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Lavrentiev regularization + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients

Authors: Andrew Knyazev and Olof Widlund
Journal: Math. Comp. 72 (2003), 17-40
MSC (2000): Primary 65N30, 35R05; Secondary 35J25, 35J70
Published electronically: July 13, 2001
MathSciNet review: 1933812
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Abstract | References | Similar Articles | Additional Information


We consider a parametric family of boundary value problems for a diffusion equation with a diffusion coefficient equal to a small constant in a subdomain. Such problems are not uniformly well-posed when the constant gets small. However, in a series of papers, Bakhvalov and Knyazev have suggested a natural splitting of the problem into two well-posed problems. Using this idea, we prove a uniform finite element error estimate for our model problem in the standard parameter-independent Sobolev norm. We also study uniform regularity of the transmission problem, needed for approximation. A traditional finite element method with only one additional assumption, namely, that the boundary of the subdomain with the small coefficient does not cut any finite element, is considered.

One interpretation of our main theorem is in terms of regularization. Our FEM problem can be viewed as resulting from a Lavrentiev regularization and a Ritz-Galerkin approximation of a symmetric ill-posed problem. Our error estimate can then be used to find an optimal regularization parameter together with the optimal dimension of the approximation subspace.

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  • 1. G. P. Astrakhantsev, Method of fictitious domains for a second-order elliptic equation with natural boundary conditions, U.S.S.R. Computational Math. and Math. Phys. 18 (1978), 114-121.
  • 2. N. S. Bakhvalov and A. V. Knyazev, Effective computation of averaged characteristics of composites of periodic structure that consist of essentially different materials, Dokl. Akad. Nauk SSSR 313 (1990), no. 4, 777–781 (Russian); English transl., Soviet Math. Dokl. 42 (1991), no. 1, 57–62. MR 1080629
  • 3. N. S. Bakhvalov and A. V. Knyazev, Fictitious domain methods and computation of homogenized properties of composites with a periodic structure of essentially different components, Numerical methods and applications, CRC, Boca Raton, FL, 1994, pp. 221–266. MR 1282311
  • 4. Nikolai S. Bakhvalov and Andrew V. Knyazev, Preconditioned iterative methods in a subspace for linear algebraic equations with large jumps in the coefficients, Domain decomposition methods in scientific and engineering computing (University Park, PA, 1993) Contemp. Math., vol. 180, Amer. Math. Soc., Providence, RI, 1994, pp. 157–162. MR 1312389, 10.1090/conm/180/01968
  • 5. N. S. Bakhvalov, A. V. Knyazev, and G. M. Kobel'kov, Iterative methods for solving equations with highly varying coefficients, Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Philadelphia, PA) (Roland Glowinski, Yuri A. Kuznetsov, Gérard A. Meurant, Jacques Périaux, and Olof Widlund, eds.), SIAM, 1991, pp. 197-205. CMP 91:12
  • 6. N. S. Bakhvalov, A. V. Knyazev, and R. R. Parashkevov, Extension theorems for Stokes and Lamé equations for nearly incompressible media and their applications to numerical solution of problems with highly discontinuous coefficients, Numerical Linear Algebra with Applications. Accepted, May 2001.
  • 7. N. Bakhvalov and G. Panasenko, Homogenisation: averaging processes in periodic media, Mathematics and its Applications (Soviet Series), vol. 36, Kluwer Academic Publishers Group, Dordrecht, 1989. Mathematical problems in the mechanics of composite materials; Translated from the Russian by D. Leĭtes. MR 1112788
  • 8. A. Bakushinsky and A. Goncharsky, Ill-posed problems: theory and applications, Mathematics and its Applications, vol. 301, Kluwer Academic Publishers Group, Dordrecht, 1994. Translated from the Russian by I. V. Kochikov. MR 1325921
  • 9. Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503330
  • 10. Oleg V. Besov, Valentin P. Il′in, and Sergey M. Nikol′skiĭ, Integral representations of functions and imbedding theorems. Vol. I, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1978. Translated from the Russian; Scripta Series in Mathematics; Edited by Mitchell H. Taibleson. MR 519341
    Oleg V. Besov, Valentin P. Il′in, and Sergey M. Nikol′skiĭ, Integral representations of functions and imbedding theorems. Vol. II, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1979. Scripta Series in Mathematics; Edited by Mitchell H. Taibleson. MR 521808
  • 11. Christoph Börgers and Olof B. Widlund, On finite element domain imbedding methods, SIAM J. Numer. Anal. 27 (1990), no. 4, 963–978. MR 1051116, 10.1137/0727055
  • 12. B. L. Buzbee, F. W. Dorr, J. A. George, and G. H. Golub, The direct solution of the discrete Poisson equation on irregular regions, SIAM J. Numer. Anal. 8 (1971), 722–736. MR 0292316
  • 13. 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
  • 14. Martin Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (1988), no. 3, 613–626. MR 937473, 10.1137/0519043
  • 15. G. R. Cowper, Gaussian quadrature formulas for triangles, Int. J. Num. Meth. Eng. 7 (1973), 405-408.
  • 16. G. C. A. DeRose and A. R. Diaz, Single scale wavelet approximations in layout optimization, Struct. Optimization 18 (1999), no. 1, 1-11.
  • 17. -, Solving three-dimensional layout optimization problems using fixed scale wavelets, Comput. Mech. 25 (2000), no. 2-3, 274-285.
