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Natroshvili}, } @preamble{ "\def\cprime{$'$} " } @incollection {MR2001e:74030, AUTHOR = {Kozhevnikov, Alexander}, TITLE = {A history of the {C}osserat spectrum}, BOOKTITLE = {The Maz\cprime ya anniversary collection, Vol. 1 (Rostock, 1998)}, PAGES = {223--234}, PUBLISHER = {Birkh\"auser}, ADDRESS = {Basel}, YEAR = {1999}, MRCLASS = {74G30 (01A60 35P05 74G25)}, MRNUMBER = {2001e:74030}, } @article {MR2001b:76030, AUTHOR = {Liu, Wensen and Plotkin, Allen}, TITLE = {Application of {C}osserat-spectrum theory to the weakly compressible {S}tokes flow past a sphere}, JOURNAL = {J. Engrg. Math.}, FJOURNAL = {Journal of Engineering Mathematics}, VOLUME = {38}, YEAR = {2000}, NUMBER = {2}, PAGES = {155--172}, ISSN = {0022-0833}, CODEN = {JLEMAU}, MRCLASS = {76D07 (76N10)}, MRNUMBER = {2001b:76030}, } @article {MR1810267, AUTHOR = {Liu, Wensen and Markenscoff, Xanthippi and Paukshto, Michael}, TITLE = {The {C}osserat spectrum theory for two-dimensional thermoelastic problems}, JOURNAL = {J. 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Comp.}, FJOURNAL = {Mathematics of Computation}, ISSN = {0025-5718}, CODEN = {MCMPAF}, } @article {MR2000k:65199, AUTHOR = {Oliveira, S.}, TITLE = {On the convergence rate of a preconditioned subspace eigensolver}, JOURNAL = {Computing}, FJOURNAL = {Computing. Archives for Scientific Computing}, VOLUME = {63}, YEAR = {1999}, NUMBER = {3}, PAGES = {219--231}, ISSN = {0010-485X}, CODEN = {CMPTA2}, MRCLASS = {65N25 (65F15)}, MRNUMBER = {2000k:65199}, MRREVIEWER = {K. Veseli{\'c}}, } @article {MR1638040, AUTHOR = {Borges, Leonardo and Oliveira, Suely}, TITLE = {A parallel {D}avidson-type algorithm for several eigenvalues}, JOURNAL = {J. Comput. Phys.}, FJOURNAL = {Journal of Computational Physics}, VOLUME = {144}, YEAR = {1998}, NUMBER = {2}, PAGES = {727--748}, ISSN = {0021-9991}, CODEN = {JCTPAH}, } @article {MR90f:65217, AUTHOR = {Pares-Sierra, Alejandro and Vallis, Geoffrey K.}, TITLE = {A fast semi-direct method for the numerical solution of nonseparable elliptic equations in irregular domains}, JOURNAL = {J. Comput. Phys.}, FJOURNAL = {Journal of Computational Physics}, VOLUME = {82}, YEAR = {1989}, NUMBER = {2}, PAGES = {398--412}, ISSN = {0021-9991}, CODEN = {JCTPAH}, } @article {MR93h:65148, AUTHOR = {Nepomnyaschikh, S. V.}, TITLE = {Mesh theorems on traces, normalizations of function traces and their inversion}, JOURNAL = {Soviet J. Numer. Anal. Math. Modelling}, FJOURNAL = {Soviet Journal of Numerical Analysis and Mathematical Modelling}, VOLUME = {6}, YEAR = {1991}, NUMBER = {3}, PAGES = {223--242}, ISSN = {0169-2895}, } @inproceedings {cpry96, AUTHOR = {Z. Cai and R. R. Parashkevov and T. F. Russell and X. Ye}, TITLE = {Overlapping Domain Decomposition for a Mixed Finite Element Method in Three Dimensions}, BOOKTITLE = {Ninth International Conference on Domain Decomposition Methods. Proceedings of the The Ninth DD meeting in Ullensvang, Norway, June 3-8 1996}, EDITOR = {Petter Bjørstad and Magne Espedal and David Keyes}, PUBLISHER = {DDM.org}, ADDRESS = {http://www.ddm.org/}, YEAR = {1998}, PAGES = {188--196}, NOTE="Available at