New convergence estimates for multigrid algorithms

Authors:
James H. Bramble and Joseph E. Pasciak

Journal:
Math. Comp. **49** (1987), 311-329

MSC:
Primary 65Nxx; Secondary 65F10

DOI:
https://doi.org/10.1090/S0025-5718-1987-0906174-X

MathSciNet review:
906174

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Abstract: In this paper, new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems. Our theory relates the convergence of multigrid algorithms to a "regularity and approximation" parameter and the number of relaxations *m*. We show that for the symmetric and nonsymmetric cycles, the multigrid iteration converges for any positive *m* at a rate which deteriorates no worse than , where *j* is the number of grid levels. We then define a generalized cycle algorithm which involves exponentially increasing (for example, doubling) the number of smoothings on successively coarser grids. We show that the resulting symmetric and nonsymmetric multigrid iterations converge for any with rates that are independent of the mesh size. The theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods.

**[1]**Ivo Babuška and A. K. Aziz,*Survey lectures on the mathematical foundations of the finite element method*, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR**0421106****[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. MR**795944**, https://doi.org/10.1137/0722038**[3]**Randolph E. Bank and Todd Dupont,*An optimal order process for solving finite element equations*, Math. Comp.**36**(1981), no. 153, 35–51. MR**595040**, https://doi.org/10.1090/S0025-5718-1981-0595040-2**[4]**D. Braess and W. Hackbusch,*A new convergence proof for the multigrid method including the 𝑉-cycle*, SIAM J. Numer. Anal.**20**(1983), no. 5, 967–975. MR**714691**, https://doi.org/10.1137/0720066**[5]**Achi Brandt,*Multi-level adaptive solutions to boundary-value problems*, Math. Comp.**31**(1977), no. 138, 333–390. MR**0431719**, https://doi.org/10.1090/S0025-5718-1977-0431719-X**[6]**Philippe G. Ciarlet,*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174****[7]**Todd Dupont, Richard P. Kendall, and H. H. Rachford Jr.,*An approximate factorization procedure for solving self-adjoint elliptic difference equations*, SIAM J. Numer. Anal.**5**(1968), 559–573. MR**0235748**, https://doi.org/10.1137/0705045**[8]**R. P. Fedorenko, "The speed of convergence of one iterative process,"*USSR Comput. Math. and Math. Phys.*, v. 1, 1961, pp. 1092-1096.**[9]**W. Hackbusch,*Multi-Grid Methods and Applications*, Springer-Verlag, New York, 1985.**[10]**J. L. Lions & E. Magenes,*Problèmes aux Limites non Homogènes et Applications*, Dunod, Paris, 1968.**[11]**J.-F. Maitre and F. Musy,*Algebraic formalisation of the multigrid method in the symmetric and positive definite case—a convergence estimation for the 𝑉-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. MR**849375****[12]**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). MR**849837**, https://doi.org/10.1016/0096-3003(86)90104-9**[13]**Jan Mandel,*Algebraic study of multigrid methods for symmetric, definite problems*, Appl. Math. Comput.**25**(1988), no. 1, 39–56. MR**923402**, https://doi.org/10.1016/0096-3003(88)90063-X**[14]**J. Mandel, S. F. McCormick & J. Ruge, "An algebraic theory for multigrid methods for variational problems." (Preprint)**[15]**S. F. McCormick,*Multigrid methods for variational problems: further results*, SIAM J. Numer. Anal.**21**(1984), no. 2, 255–263. MR**736329**, https://doi.org/10.1137/0721018**[16]**S. F. McCormick,*Multigrid methods for variational problems: general theory for the 𝑉-cycle*, SIAM J. Numer. Anal.**22**(1985), no. 4, 634–643. MR**795945**, https://doi.org/10.1137/0722039**[17]**J. Nečas,*Les Méthodes Directes en Théorie des Équations Elliptiques*, Academia, Prague, 1967.**[18]**R. A. Nicolaides,*On the 𝑙² convergence of an algorithm for solving finite element equations*, Math. Comp.**31**(1977), no. 140, 892–906. MR**0488722**, https://doi.org/10.1090/S0025-5718-1977-0488722-3

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DOI:
https://doi.org/10.1090/S0025-5718-1987-0906174-X

Article copyright:
© Copyright 1987
American Mathematical Society