# zbMATH — the first resource for mathematics

The analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems. (English) Zbl 0699.65075
Summary: We prove some new estimates for the convergence of multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems. We provide results for the so-called ‘symmetric’ multigrid schemes. We show that for the variable $${\mathcal V}$$-cycle and the $${\mathcal W}$$-cycle schemes, multigrid algorithms with any amount of smoothing on the finest grid converge at a rate that is independent of the number of levels or unknowns, provided that the initial grid is sufficiently fine. We show that the $${\mathcal V}$$-cycle algorithm also converges (under appropriate assumptions on the coarsest grid) but at a rate which may deteriorate as the number of levels increases. This deterioration for the $${\mathcal V}$$- cycle may occur even in the case of full elliptic regularity. Finally, the results of numerical experiments are given which illustrate the convergence behavior suggested by the theory.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 35J25 Boundary value problems for second-order elliptic equations
Full Text:
##### References:
 [1] Randolph E. Bank, A comparison of two multilevel iterative methods for nonsymmetric and indefinite elliptic finite element equations, SIAM J. Numer. Anal. 18 (1981), no. 4, 724 – 743. · Zbl 0471.65074 · doi:10.1137/0718048 · doi.org [2] Randolph E. Bank and Craig C. Douglas, Sharp estimates for multigrid rates of convergence with general smoothing and acceleration, SIAM J. Numer. Anal. 22 (1985), no. 4, 617 – 633. · Zbl 0578.65025 · doi:10.1137/0722038 · doi.org [3] Randolph E. Bank and Todd Dupont, An optimal order process for solving finite element equations, Math. Comp. 36 (1981), no. 153, 35 – 51. · Zbl 0466.65059 [4] D. Braess and W. Hackbusch, A new convergence proof for the multigrid method including the \?-cycle, SIAM J. Numer. Anal. 20 (1983), no. 5, 967 – 975. · Zbl 0521.65079 · doi:10.1137/0720066 · doi.org [5] James H. Bramble and Joseph E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), no. 180, 311 – 329. · Zbl 0659.65098 [6] Achi Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comp. 31 (1977), no. 138, 333 – 390. · Zbl 0373.65054 [7] Pierre Grisvard, Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 207 – 274. [8] W. Hackbusch, Multi-Grid Methods and Applications, Springer-Verlag, New York, 1985. · Zbl 0595.65106 [9] Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. · Zbl 0342.47009 [10] V. A. Kondrat$$^{\prime}$$ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč. 16 (1967), 209 – 292 (Russian). [11] S. G. Kreĭn and Ju. I. Petunin, Scales of Banach spaces, Uspehi Mat. Nauk 21 (1966), no. 2 (128), 89 – 168 (Russian). · Zbl 0173.15702 [12] J. L. Lions & E. Magenes, Problèmes aux Limites non Homogènes et Applications, Dunod, Paris, 1968. · Zbl 0165.10801 [13] J.-F. Maitre and F. Musy, Algebraic formalisation of the multigrid method in the symmetric and positive definite case — a convergence estimation for the \?-cycle, Multigrid methods for integral and differential equations (Bristol, 1983) Inst. Math. Appl. Conf. Ser. New Ser., vol. 3, Oxford Univ. Press, New York, 1985, pp. 213 – 223. · Zbl 0577.65097 [14] Jan Mandel, Multigrid convergence for nonsymmetric, indefinite variational problems and one smoothing step, Appl. Math. Comput. 19 (1986), no. 1-4, 201 – 216. Second Copper Mountain conference on multigrid methods (Copper Mountain, Colo., 1985). · Zbl 0614.65031 · doi:10.1016/0096-3003(86)90104-9 · doi.org [15] Jan Mandel, Algebraic study of multigrid methods for symmetric, definite problems, Appl. Math. Comput. 25 (1988), no. 1, 39 – 56. · Zbl 0636.65026 · doi:10.1016/0096-3003(88)90063-X · doi.org [16] J. Mandel, S. F. McCormick & J. Ruge, An Algebraic Theory for Multigrid Methods for Variational Problems. (Preprint.) · Zbl 0647.65070 [17] S. F. McCormick, Multigrid methods for variational problems: further results, SIAM J. Numer. Anal. 21 (1984), no. 2, 255 – 263. · Zbl 0534.65063 · doi:10.1137/0721018 · doi.org [18] S. F. McCormick, Multigrid methods for variational problems: general theory for the \?-cycle, SIAM J. Numer. Anal. 22 (1985), no. 4, 634 – 643. · Zbl 0602.65038 · doi:10.1137/0722039 · doi.org [19] J. Neoas, Les Méthodes Directes en Théorie des Équations Elliptiques, Academia, Prague, 1967. [20] Alfred H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp. 28 (1974), 959 – 962. · Zbl 0321.65059 [21] Harry Yserentant, The convergence of multilevel methods for solving finite-element equations in the presence of singularities, Math. Comp. 47 (1986), no. 176, 399 – 409. · Zbl 0615.65115
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.