http://www.ddm.org/DD9/Cai.pdf" } @article {MR1808695, AUTHOR = {Yang, Daoqi}, TITLE = {Finite elements for elliptic problems with wild coefficients}, NOTE = {1999 International Symposium on Computational Sciences, to honor John R. Rice (West Lafayette, IN)}, JOURNAL = {Math. Comput. Simulation}, FJOURNAL = {Mathematics and Computers in Simulation}, VOLUME = {54}, YEAR = {2000}, NUMBER = {4-5}, PAGES = {383--395}, ISSN = {0378-4754}, CODEN = {MCSIDR}, MRCLASS = {65N30}, MRNUMBER = {1 808 695}, } @PHDTHESIS{derose, AUTHOR="Giuseppe DeRose", SCHOOL="{Michigan State University}", TITLE="Solving Topology Optimization Problems Using Wavelet-{G}alerkin Techniques", YEAR="1998", ADDRESS="{Mechanical Engineering Department, 2440 Engineering Building, East Lansing MI 48824}" } @ARTICLE{diaz99, AUTHOR = {D{\'\i}az, Alejandro R.}, TITLE = {A wavelet-{G}alerkin scheme for analysis of large-scale problems on simple domains}, JOURNAL = {Internat. J. Numer. Methods Engrg.}, FJOURNAL = {International Journal for Numerical Methods in Engineering}, VOLUME = {44}, YEAR = {1999}, NUMBER = {11}, PAGES = {1599--1616}, ISSN = {0029-5981}, CODEN = {IJNMBH}, MRCLASS = {80-08 (65N30 80A20)}, MRNUMBER = {99m:80002}, } @ARTICLE{dd99, AUTHOR="DeRose, G. C. A. and Diaz, A. R.", TITLE="Single scale wavelet approximations in layout optimization", JOURNAL="Struct. Optimization", YEAR="1999", VOLUME="18", NUMBER="1", PAGES="1--11" } @ARTICLE{dd00, AUTHOR="DeRose, G. C. A. and Diaz, A. R.", TITLE="Solving three-dimensional layout optimization problems using fixed scale wavelets ", JOURNAL="Comput. Mech.", YEAR="2000", VOLUME="25", NUMBER="2-3", PAGES="274-285" } @article{kw96, author = "Kobelkov, G. M. and Weinstein, P. A.", title = "{Iterative methods for solving boundary value problems for elliptic equations with varying coefficients.}", language = "{English}", JOURNAL="Nova J. Math. Game Theory Algebra", VOLUME="4", YEAR="1996", NUMBER="3--4", PAGES="277-286", source = "{Nova J. Math. Game Theory Algebra 4, No.3-4, 277-286 (1996). [ISSN 1060-9881 ]}", abstract = "{The numerical solution of the Dirichlet boundary value problem for an elliptic equation with strongly varying coefficients is considered. Special iterative methods are proposed with convergence rate independent of the diversity coefficient values. The symmetrized version of the iterative algorithm is also considered.}", reviewer = "{Michael Sever (Jerusalem)}", keywords = "{strongly varying coefficients; convergence rate}", classmath = "{*35A40 Numerical approximation to solutions of PDE 65N06 Finite difference methods (BVP of PDE) 35J15 Second order elliptic equations, general }", } @article{vks93, author = "Veretentseva, T. V. and Kobel'kov, G. M. and Sushko, V. G.", title = "{On approximate solutions to some problems with rapidly varying coefficients.}", language = "{English. Russian original}", JOURNAL="Mosc. Univ. Comput. Math. Cybern.", YEAR="1993", NUMBER="3", PAGES="18--23", source = "{Mosc. Univ. Comput. Math. Cybern. 1993, No.3, 18-23 (1993); translation from Vestn. Mosk. Univ., Ser. XV 1993, No.3, 22-27 (1993). [ISSN 0278-6419]}", abstract = "{An asymptotic parametric expansion of the solution of a boundary value problem is constructed for an elliptic equation with piecewise continuous sharply varying coefficients.