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Some error estimates for the lumped mass finite element method for a parabolic problem. (English) Zbl 1251.65129

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.

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
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[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
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