Some error estimates for the lumped mass finite element method for a parabolic problem

Authors:
P. Chatzipantelidis, R. D. Lazarov and V. Thomée

Journal:
Math. Comp. **81** (2012), 1-20

MSC (2010):
Primary 65M60, 65M15

DOI:
https://doi.org/10.1090/S0025-5718-2011-02503-2

Published electronically:
May 23, 2011

MathSciNet review:
2833485

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Abstract | References | Similar Articles | Additional Information

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.

**1.**P. Chatzipantelidis, R. D. Lazarov, and V. Thomée,*Error estimates for a finite volume element method for parabolic equations in convex polygonal domains*, Numerical Methods for Partial Differential Equations, 20(5), (2004), 650-674. MR**2076342 (2005g:65122)****2.**P. Chatzipantelidis, R. D. Lazarov, V. Thomée, and L. B. Wahlbin,*Parabolic finite element equations in nonconvex polygonal domains*, BIT Numerical Mathematics, 46 (suppl. 5), (2006), 113-143. MR**2283311 (2007m:65084)****3.**C. M. Chen and V. Thomée,*The lumped mass finite element method for a parabolic problem*, J. Austral. Math. Soc. Ser. B, 26(3), (1985), 329-354. MR**776320 (86m:65117)****4.**G. J. Fix,*Effects of quadrature errors in finite element approximations of steady state, eigenvalue and parabolic problem*, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Ed. A. K. Aziz, Academic Press, New York, 1972, pp. 525-556. MR**0413546 (54:1660)****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(2), (1970), 241-252. MR**0251935 (40:5160)****6.**N. Saito,*A holomorphic semigroup approach to the lumped mass finite element method*, J. Comput. Appl. Math., 169(1), (2004), 71-85. MR**2071261 (2005e:65149)****7.**V. Thomée,*Galerkin Finite Element Methods for Parabolic Problems*, Springer-Verlag, Second Edition, Berlin, 2006. MR**2249024 (2007b:65003)**

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Additional Information

**P. Chatzipantelidis**

Affiliation:
Department of Mathematics, University of Crete, GR–71409 Heraklion, Greece

Email:
chatzipa@math.uoc.gr

**R. D. Lazarov**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA, and Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

Email:
lazarov@math.tamu.edu

**V. Thomée**

Affiliation:
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Göteborg, Sweden, and Institute of Applied and Computational Mathematics, FORTH, Heraklion GR–71110, Greece

Email:
thomee@chalmers.se

DOI:
https://doi.org/10.1090/S0025-5718-2011-02503-2

Keywords:
Lumped mass method,
parabolic partial differential equations,
nonsmooth initial data,
error estimates

Received by editor(s):
November 5, 2009

Received by editor(s) in revised form:
November 2, 2010

Published electronically:
May 23, 2011

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.