}", keywords = "{asymptotic parametric expansion}", classmath = "{*35A35 Theoretical approximation to solutions of PDE 35J25 Second order elliptic equations, boundary value problems 35C20 Asymptotic expansions of solutions of PDE }", } @article{ak93, author = "Zh. Arbash and G. M. Kobel'kov", title = "{Numerical methods for solving elasticity theory problems with strongly varying coefficients.}", language = "{English}", JOURNAL="Russ. J. Numer. Anal. Math. Model.", YEAR="1993", NUMBER="5", VOLUME=8, PAGES="371--383", source = "{Russ. J. Numer. Anal. Math. Model. 8, No.5, 371-383 (1993). [ISSN 0927-6467]}", abstract = "{For the solution of linear elasticity equations under rigid-contact conditions on the boundary of a rectangle (a cube), we suggest iterative methods whose convergence rates do not depend on the variation of the Lame coefficients $\lambda$ and $\mu$ in the domain under study.}", keywords = "{convergence rate; periodic boundary conditions; rigid-contact conditions; rectangle; cube; iterative methods; Lame coefficients}", classmath = "{*73V15 Finite difference methods 73C35 Mixed boundary value problems in elasticity 73C02 Classical linear elasticity }", } @article{kob91, author = "G. M. Kobel'kov", title = "{Efficient methods for solving elasticity theory equations.}", language = "{English}", JOURNAL="Sov. J. Numer. Anal. Math. Model.", YEAR="1991", NUMBER="5", VOLUME=6, PAGES="361--375", source = "{Sov. J. Numer. Anal. Math. Model. 6, No.5, 361-375 (1991). [ISSN 0169-2895]}", abstract = "{Iteration techniques are discussed for solving the mesh equations approximating the boundary value problems of the linear elasticity theory. It has been assumed that $\mu\sb 0 \leq \mu = \mu (x) \le \mu\sb 1$, $ 0 \leq \lambda = \lambda (x) \leq \infty$, where $\lambda$ and $\mu$ are Lame constants, and that the ratio $\mu\sb 1 / \mu\sb 0$ is small. Under these conditions the convergence rate of the iteration techniques is independent of $\mu$ and $\lambda$, and it is finite for the boundary value problem problems of the same type. For the boundary value problems of different types, the convergence rate is $\gamma = 1 - c \surd h$. The results obtained in the paper are valid both in two- and three-dimensional cases.}", keywords = "{one-level iterative method; mesh equations; Lame constants; convergence rate}", classmath = "{*73V15 Finite difference methods 73C35 Mixed boundary value problems in elasticity 73C02 Classical linear elasticity }", } @article{kob96, author = "G. M. Kobel'kov", title = "{On the solution of the boundary value problem for the diffusion equation with highly varying coefficient.}", language = "{English}", JOURNAL="Russ. J. Numer. Anal. Math. Model.", YEAR="1996", NUMBER="6", VOLUME="11", PAGES="487--495", source = "{Russ. J. Numer. Anal. Math. Model. 