Applications of Mathematics, Vol. 66, No. 1, pp. 21-42, 2021

Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method

Martin Ladecký, Ivana Pultarová, Jan Zeman

Received August 27, 2019.   Published online October 21, 2020.

Abstract:  A method of characterizing all eigenvalues of a preconditioned discretized scalar diffusion operator with Dirichlet boundary conditions has been recently introduced in Gergelits, Mardal, Nielsen, and Strakoš (2019). Motivated by this paper, we offer a slightly different approach that extends the previous results in some directions. Namely, we provide bounds on all increasingly ordered eigenvalues of a general diffusion or elasticity operator with tensor data, discretized with the conforming finite element method, and preconditioned by the inverse of a matrix of the same operator with different data. Our results hold for mixed Dirichlet and Robin or periodic boundary conditions applied to the original and preconditioning problems. The bounds are two-sided, guaranteed, easily accessible, and depend solely on the material data.
Keywords:  bound on eigenvalues; preconditioning; elliptic differential equation
Classification MSC:  65F08, 65N30
DOI:  10.21136/AM.2020.0217-19

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[1] R. Blaheta: Displacement decomposition - incomplete factorization preconditioning techniques for linear elasticity problems. Numer. Linear Algebra Appl. 1 (1994), 107-128. DOI 10.1002/nla.1680010203 | MR 1277796 | Zbl 0837.65021
[2] P. G. Ciarlet: Mathematical Elasticity. Volume I: Three-Dimensional Elasticity. Studies in Mathematics and Its Applications 20. North-Holland, Amsterdam (1988). DOI 10.1016/S0168-2024(08)70055-9 | MR 0936420 | Zbl 0648.73014
[3] A. Ern, J.-L. Guermond: Theory and Practice of Finite Elements. Applied Mathematical Sciences 159. Springer, New York (2004). DOI 10.1007/978-1-4757-4355-5 | MR 2050138 | Zbl 1059.65103
[4] T. Gergelits, K.-A. Mardal, B. F. Nielsen, Z. Strakoš: Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator. SIAM J. Numer. Anal. 57 (2019), 1369-1394. DOI 10.1137/18M1212458 | MR 3961990 | Zbl 07100344
[5] T. Gergelits, B. F. Nielsen, Z. Strakoš: Generalized spectrum of second order differential operators. SIAM J. Numer. Anal. 58 (2020), 2193-2211. DOI 10.1137/20M1316159 | MR 4128499 | Zbl 07236291
[6] T. Gergelits, Z. Strakoš: Composite convergence bounds based on Chebyshev polynomials and finite precision conjugate gradient computations. Numer. Algorithms 65 (2014), 759-782. DOI 10.1007/s11075-013-9713-z | MR 3187962 | Zbl 1298.65054
[7] G. H. Golub, C. F. Van Loan: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences. The John Hopkins University Press, Baltimore (1996). MR 1417720 | Zbl 0865.65009
[8] J. Liesen, Z. Strakoš: Krylov Subspace Methods: Principles and Analysis. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013). DOI 10.1093/acprof:oso/9780199655410.001.0001 | MR 3024841 | Zbl 1263.65034
[9] G. Meurant, Z. Strakoš: The Lanczos and conjugate gradient algorithms in finite precision arithmetic. Acta Numerica 15 (2006), 471-542. DOI 10.1017/S096249290626001X | MR 2269746 | Zbl 1113.65032
[10] G. Meurant, P. Tichý: On computing quadrature-based bounds for the $A$-norm of the error in conjugate gradients. Numer. Algorithms 62 (2013), 163-191. DOI 10.1007/s11075-012-9591-9 | MR 3011386 | Zbl 1261.65034
[11] J. Nečas, I. Hlaváček: Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction. Studies in Applied Mechanics 3. Elsevier, Amsterdam (1981). DOI 10.1016/c2009-0-12554-0 | MR 0600655 | Zbl 0448.73009
[12] B. F. Nielsen, A. Tveito, W. Hackbusch: Preconditioning by inverting the Laplacian: An analysis of the eigenvalues. IMA J. Numer. Anal. 29 (2009), 24-42. DOI 10.1093/imanum/drm018 | MR 2470938 | Zbl 1167.65066
[13] Y. Saad: Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics, Philadelphia (2003). DOI 10.1137/1.9780898718003 | MR 1990645 | Zbl 1031.65046
[14] D. Serre: Matrices: Theory and Applications. Graduate Texts in Mathematics 216. Springer, New York (2010). DOI 10.1007/978-1-4419-7683-3 | MR 2744852 | Zbl 1206.15001
[15] Z. Strakoš: On the real convergence rate of the conjugate gradient method. Linear Algebra Appl. 154-156 (1991), 535-549. DOI 10.1016/0024-3795(91)90393-B | MR 1113159 | Zbl 0732.65021
[16] A. van der Sluis, H. A. van der Vorst: The rate of convergence of conjugate gradients. Numer. Math. 48 (1986), 543-560. DOI 10.1007/BF01389450 | MR 0839616 | Zbl 0596.65015
[17] H. A. van der Vorst: Iterative Krylov Methods for Large Linear Systems. Cambridge Monographs on Applied and Computational Mathematics 13. Cambridge University Press, Cambridge (2003). DOI 10.1017/CBO9780511615115 | MR 1990752 | Zbl 1023.65027

Affiliations:   Martin Ladecký, Ivana Pultarová (corresponding author), Jan Zeman, Czech Technical University in Prague, Jugoslávských partyzánů 1580/3, 160 00 Praha 6, Czech Republic, e-mail:,,

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