A least-squares approach based on a discrete minus one inner product for first order systems
Authors:
James H. Bramble, Raytcho D. Lazarov and Joseph E. Pasciak
Journal:
Math. Comp. 66 (1997), 935-955
MSC (1991):
Primary 65N30; Secondary 65F10
DOI:
https://doi.org/10.1090/S0025-5718-97-00848-X
MathSciNet review:
1415797
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: The purpose of this paper is to develop and analyze a least-squares approximation to a first order system. The first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and/or nonsymmetric. The approach taken here is novel in that the least-squares functional employed involves a discrete inner product which is related to the inner product in (the Sobolev space of order minus one on
). The use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions. In addition, the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned, thus providing an efficient method for solving the algebraic equations. The preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems, a task which is well understood.
- 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. A. K. Aziz, R. B. Kellogg, and A. B. Stephens, Least squares methods for elliptic systems, Math. Comp. 44 (1985), no. 169, 53–70. MR 771030, https://doi.org/10.1090/S0025-5718-1985-0771030-5
- 3. I. Babu\v{s}ka, On the Schwarz algorithm in the theory of differential equations of mathematical physics, Tchecosl. Math. J. 8 (1958), 328-342 (in Russian).
- 4. 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
- 5. James H. Bramble and Joseph E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), no. 180, 311–329. MR 906174, https://doi.org/10.1090/S0025-5718-1987-0906174-X
- 6. James H. Bramble and Ridgway Scott, Simultaneous approximation in scales of Banach spaces, Math. Comp. 32 (1978), no. 144, 947–954. MR 501990, https://doi.org/10.1090/S0025-5718-1978-0501990-5
- 7. James H. Bramble and Jinchao Xu, Some estimates for a weighted 𝐿² projection, Math. Comp. 56 (1991), no. 194, 463–476. MR 1066830, https://doi.org/10.1090/S0025-5718-1991-1066830-3
- 8. Pavel B. Bochev and Max D. Gunzburger, Accuracy of least-squares methods for the Navier-Stokes equations, Comput. & Fluids 22 (1993), no. 4-5, 549–563. MR 1230751, https://doi.org/10.1016/0045-7930(93)90025-5
- 9. Pavel B. Bochev and Max D. Gunzburger, Analysis of least squares finite element methods for the Stokes equations, Math. Comp. 63 (1994), no. 208, 479–506. MR 1257573, https://doi.org/10.1090/S0025-5718-1994-1257573-4
- 10. F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. no. , no. R-2, 129–151 (English, with French summary). MR 365287
- 11. Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205
- 12. Z. Cai, R. Lazarov, T. A. Manteuffel, and S. F. McCormick, First-order system least squares for second-order partial differential equations. I, SIAM J. Numer. Anal. 31 (1994), no. 6, 1785–1799. MR 1302685, https://doi.org/10.1137/0731091
- 13. P.G. Ciarlet, Basic error estimates for elliptic problems, Finite Element Methods : Handbook of Numerical Analysis, (P.G. Ciarlet and J.L. Lions, eds.), vol. II, North-Holland, New York, 1991, pp. 18-352. CMP 91:14
- 14. G. F. Carey and Y. Shen, Convergence studies of least-squares finite elements for first order systems, Comm. Appl. Numer. Meth. 5 (1989), 427-434.
- 15. Ching Lung Chang, Finite element approximation for grad-div type systems in the plane, SIAM J. Numer. Anal. 29 (1992), no. 2, 452–461. MR 1154275, https://doi.org/10.1137/0729027
- 16. Tsu-Fen Chen, On least-squares approximations to compressible flow problems, Numer. Methods Partial Differential Equations 2 (1986), no. 3, 207–228. MR 925373, https://doi.org/10.1002/num.1690020305
- 17. T. F. Chen and G. J. Fix, Least-squares finite element simulation of transonic flows, Appl. Numer. Math. 2 (1986), 399-408.