11, No.6, 487-495 (1996). [ISSN 0927-6467 ]}", abstract = "{The Dirichlet problem for a linear elliptic equation of the form $- \text{div} (\alpha \nabla u) = f$ in a domain $\Omega$ is considered, where the diffusion coefficient $\alpha = k a$, $k$ being a highly varying piecewise constant coefficient and $a$ is a bounded piecewise smooth function. In order to solve this problem, an iterative method, which is a generalization of the one earlier proposed by the author [Ficitiuos domain method and the solution of elliptic equations with highly varying coefficients, Sov. J. Numer. Anal. Math. Model 2, No. 6, 407-419 (1987; Zbl 825.65076)] is presented. The iterated method is based on an implicit two-level scheme and it requires at every step a single solution of the Dirichlet problem for the Poisson equation in $\Omega$ and additional calculations by explicit formulae. \par In the main result, it is shown that the iterative method is convergent to the solution of the original problem and the convergence rate is of a geometric progression whose ratio is independent of the jump of the coefficient $k$. This result can be extended to the case of a general elliptic operator of divergence form.}", reviewer = "{S.Migorski (Krakow)}", keywords = "{Dirichlet problem; linear elliptic equation; highly varying coefficient; convergence; iterative method.; implicit two-level scheme; Poisson equation}", classmath = "{*65N55 Multigrid methods; domain decomposition (BVP of PDE) 65F10 Iterative methods for linear systems 35J25 Second order elliptic equations, boundary value problems }", } @article {MR97g:65077, AUTHOR = {Edelman, Alan and Smith, Steven T.}, TITLE = {On conjugate gradient-like methods for eigen-like problems}, NOTE = {International Linear Algebra Year (Toulouse, 1995)}, JOURNAL = {BIT}, FJOURNAL = {BIT. Numerical Mathematics}, VOLUME = {36}, YEAR = {1996}, NUMBER = {3}, PAGES = {494--508}, ISSN = {0006-3835}, CODEN = {NBITAB}, MRCLASS = {65F15 (90C30)}, MRNUMBER = {97g:65077}, } @article {eas99, AUTHOR = {Edelman, Alan and Arias, Tom{\'a}s A. and Smith, Steven T.}, TITLE = {The geometry of algorithms with orthogonality constraints}, JOURNAL = {SIAM J. Matrix Anal. Appl.}, FJOURNAL = {SIAM Journal on Matrix Analysis and Applications}, VOLUME = {20}, YEAR = {1999}, NUMBER = {2}, PAGES = {303--353 (electronic)}, ISSN = {1095-7162}, MRCLASS = {65F30 (15A18 51F20 65H10 90C30)}, MRNUMBER = {99h:65075}, MRREVIEWER = {Michael M. Dediu}, } @article {MR30:5037, AUTHOR = {C{\'e}a, Jean}, TITLE = {Approximation variationnelle des probl\`emes aux limites}, JOURNAL = {Ann. Inst. Fourier (Grenoble)}, VOLUME = {14}, YEAR = {1964}, NUMBER = {fasc. 2}, PAGES = {345--444}, } @article {MR44:6166, AUTHOR = {Babu{\v{s}}ka, Ivo}, TITLE = {Error-bounds for finite element method}, JOURNAL = {Numer. Math.}, VOLUME = {16}, YEAR = {1970/1971}, PAGES = {322--333}, } @article {MR38:1823, AUTHOR = {Nitsche, J.}, TITLE = {Ein {K}riterium f\"ur die {Q}uasi-{O}ptimalit\"at des {R}itzschen {V}erfahrens}, JOURNAL = {Numer. Math.}, VOLUME = {11}, YEAR = {1968}, PAGES = {346--348}, } @article {MR36:6923, AUTHOR = {Aubin, Jean-Pierre}, TITLE = {Evaluations des erreurs de troncature des approximations des espaces de {S}obolev}, JOURNAL = {J. Math. Anal. Appl.}, VOLUME = {21}, YEAR = {1968}, PAGES = {356--368}, } @article {lm98, AUTHOR = {R. B. Lehoucq and Karl Meerbergen }, TITLE = {Using Generalized {C}ayley Transformations within an Inexact Rational {K}rylov Sequence Method }, JOURNAL = {SIAM J. Matrix Anal. Appl.