 Heat equation 34A30 Linear ordinary differential equations and systems Keywords:semidiscretization; domain decomposition; Large scale systems; heat equations; finite element methods; subcycling algorithms; stability; first order convergence PDFBibTeX XMLCite \textit{T. J. R. Hughes} et al., Numer. Methods Partial Differ. Equations 3, No. 2, 131--137 (1987; Zbl 0653.65054) Full Text: DOI References: [1] Gear, Bit. 24 pp 484– (1984) [2] Belytschki, Computer Methods in Applied Mechanics and Engineering 49 pp 281– (1985) [3] Analysis of Transient Algorithms with Particular Reference to Stability Behavior, in Computational Methods for Transient Analysis. and , Eds., North-Holland, Amsterdam. The Netherlands, 1983, pp. 67-155. 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.