Uniform substructuring preconditioners for high order FEM on triangles and the influence of nodal basis functions. (English) Zbl 07882246
The authors introduce a new robust substructuring preconditioner for high-order FEM discretizations of problems in the form \(A_\kappa := (1 - \kappa)L + \kappa M\) where \(L\) is the stiffness matrix and \(M\) is the mass matrix.
Section 3 investigates the influence of the choice of the basis functions on the preconditioner for the pure mass matrix case.
For arbitrary \(0 \le \kappa \le 1\), Section 4 presents a preconditioner that is robust with respect to \(\kappa\) and results in a bound of \(\mathcal{O}(1 + \log^2(p))\) for the condition number. This is achieved by combining an existing stiffness matrix substructuring preconditioner [I. Babuška et al., SIAM J. Numer. Anal. 28, No. 3, 624–661 (1991; Zbl 0754.65083)] with an appropriate Jacobi smoothing step.
Numerical experiments in Section 5 support the theoretical findings and are followed by the proofs of the main results in Sections 6–8. A summary in Section 9 concludes the paper.
Section 3 investigates the influence of the choice of the basis functions on the preconditioner for the pure mass matrix case.
For arbitrary \(0 \le \kappa \le 1\), Section 4 presents a preconditioner that is robust with respect to \(\kappa\) and results in a bound of \(\mathcal{O}(1 + \log^2(p))\) for the condition number. This is achieved by combining an existing stiffness matrix substructuring preconditioner [I. Babuška et al., SIAM J. Numer. Anal. 28, No. 3, 624–661 (1991; Zbl 0754.65083)] with an appropriate Jacobi smoothing step.
Numerical experiments in Section 5 support the theoretical findings and are followed by the proofs of the main results in Sections 6–8. A summary in Section 9 concludes the paper.
Reviewer: Julian Streitberger (Wien)
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 65F08 | Preconditioners for iterative methods |
| 65F35 | Numerical computation of matrix norms, conditioning, scaling |
| 65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
| 65F10 | Iterative numerical methods for linear systems |
Citations:
Zbl 0754.65083Software:
DLMFReferences:
| [1] | Ainsworth, M., A hierarchical domain decomposition preconditioner for \(h-p\) finite element approximation on locally refined meshes, SIAM J. Sci. Comput., 17 (1996), pp. 1395-1413, doi:10.1137/S1064827594272578. · Zbl 0860.65111 |
| [2] | Ainsworth, M., A preconditioner based on domain decomposition for \(h-p\) finite-element approximation on quasi-uniform meshes, SIAM J. Numer. Anal., 33 (1996), pp. 1358-1376, doi:10.1137/S0036142993258221. · Zbl 0855.65044 |
| [3] | Ainsworth, M. and Coyle, J., Conditioning of hierarchic \(p\)-version Nédélec elements on meshes of curvilinear quadrilaterals and hexahedra, SIAM J. Numer. Anal., 41 (2003), pp. 731-750, doi:10.1137/S003614290239590X. · Zbl 1091.78014 |
| [4] | Ainsworth, M. and Demkowicz, L., Explicit polynomial preserving trace liftings on a triangle, Math. Nachr., 282 (2009), pp. 640-658. · Zbl 1175.46019 |
| [5] | Ainsworth, M. and Jiang, S., Preconditioning the mass matrix for high order finite element approximation on triangles, SIAM J. Numer. Anal., 57 (2019), pp. 355-377, doi:10.1137/18M1182450. · Zbl 1426.65168 |
| [6] | Ainsworth, M., Jiang, S., and Sanchéz, M. A., An \(\mathcal{O}(p^3)hp\)-version FEM in two dimensions: Preconditioning and post-processing, Comput. Methods Appl. Mech. Engrg., 350 (2019), pp. 766-802. · Zbl 1441.65089 |
| [7] | Babuška, I., Craig, A., Mandel, J., and Pitkäranta, J., Efficient preconditioning for the \(p\)-version finite element method in two dimensions, SIAM J. Numer. Anal., 28 (1991), pp. 624-661, doi:10.1137/0728034. · Zbl 0754.65083 |
| [8] | Bernardi, C., Dauge, M., and Maday, Y., Polynomials in the Sobolev World, 2007. |
| [9] | Bramble, J. H., Pasciak, J. E., and Schatz, A. H., The construction of preconditioners for elliptic problems by substructuring, I, Math. Comp., 47 (1986), pp. 103-134, doi:10.2307/2008084. · Zbl 0615.65112 |
| [10] | Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A., Fundamentals in single domains, in Spectral Methods, Scientific Computation, Springer-Verlag, Berlin, 2006. · Zbl 1093.76002 |
| [11] | Casarin, M. A., Quasi-optimal Schwarz methods for the conforming spectral element discretization, SIAM J. Numer. Anal., 34 (1997), pp. 2482-2502, doi:10.1137/S0036142995292281. · Zbl 0889.65123 |