}, FJOURNAL = {SIAM Journal on Matrix Analysis and Applications}, VOLUME = {20}, YEAR = {1998}, PAGES = {131--148}, NUMBER=1, } @article {MR99j:65062, AUTHOR = {Stathopoulos, Andreas and Saad, Yousef}, TITLE = {Restarting techniques for the ({J}acobi-){D}avidson symmetric eigenvalue methods}, NOTE = {Large scale eigenvalue problems (Argonne, IL, 1997)}, JOURNAL = {Electron. Trans. Numer. Anal.}, FJOURNAL = {Electronic Transactions on Numerical Analysis}, VOLUME = {7}, YEAR = {1998}, PAGES = {163--181 (electronic)}, ISSN = {1068-9613}, MRCLASS = {65F15 (65F10)}, MRNUMBER = {99j:65062}, } @article {MR99i:65042, AUTHOR = {Wu, Kesheng and Saad, Yousef and Stathopoulos, Andreas}, TITLE = {Inexact {N}ewton preconditioning techniques for large symmetric eigenvalue problems}, NOTE = {Large scale eigenvalue problems (Argonne, IL, 1997)}, JOURNAL = {Electron. Trans. Numer. Anal.}, FJOURNAL = {Electronic Transactions on Numerical Analysis}, VOLUME = {7}, YEAR = {1998}, PAGES = {202--214 (electronic)}, ISSN = {1068-9613}, MRCLASS = {65F15}, MRNUMBER = {99i:65042}, } @inproceedings {dd10, TITLE = {Domain Decomposition Methods 10}, BOOKTITLE = {Proceedings of the Tenth International Conference on Domain Decomposition Methods held at the University of Colorado at Boulder, August 10-14, 1997}, EDITOR = {Jan Mandel and Charbel Farhat and Xiao-Chuan Cai}, SERIES = {Contemporary Mathematics}, VOLUME = {218}, PUBLISHER = {American Mathematical Society}, ADDRESS = {Providence, RI}, YEAR = {1998}, PAGES = {554}, ISBN = {0-8218-0988-1 }, } @article {MR1484080, AUTHOR = {Jia, Zhongxiao}, TITLE = {A refined iterative algorithm based on the block {A}rnoldi process for large unsymmetric eigenproblems}, JOURNAL = {Linear Algebra Appl.}, FJOURNAL = {Linear Algebra and its Applications}, VOLUME = {270}, YEAR = {1998}, PAGES = {171--189}, ISSN = {0024-3795}, CODEN = {LAAPAW}, } @article {MR98c:65060, AUTHOR = {Jia, Zhongxiao}, TITLE = {Refined iterative algorithms based on {A}rnoldi's process for large unsymmetric eigenproblems}, JOURNAL = {Linear Algebra Appl.}, FJOURNAL = {Linear Algebra and its Applications}, VOLUME = {259}, YEAR = {1997}, PAGES = {1--23}, ISSN = {0024-3795}, CODEN = {LAAPAW}, } @article {MR96d:65062, AUTHOR = {Jia, Zhong Xiao}, TITLE = {The convergence of generalized {L}anczos methods for large unsymmetric eigenproblems}, JOURNAL = {SIAM J. Matrix Anal. Appl.}, FJOURNAL = {SIAM Journal on Matrix Analysis and Applications}, VOLUME = {16}, YEAR = {1995}, NUMBER = {3}, PAGES = {843--862}, ISSN = {0895-4798}, CODEN = {SJMAEL}, } @article {MR96b:65038, AUTHOR = {Nechepurenko, Yu. M.}, TITLE = {The singular function method for computing the eigenvalues of polynomial matrices}, JOURNAL = {Comput. Math. Math. Phys. }, FJOURNAL = {Computational Mathematics and Mathematical Physics }, VOLUME = {35}, YEAR = {1995}, NUMBER = {5}, PAGES = {507--517}, } @article {k87a, AUTHOR = {Knyazev, A. V.