Geodesic
General results for lumped mass approximation of isoparametric eigenvalue problem on triangular meshes
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 8 (2005) no. 3, pp. 189-205.

Voir la notice de l'article provenant de la source Math-Net.Ru

This paper deals with a FE-numerical quadrature method giving a diagonalization of the mass matrix (lumped mass matrix). The method is applied for a class of second order selfadjoint elliptic operators defined on a bounded domain in the plane. The isoparametric finite element transformations and triangular Lagrange finite elements are used. The paper concludes with the investigation started by the authors in [2,3] for the isoparametric variant of the lumped mass modification for second order planar eigenvalue problems. Thus the relationship between the possible quadrature formulas and the precision of the method is proved. The effect of these numerical integrations on the error in eigenvalues and eigenfunctions is estimated. At the end of the paper, the numerical results confirming the theory are presented. 
@article{SJVM_2005_8_3_a1,
     author = {A. B. Andreev and M. S. Petrov and T. D. Todorov},
     title = {General results for lumped mass approximation of isoparametric eigenvalue problem on triangular meshes},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {189--205},
     publisher = {mathdoc},
     volume = {8},
     number = {3},
     year = {2005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2005_8_3_a1/}
}
TY  - JOUR
AU  - A. B. Andreev
AU  - M. S. Petrov
AU  - T. D. Todorov
TI  - General results for lumped mass approximation of isoparametric eigenvalue problem on triangular meshes
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2005
SP  - 189
EP  - 205
VL  - 8
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJVM_2005_8_3_a1/
LA  - en
ID  - SJVM_2005_8_3_a1
ER  - 
%0 Journal Article
%A A. B. Andreev
%A M. S. Petrov
%A T. D. Todorov
%T General results for lumped mass approximation of isoparametric eigenvalue problem on triangular meshes
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2005
%P 189-205
%V 8
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJVM_2005_8_3_a1/
%G en
%F SJVM_2005_8_3_a1
A. B. Andreev; M. S. Petrov; T. D. Todorov. General results for lumped mass approximation of isoparametric eigenvalue problem on triangular meshes. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 8 (2005) no. 3, pp. 189-205. http://geodesic.mathdoc.fr/item/SJVM_2005_8_3_a1/

[1] Andreev A. B., Kascieva V. A., Vanmaele M., “Some results in lumped mass finite element approximation of eigenvalue problems using numerical quadrature formulas”, J. Comp. Appl. Math., 43 (1992), 291–311 | DOI | MR | Zbl

[2] Andreev A. B., Todorov N. D., “Lumped mass approximation for an isoparametric finite element eigenvalue problem”, Sib. J. Num. Math. / Sib. Branch of Russ. Acad. Sci., 2:4 (1999), 295–308

[3] Andreev A. B., Todorov T. D., “Lumped mass error estimates for an isoparametric finite element eigenvalue problem”, Sib. J. Num. Math. / Sib. Branch of Russ. Acad. Sci., 3:3 (2000), 215–228 | Zbl

[4] Andreev A. B., Todorov T. D., “Isoparametric numerical Integration on triangular finite element meshes”, Comptes rendus de l'Academie bulgare des Sciences, 57:7 (2004), 34–44 | MR

[5] Andreev A. B., Todorov T. D., “Isoparametric finite-element approximation of a Steklov eigenvalue problem”, IMA J. Num. Analysis, 24:2 (2004), 309–322 | DOI | MR | Zbl

[6] Bhattacharyya P. K., “Neela Nataraj Isoparametric mixed finite element approximation of eigenvalues and eigenvectors of 4-th order eigenvalue problems with variable coefficients”, Math. Model. and Num. Analysis, 36:1 (2002), 1–32 | DOI | MR | Zbl

[7] Brenner S. C., Scott L. R., The mathematical theory of finite element methods, Texts in Appl. Math., 15, Springer, 1994 | MR | Zbl

[8] Ciarlet P. G., The Finite Element Method for Elliptic Problems, Amsterdam, 1978 | MR

[9] Ciarlet P. G., Raviart P. A., “Interpolation theory over curved elements, with applications to finite element methods”, Comp. Meth. Appl. Mech. Eng., 1 (1972), 217–249 | DOI | MR | Zbl

[10] Ciarlet P. G., Raviart P. A., “The combined effect of curved boundaries and numerical integration in isoparametric finite element method”, Math. Foundation of the FEM with Applications to PDE, ed. A. K. Aziz, Academic Press, New York, 1972, 409–474 | MR

[11] Chen C. M., Thomée V., “The lumped mass finite element method for a parabolic problem”, J. Austral. Math. Soc. B, 26 (1985), 329–354 | DOI | MR | Zbl

[12] Irons B. M., “Engineering application of numerical integration in stiffness methods”, AIAA J., 11 (1970), 2035–2037

[13] Lebaud M. P., “Error estimate in an isoparametric finite element eigenvalue problem”, Math. Comp., 63:207 (1994), 19–40 | DOI | MR | Zbl

[14] Shajdurov V. V., Finite Element Multigrid Methods, Nauka, Moscow, 1989 (In Russian) | Zbl

[15] Todorov T. D., “21-node 3-simplex isoparametric finite element”, Comptes rendus de l'Academie bulgare des Sciences, 56:1 (2003), 17–24 | MR | Zbl

[16] Todorov T. D., “Convergence analysis for eigenvalue approximations on triangular finite element meshes”, Lecture Notes in Computer Science (to appear)

[17] Vanselow R., “Error estimates of a FEM with lumping for parabolic PDEs”, Computing, 68 (2002), 131–141 | DOI | MR | Zbl