| [12] | Chan, T. F. and Mathew, T. P., Domain decomposition algorithms, in Acta Numerica, 1994, , Cambridge University Press, Cambridge, 1994, pp. 61-143, doi:10.1017/S0962492900002427. · Zbl 0809.65112 |
| [13] | Demkowicz, L., Gopalakrishnan, J., and Schöberl, J., Polynomial extension operators. Part I, SIAM J. Numer. Anal., 46 (2008), pp. 3006-3031, doi:10.1137/070698786. · Zbl 1189.46025 |
| [14] | Ditzian, Z., Multivariate Bernstein and Markov inequalities, J. Approx. Theory, 70 (1992), pp. 273-283, doi:10.1016/0021-9045(92)90061-R. · Zbl 0773.41016 |
| [15] | Olver, F. W. J., Olde Daalhuis, A. B., Lozier, D. W., Schneider, B. I., Boisvert, R. F., Clark, C. W., Miller, B. R., Saunders, B. V., Cohl, H. S., and McClain, M. A., eds., NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.24 of 2019-09-15. |
| [16] | Dodds,, R. H. Jr. and Lopez, L., Substructuring in linear and nonlinear analysis, Internat. J. Numer. Methods Engrg., 15 (1980), pp. 583-597. · Zbl 0434.73080 |
| [17] | Dryja, M., Smith, B. F., and Widlund, O. B., Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, SIAM J. Numer. Anal., 31 (1994), pp. 1662-1694, doi:10.1137/0731086. · Zbl 0818.65114 |
| [18] | Duffy, M. G., Quadrature over a pyramid or cube of integrands with a singularity at a vertex, SIAM J. Numer. Anal., 19 (1982), pp. 1260-1262, doi:10.1137/0719090. · Zbl 0493.65011 |
| [19] | Gray, P. and Scott, S., Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system \(a+ 2b\rightarrow 3b; b\rightarrow c\), Chem. Eng. Sci., 39 (1984), pp. 1087-1097. |
| [20] | Guo, B. and Cao, W., An additive Schwarz method for the \(h\text{-}p\) version of the finite element method in three dimensions, SIAM J. Numer. Anal., 35 (1998), pp. 632-654, doi:10.1137/S0036142996299435. · Zbl 0911.65105 |
| [21] | Heuer, N. and Leydecker, F., An extension theorem for polynomials on triangles, Calcolo, 45 (2008), pp. 69-85. · Zbl 1175.65143 |
| [22] | Maitre, J.-F. and Pourquier, O., Conditionnements et préconditionnements diagonaux pour la \(p\)-version des méthodes d’éléments finis pour des problèmes elliptiques du second ordre, C. R. Acad. Sci. Paris Sér. I Math., 318 (1994), pp. 583-586. · Zbl 0802.65109 |
| [23] | Mandel, J., Iterative solvers by substructuring for the \(p\)-version finite element method, Comput. Methods Appl. Mech. Engrg., 80 (1990), pp. 117-128, doi:10.1016/0045-7825(90)90017-G. · Zbl 0754.73086 |
| [24] | Muñoz-Sola, R., Polynomial liftings on a tetrahedron and applications to the hp version of the finite element method in three dimensions, SIAM J. Numer. Anal., 34 (1997), pp. 282-314, doi:10.1137/S0036142994267552. · Zbl 0871.46016 |
| [25] | Olsen, E. T. and Douglas, J. Jr., Bounds on spectral condition numbers of matrices arising in the \(p\)-version of the finite element method, Numer. Math., 69 (1995), pp. 333-352, doi:10.1007/s002110050096. · Zbl 0834.65034 |
| [26] | Patera, A. T., A spectral element method for fluid dynamics: Laminar flow in a channel expansion, J. Comput. Phys., 54 (1984), pp. 468-488. · Zbl 0535.76035 |
| [27] | Pavarino, L. F., Additive Schwarz methods for the \(p\)-version finite element method, Numer. Math., 66 (1994), pp. 493-515, doi:10.1007/BF01385709. · Zbl 0791.65083 |
| [28] | Pearson, J. E., Complex patterns in a simple system, Science, 261 (1993), pp. 189-192. |
| [29] | Ruuth, S. J., Implicit-explicit methods for reaction-diffusion problems in pattern formation, J. Math. Biol., 34 (1995), pp. 148-176, doi:10.1007/BF00178771. · Zbl 0835.92006 |
| [30] | Schöberl, J., Melenk, J. M., Pechstein, C., and Zaglmayr, S., Additive Schwarz preconditioning for \(p\)-version triangular and tetrahedral finite elements, IMA J. Numer. Anal., 28 (2008), pp. 1-24, doi:10.1093/imanum/drl046. · Zbl 1153.65113 |
| [31] | Schwab, Ch., \(p\)- and \(hp\)-finite element methods, in Theory and Applications in Solid and Fluid Mechanics, , The Clarendon Press, Oxford University Press, New York, 1998. · Zbl 0910.73003 |
| [32] | Smith, B., Bjorstad, P., and Gropp, W., Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, 2004. |
| [33] | Szabó, B. and Babuška, I., Finite Element Analysis, A Wiley-Interscience Publication, John Wiley & Sons, New York, 1991. · Zbl 0792.73003 |
| [34] | Toselli, A. and Widlund, O., Domain Decomposition Methods—Algorithms and Theory, , Springer-Verlag, Berlin, 2005, doi:10.1007/b137868. · Zbl 1069.65138 |
| [35] | Wilson, E. L., The static condensation algorithm, Internat. J. Numer. Methods Engrg., 8 (1974), pp. 198-203. |
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