}, TITLE = {On modified gradient methods for spectral problems}, NOTE = {{(In Russian)}}, JOURNAL = {Differentsial\cprime nye Uravneniya}, FJOURNAL = {Differentsial\cprime nye Uravneniya}, VOLUME = {23}, YEAR = {1987}, NUMBER = {4}, PAGES = {715--717, 735}, ISSN = {0374-0641}, MRCLASS = {65F15 (49D07 49D10)}, MRNUMBER = {88e:65043}, } @incollection {MR88m:65060, AUTHOR = {Ru{\`e}, Aksel}, TITLE = {Krylov rational sequence methods for computing eigenvalues}, BOOKTITLE = {Computational methods in linear algebra (Russian) (Moscow, 1982)}, PAGES = {203--218}, PUBLISHER = {Akad. Nauk SSSR Otdel Vychisl. Mat.}, ADDRESS = {Moscow}, YEAR = {1983}, NOTE={(In Russian)} } @article {MR1618804, AUTHOR = {Ruhe, Axel}, TITLE = {Rational {K}rylov: a practical algorithm for large sparse nonsymmetric matrix pencils}, JOURNAL = {SIAM J. Sci. Comput.}, FJOURNAL = {SIAM Journal on Scientific Computing}, VOLUME = {19}, YEAR = {1998}, NUMBER = {5}, PAGES = {1535--1551 (electronic)}, ISSN = {1095-7197},} @article {MR92a:65115, AUTHOR = {Grzeg{\'o}rski, S. M.}, TITLE = {On the scaled {N}ewton method for the symmetric eigenvalue problem}, JOURNAL = {Computing}, FJOURNAL = {Computing. Archiv f\"ur Informatik und Numerik}, VOLUME = {45}, YEAR = {1990}, NUMBER = {3}, PAGES = {277--282}, ISSN = {0010-485X}, CODEN = {CMPTA2},} @article {MR48:5356, AUTHOR = {Kublanovskaja, V. N.}, TITLE = { A Newton's method for finding the eigenvalues and eigenvectors of a matrix}, JOURNAL = {U.S.S.R. Comput. Math. Math. Phys. }, VOLUME = {12}, YEAR = {1973}, NUMBER = 6, PAGES = {1--12},} @article {MR94m:65058, AUTHOR = {Szyld, Daniel B. and Widlund, Olof B.}, TITLE = {Variational analysis of some conjugate gradient methods}, JOURNAL = {East-West J. Numer. Math.}, FJOURNAL = {East-West Journal of Numerical Mathematics}, VOLUME = {1}, YEAR = {1993}, NUMBER = {1}, PAGES = {51--74}, ISSN ={0928-0200}, } @incollection {MR82b:65026, AUTHOR = {Szyld, Daniel B. and Widlund, Olof B.}, TITLE = {Applications of conjugate gradient type methods to eigenvalue calculations}, BOOKTITLE = {Advances in computer methods for partial differential equations, III (Proc. Third IMACS Internat. Sympos., Lehigh Univ., Bethlehem, Pa., 1979)}, PAGES = {167--173}, PUBLISHER = {IMACS}, ADDRESS = {New Brunswick, N.J.}, YEAR = {1979}, } @article {MR1445734, AUTHOR = {Cai, Zhiqiang and Mandel, Jan and McCormick, Steve}, TITLE = {Multigrid methods for nearly singular linear equations and eigenvalue problems}, JOURNAL = {SIAM J. Numer. Anal.}, FJOURNAL = {SIAM Journal on Numerical Analysis}, VOLUME = {34}, YEAR = {1997}, NUMBER = {1}, PAGES = {178--200}, ISSN = {0036-1429}, CODEN = {SJNAAM}, } @book{temam, AUTHOR = {Temam, Roger}, TITLE = {Probl\`emes math\'ematiques en plasticit\'e}, SERIES = {M\'ethodes Math\'ematiques de l'Informatique [Mathematical Methods of Information Science]}, VOLUME = {12}, PUBLISHER = {Gauthier-Villars}, ADDRESS = {Paris}, YEAR = {1983}, PAGES = {vii+353}, ISBN = {2-04-015429-9}, } @ARTICLE{c73, AUTHOR="G. R. Cowper", TITLE="Gaussian quadrature formulas for triangles", JOURNAL="Int. J. Num. Meth. Eng.", VOLUME="7", YEAR="1973", PAGES="405-408" } @INPROCEEDINGS{mms95, AUTHOR="T.A. Manteuffel and S. McCormick and G. Starke", TITLE="First-Order Systems Least-Squares for Second-Order Elliptic problems with Discontinuous