- 18. Monique Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. MR 961439
- 19. P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
- 20. Thomas J. R. Hughes and Michel Mallet, A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 58 (1986), no. 3, 305–328. MR 865671, https://doi.org/10.1016/0045-7825(86)90152-0
- 21. Thomas J. R. Hughes and Michel Mallet, A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 58 (1986), no. 3, 305–328. MR 865671, https://doi.org/10.1016/0045-7825(86)90152-0
- 22. Dennis C. Jespersen, A least squares decomposition method for solving elliptic equations, Math. Comp. 31 (1977), no. 140, 873–880. MR 461948, https://doi.org/10.1090/S0025-5718-1977-0461948-0
- 23. Bo-Nan Jiang and C. L. Chang, Least-squares finite elements for the Stokes problem, Comput. Methods Appl. Mech. Engrg. 78 (1990), no. 3, 297–311. MR 1039687, https://doi.org/10.1016/0045-7825(90)90003-5
- 24. Bo-Nan Jiang and Louis A. Povinelli, Optimal least-squares finite element method for elliptic problems, Comput. Methods Appl. Mech. Engrg. 102 (1993), no. 2, 199–212. MR 1199921, https://doi.org/10.1016/0045-7825(93)90108-A
- 25. O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. MR 0254401
- 26. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
- 27. J. Mandel, S. McCormick and R. Bank, Variational multigrid theory, Multigrid Methods (S. McCormick, ed.), SIAM, Philadelphia, Penn., 1987, pp. 131-178. CMP 21:05
- 28. P. Neittaanmäki and J. Saranen, On finite element approximation of the gradient for solution of Poisson equation, Numer. Math. 37 (1981), no. 3, 333–337. MR 627107, https://doi.org/10.1007/BF01400312
- 29. Jindřich Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967 (French). MR 0227584
- 30. A. I. Pehlivanov, G. F. Carey, and R. D. Lazarov, Least-squares mixed finite elements for second-order elliptic problems, SIAM J. Numer. Anal. 31 (1994), no. 5, 1368–1377. MR 1293520, https://doi.org/10.1137/0731071
- 31. A. I. Pehlivanov, G. F. Carey, R. D. Lazarov, and Y. Shen, Convergence analysis of least-squares mixed finite elements, Computing 51 (1993), no. 2, 111–123 (English, with English and German summaries). MR 1248894, https://doi.org/10.1007/BF02243846
- 32. A. I. Pehlivanov, G. F. Carey and P. S. Vassilevski, Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates, Numerische Mathematik 72 (1996), 502-522. CMP 96:08
- 33. P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 292–315. Lecture Notes in Math., Vol. 606. MR 0483555
- 34. W. L. Wendland, Elliptic systems in the plane, Monographs and Studies in Mathematics, vol. 3, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. MR 518816
Retrieve articles in Mathematics of Computation with MSC (1991): 65N30, 65F10
Retrieve articles in all journals with MSC (1991): 65N30, 65F10
Additional Information
James H. Bramble
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853 and Department of Mathematics, Texas A&M University, College Station, Texas 77843-3404
Email:
bramble@math.tamu.edu
Raytcho D. Lazarov
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843-3404
Email:
lazarov@math.tamu.edu
Joseph E. Pasciak
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843-3404
Email:
pasciak@math.tamu.edu
DOI:
https://doi.org/10.1090/S0025-5718-97-00848-X
Received by editor(s):
October 9, 1995
Received by editor(s) in revised form:
June 5, 1996
Additional Notes:
This manuscript has been authored under contract number DE-AC02-76CH00016 with the U.S. Department of Energy. Accordingly, the U.S. Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. This work was also supported in part under the National Science Foundation Grant No. DMS-9007185 and by the U.S. Army Research Office through the Mathematical Sciences Institute, Cornell University.
Article copyright:
© Copyright 1997
American Mathematical Society