Chatzipantelidis, P.; Lazarov, R. D.; Thomée, V. Some error estimates for the lumped mass finite element method for a parabolic problem. (English) Zbl 1251.65129 Math. Comput. 81, No. 277, 1-20 (2012). Authors’ abstract: We study the spatially semidiscrete lumped mass method for the model homogeneous heat equation with homogeneous Dirichlet boundary conditions. Improving earlier results we show that known optimal order smooth initial data error estimates for the standard Galerkin method carry over to the lumped mass method whereas nonsmooth initial data estimates require special assumptions on the triangulation. We also discuss the application to time discretization by the backward Euler and Crank-Nicolson methods. Reviewer: Weizhong Dai (Ruston) Cited in 1 ReviewCited in 10 Documents MSC: 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 35K05 Heat equation 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs Keywords:lumped mass method; nonsmooth initial data; error estimates; semidiscretization; backward Euler method; heat equation; Galerkin method; Crank-Nicolson methods PDFBibTeX XMLCite \textit{P. Chatzipantelidis} et al., Math. Comput. 81, No. 277, 1--20 (2012; Zbl 1251.65129) Full Text: DOI References: [1] P. Chatzipantelidis, R. D. Lazarov, and V. Thomée, Error estimates for a finite volume element method for parabolic equations in convex polygonal domains, Numer. Methods Partial Differential Equations 20 (2004), no. 5, 650 – 674. · Zbl 1067.65092 · doi:10.1002/num.20006 [2] P. Chatzipantelidis, R. D. Lazarov, V. Thomée, and L. B. Wahlbin, Parabolic finite element equations in nonconvex polygonal domains, BIT 46 (2006), no. suppl., S113 – S143. · Zbl 1108.65097 · doi:10.1007/s10543-006-0087-7 [3] C. M. Chen and V. Thomée, The lumped mass finite element method for a parabolic problem, J. Austral. Math. Soc. Ser. B 26 (1985), no. 3, 329 – 354. · Zbl 0576.65110 · doi:10.1017/S0334270000004549 [4] George J. Fix, Effects of quadrature errors in finite element approximation of steady state, eigenvalue and parabolic problems, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 525 – 556. · Zbl 0282.65081 [5] H.-O. Kreiss, V. Thomée, and O. Widlund, Smoothing of initial data and rates of convergence for parabolic difference equations, Comm. Pure Appl. Math. 23 (1970), 241 – 259. · Zbl 0188.41001 · doi:10.1002/cpa.3160230210 [6] Norikazu Saito, A holomorphic semigroup approach to the lumped mass finite element method, J. Comput. Appl. Math. 169 (2004), no. 1, 71 – 85. · Zbl 1058.65108 · doi:10.1016/j.cam.2003.11.003 [7] Vidar Thomée, Galerkin finite element methods for parabolic problems, 2nd ed., Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 2006. · Zbl 1105.65102 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.