Coefficients", BOOKTITLE=" Proceedings of the Seventh Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO, April 3-7, 1995", YEAR="1995", PAGES="551", PUBLISHER="NASA Conference Publication 3339, Part 2", } @techreport{bkp98, AUTHOR = {N. S. Bakhvalov and A. V. Knyazev and R. R. Parashkevov}, TITLE = {An Efficient Iterative Method for solving {L}am\'{e} equations for nearly incompressible media and {S}tokes equations with highly discontinuous coefficients}, type = "Preprint", institution = "Center for Computational Mathematics, University of Colorado at Denver", YEAR="1997", number = "UCD-CCM 120", address = "Denver, CO", NOTE="Available at http://www-math.cudenver.edu/ccmreports/rep120.ps.gz " } @ARTICLE{bkp02, AUTHOR = {N. S. Bakhvalov and A. V. Knyazev and R. R. Parashkevov}, TITLE = { Extension Theorems for {Stokes and Lame} equations for nearly incompressible media and their applications to numerical solution of problems with highly discontinuous coefficients}, JOURNAL="Numerical Linear Algebra with Applications", VOLUME="9", NUMBER=2, YEAR="2002", PAGES="115--139", } @ARTICLE{kw98, AUTHOR = {A. V. Knyazev and O. Widlund}, TITLE = {{L}avrentiev Regularization + {R}itz Approximation = Uniform Finite Element Error Estimates for Differential Equations with Rough Coefficients}, JOURNAL="Mathematics of Computation", VOLUME="", YEAR="2001", PAGES="", NOTE= "Posted on July 13, 2001, S0025-5718-01-01378-3 (to appear in print)." } @ARTICLE{kw03, AUTHOR = {A. V. Knyazev and O. Widlund}, TITLE = {{L}avrentiev Regularization + {R}itz Approximation = Uniform Finite Element Error Estimates for Differential Equations with Rough Coefficients}, JOURNAL="Mathematics of Computation", VOLUME="72", YEAR="2003", PAGES="17-40", NOTE= "Posted on July 13, 2001, S0025-5718-01-01378-3." } @book {etemplates, TITLE = {Templates for the solution of algebraic eigenvalue problems}, EDITOR = {Bai, Zhaojun and Demmel, James and Dongarra, Jack and Ruhe, Axel and van der Vorst, Henk}, PUBLISHER = {Society for Industrial and Applied Mathematics (SIAM)}, ADDRESS = {Philadelphia, PA}, YEAR = {2000}, PAGES = {xxx+410}, ISBN = {0-89871-471-0}, MRCLASS = {65-02 (65F15)}, MRNUMBER = {1 792 141}, } @incollection{k99, AUTHOR = {A. V. Knyazev}, TITLE = {Preconditioned eigensolvers: practical algorithms}, BOOKTITLE = {Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide}, PUBLISHER = {SIAM}, YEAR="2000", PAGES="352--368", ADDRESS = {Philadelphia}, EDITOR="Z. Bai and J. Demmel and J. Dongarra and A. Ruhe and H. van der Vorst", NOTE="Section 11.3. An extended version published as a technical report UCD-CCM 143, 1999, at the Center for Computational Mathematics, University of Colorado at Denver, http://www-math.cudenver.edu/ccmreports/rep143.ps.gz" } @ARTICLE{k98, AUTHOR = {A. V. Knyazev}, TITLE = {Preconditioned eigensolvers---an oxymoron?}, NOTE = {Large scale eigenvalue problems (Argonne, IL, 1997)}, JOURNAL = {Electron. Trans. Numer. Anal.