  • 18. Alejandro R. Díaz, A wavelet-Galerkin scheme for analysis of large-scale problems on simple domains, Internat. J. Numer. Methods Engrg. 44 (1999), no. 11, 1599–1616. MR 1680220, 10.1002/(SICI)1097-0207(19990420)44:11<1599::AID-NME556>3.3.CO;2-G
  • 19. L. Escauriaza, E. B. Fabes, and G. Verchota, On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries, Proc. Amer. Math. Soc. 115 (1992), no. 4, 1069–1076. MR 1092919, 10.1090/S0002-9939-1992-1092919-1
  • 20. Richard S. Falk and John E. Osborn, Remarks on mixed finite element methods for problems with rough coefficients, Math. Comp. 62 (1994), no. 205, 1–19. MR 1203735, 10.1090/S0025-5718-1994-1203735-1
  • 21. P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
  • 22. P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR 1173209
  • 23. N. Heuer and E. P. Stephan, The Poincaré-Steklov operator within countably normed spaces, Mathematical aspects of boundary element methods (Palaiseau, 1998), Chapman & Hall/CRC Res. Notes Math., vol. 414, Chapman & Hall/CRC, Boca Raton, FL, 2000, pp. 152–164. MR 1719852
  • 24. V. V. Jikov, S. M. Kozlov, and O. A. Oleĭnik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosif′yan]. MR 1329546
  • 25. R. Bruce Kellogg, On the Poisson equation with intersecting interfaces, Applicable Anal. 4 (1974/75), 101–129. Collection of articles dedicated to Nikolai Ivanovich Muskhelishvili. MR 0393815
  • 26. Carlos E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, vol. 83, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1282720
  • 27. A. V. Knyazev, Iterative solution of PDE with strongly varying coefficients: algebraic version, Iterative Methods in Linear Algebra (Amsterdam) (R. Beauwens and P. de Groen, eds.), Elsevier, 1992, Proceedings IMACS Symp. Iterative Methods in Linear Algebra, Brussels, 1991, pp. 85-89. CMP 92:11
  • 28. Serge Levendorskii, Degenerate elliptic equations, Mathematics and its Applications, vol. 258, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 1247957
  • 29. J.-L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lecture Notes in Mathematics, Vol. 323, Springer-Verlag, Berlin-New York, 1973 (French). MR 0600331
  • 30. T.A. Manteuffel, S. McCormick, and G. Starke, First-order systems least-squares for second-order elliptic problems with discontinuous coefficients, Proceedings of the Seventh Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO, April 3-7, 1995, NASA Conference Publication 3339, Part 2, 1995, p. 551.
  • 31. G. I. Marchuk, Yu. A. Kuznetsov, and A. M. Matsokin, Fictitious domain and domain decomposition methods, Soviet J. Numerical Analysis and Math. Modelling 1 (1986), 1-82.
  • 32. William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312
  • 33. Serge Nicaise and Anna-Margarete Sändig, Transmission problems for the Laplace and elasticity operators: regularity and boundary integral formulation, Math. Models Methods Appl. Sci. 9 (1999), no. 6, 855–898. MR 1702865, 10.1142/S0218202599000403
  • 34. O. A. Oleĭnik, A. S. Shamaev, and G. A. Yosifian, Mathematical problems in elasticity and homogenization, Studies in Mathematics and its Applications, vol. 26, North-Holland Publishing Co., Amsterdam, 1992. MR 1195131
  • 35. R. Plato and G. Vainikko, On the regularization of projection methods for solving ill-posed problems, Numer. Math. 57 (1990), no. 1, 63–79. MR 1043802, 10.1007/BF01386397
  • 36. W. Proskurowski and O. Widlund, A finite element - capacitance matrix method for the Neumann problem for Laplace's equation, SIAM Stat. and Sci. Comput. 1 (1980), 410-425.
  • 37. Włodzimierz Proskurowski and Olof Widlund, A finite element-capacitance matrix method for the Neumann problem for Laplace’s equation, SIAM J. Sci. Statist. Comput. 1 (1980), no. 4, 410–425. MR 610753, 10.1137/0901029
  • 38. Enrique Sánchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Physics, vol. 127, Springer-Verlag, Berlin-New York, 1980. MR 578345
  • 39. V. K. Saul'ev, On solving boundary value problems with high performance computers by a fictitious domain method, Siberian Math. J. 4 (1963), no. 4, 912, (In Russian).
  • 40. Giuseppe Savaré, Regularity results for elliptic equations in Lipschitz domains, J. Funct. Anal. 152 (1998), no. 1, 176–201. MR 1600081, 10.1006/jfan.1997.3158
  • 41. V. V. Vasin and A. L. Ageev, Ill-posed problems with a priori information, Inverse and Ill-posed Problems Series, VSP, Utrecht, 1995. MR 1374573
  • 42. Olof B. Widlund, An extension theorem for finite element spaces with three applications, Numerical Techniques in Continuum Mechanics (Braunschweig/Wiesbaden) (Wolfgang Hackbusch and Kristian Witsch, eds.), Notes on Numerical Fluid Mechanics, v. 16, Friedr. Vieweg und Sohn, 1987, Proceedings of the Second GAMM-Seminar, Kiel, January, 1986, pp. 110-122.

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Additional Information

Andrew Knyazev
Affiliation: Department of Mathematics, University of Colorado at Denver P.O. Box 173364, Campus Box 170, Denver, Colorado 80217-3364

Olof Widlund
Affiliation: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012

Keywords: Galerkin, Lavrentiev, Ritz, Tikhonov, discontinuous coefficients, error estimate, finite elements, regularization, regularity, transmission problem, fictitious domain, embedding
Received by editor(s): May 19, 1998
Received by editor(s) in revised form: December 28, 2000
Published electronically: July 13, 2001
Additional Notes: The first author was supported by NSF Grant DMS-9501507
The second author was supported in part by NSF Grant CCR-9732208 and in part by the U.S. Department of Energy under contract DE-FG02-92ER25127
Article copyright: © Copyright 2001 American Mathematical Society