Shapira, Yair; Israeli, Moshe; Sidi, Avram; Zrahia, Uzi Preconditioning spectral element schemes for definite and indefinite problems. (English) Zbl 0939.65128 Numer. Methods Partial Differ. Equations 15, No. 5, 535-543 (1999). Authors’ abstract: Spectral element schemes for the solution of elliptic boundary value problems are considered. Preconditioning methods based on finite difference and finite element schemes are implemented. Numerical experiments show that inverting the preconditioner by a single multigrid iteration is most efficient and that the finite difference preconditioner is superior to the finite element one for both definite and indefinite problems. A multigrid preconditioner is also derived from the finite difference preconditioner and is found suitable for the CGS acceleration method. It is pointed out that, for the finite difference and finite element preconditioners, CGS does not always converge to the accurate algebraic solution. Reviewer: Gunther Schmidt (Berlin) Cited in 1 Document MSC: 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 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 65F35 Numerical computation of matrix norms, conditioning, scaling 65N06 Finite difference methods for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 35J25 Boundary value problems for second-order elliptic equations Keywords:spectral elements; preconditioning; definite and indefinite elliptic problems; conjugate gradient method; convergence; finite difference; finite element; numerical experiments; multigrid; CGS acceleration method PDFBibTeX XMLCite \textit{Y. Shapira} et al., Numer. Methods Partial Differ. Equations 15, No. 5, 535--543 (1999; Zbl 0939.65128) Full Text: DOI References: [1] and Spectral methods in fluid dynamics, Springer-Verlag, Berlin, Heidelberg, 1988. · doi:10.1007/978-3-642-84108-8 [2] and Numerical analysis of spectral methods, Soc Indust Appl Math, Philadelphia 1977. · doi:10.1137/1.9781611970425 [3] Patera, J Comp Phys 54 (1984) · Zbl 0535.76035 · doi:10.1016/0021-9991(84)90128-1 [4] Orszag, J Comp Phys 37 pp 70– (1980) · Zbl 0476.65078 · doi:10.1016/0021-9991(80)90005-4 [5] Deville, J Comp Phys 60 pp 517– (1985) · Zbl 0585.65073 · doi:10.1016/0021-9991(85)90034-8 [6] Quarteroni, SIAM J Num Anal 29 pp 917– (1992) · Zbl 0753.65087 · doi:10.1137/0729056 [7] Sonneveld, SIAM J Sci Statist Comp 10 pp 36– (1989) · Zbl 0666.65029 · doi:10.1137/0910004 [8] Freund, SIAM J Sci Statist Comp 13 pp 425– (1992) · Zbl 0761.65018 · doi:10.1137/0913023 [9] Space time spectral elements for dynamic analysis of composite laminates, D.Sc. thesis, Technion, Israel Inst Tech, Haifa, Israel, 1993. [10] ?Extrapolating to the limit of a vector sequence,? Information linkage between applied mathematics and industry, (Editor), Academic, New York, 1970, pp. 387-396. [11] Me?sina, Comp Meth Appl Mech Eng 10 pp 165– (1977) · Zbl 0344.65019 · doi:10.1016/0045-7825(77)90004-4 [12] Sidi, J Comp Appl Math 36 pp 305– (1991) · Zbl 0747.65002 · doi:10.1016/0377-0427(91)90013-A [13] Shapira, SIAM J Sci Comp 17 pp 439– (1996) · Zbl 0851.65086 · doi:10.1137/S1064827593258838 [14] Iterative solution of elliptic PDEs and implementation on parallel computers, D.Sc. thesis, Technion, Israel Inst Tech, Haifa, Israel, 1993. [15] Shapira, Appl Numer Math 26 pp 377– (1998) · Zbl 0894.65060 · doi:10.1016/S0168-9274(97)00070-6 [16] Shapira, Numer Linear Algebra Appl 5 pp 165– (1998) · Zbl 0937.65134 · doi:10.1002/(SICI)1099-1506(199805/06)5:3<165::AID-NLA132>3.0.CO;2-N [17] Multigrid methods and applications, Springer-Verlag, Berlin, Heidelberg, 1985. · doi:10.1007/978-3-662-02427-0 [18] and An automatic multigrid method for the solution of sparse linear systems, Sixth Copper Mnt Multigrid Meth, and (Editors), NASA, Langley Research Center, Hampton, VA, 1993, pp. 567-582. [19] Freund, SIAM J Sci Statist Comp 14 pp 470– (1993) · Zbl 0781.65022 · doi:10.1137/0914029 [20] Sidi, Numer Algorithms 18 pp 113– (1998) · Zbl 0917.65029 · doi:10.1023/A:1019113314010 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.