}, FJOURNAL = {Electronic Transactions on Numerical Analysis}, VOLUME = {7}, YEAR = {1998}, PAGES = {104--123 (electronic)}, ISSN = {1068-9613}, MRCLASS = {65F15 (65F35 65N25)}, MRNUMBER = {99h:65068}, } @ARTICLE{a78, AUTHOR="G. P. Astrakhantsev", TITLE="Method of fictitious domains for a second-order elliptic equation with natural boundary conditions", JOURNAL="U.S.S.R. Computational Math. and Math. Phys.", VOLUME="18", YEAR="1978", PAGES="114--121" } @article {MR54:11760, author ={G.P. Astrakhantsev}, title ={{The iterative improvement of eigenvalues}}, journal = {USSR Comput. Math. and Math. Physics}, volume = {16,1}, pages = {123--132}, year = 1976} @article{STR1976, author ={L.G. Strakhovskaya}, title ={{An iterative method for evaluating the first eigenvalue of an elliptic operator}}, journal = {USSR Comput. Math. and Math. Physics}, volume = {17,3}, pages = {88--101}, year = 1977} @article {MR97h:65146, AUTHOR = {Astrakhantsev, G. P.}, TITLE = {A decomposition method for the solution of elliptic problems in a three-dimensional domain}, JOURNAL = {Zh. Vychisl. Mat. i Mat. Fiz.}, FJOURNAL = {Rossi\u\i skaya Akademiya Nauk. Zhurnal Vychislitel{\cprime} no\u\i\ Matematiki i Matematichesko\u\i\ Fiziki}, VOLUME = {36}, YEAR = {1996}, NUMBER = {10}, PAGES = {87--96}, ISSN = {0044-4669}, } @article {MR90m:73050, AUTHOR = {Astrakhantsev, G. P.}, TITLE = {On a mixed finite element method in problems in the theory of shells}, JOURNAL = {Zh. Vychisl. Mat. i Mat. Fiz.}, FJOURNAL = {Akademiya Nauk SSSR. Zhurnal Vychislitel{\cprime} no\u\i\ Matematiki i Matematichesko\u\i\ Fiziki}, VOLUME = {29}, YEAR = {1989}, NUMBER = {10}, PAGES = {1492--1504, 1598}, ISSN = {0044-4669}, } @article {MR88d:65153, AUTHOR = {Astrakhantsev, G. P. and Rukhovets, L. A.}, TITLE = {Fictitious component method for solving grid equations used to approximate higher-order elliptic equations with natural boundary conditions}, JOURNAL = {Soviet J. Numer. Anal. Math. Modelling}, FJOURNAL = {Soviet Journal of Numerical Analysis and Mathematical Modelling}, VOLUME = {1}, YEAR = {1986}, NUMBER = {1}, PAGES = {37--46}, ISSN = {0169-2895}, } @book {MR95b:35079, AUTHOR = {Levendorskii, Serge}, TITLE = {Degenerate elliptic equations}, SERIES = {Mathematics and its Applications}, VOLUME = {258}, PUBLISHER = {Kluwer Academic Publishers Group}, ADDRESS = {Dordrecht}, YEAR = {1993}, PAGES = {xii+431}, ISBN = {0-7923-2305-X}, } @book {MR96h:35003b, AUTHOR = {Jikov, V. V. and Kozlov, S. M. and Ole{\u\i}nik, O. A.}, TITLE = {Homogenization of differential operators and integral functionals}, NOTE = {Translated from the Russian by G. A. Yosifian [G. A. Iosifyan]}, PUBLISHER = {Springer-Verlag}, ADDRESS = {Berlin}, YEAR = {1994}, PAGES = {xii+570}, ISBN = {3-540-54809-2}, } @book {MR82h:35001, AUTHOR = {Bensoussan, Alain and Lions, Jacques-Louis and Papanicolaou, George}, TITLE = {Asymptotic analysis for periodic structures}, SERIES = {Studies in Mathematics and its Applications}, VOLUME = {5}, PUBLISHER = {North-Holland Publishing Co.}, ADDRESS = {Amsterdam}, YEAR = {1978}, PAGES = {xxiv+700}, ISBN = {0-444-85172-0}, } @book {MR82j:35010, AUTHOR = {S{\'a}nchez-Palencia, Enrique}, TITLE = {Nonhomogeneous media and vibration theory}, SERIES = {Lecture Notes in Physics}, VOLUME = {127}, PUBLISHER = {Springer-Verlag}, ADDRESS = {Berlin}, YEAR = {1980}, PAGES = {ix+398}, ISBN = {3-540-10000-8}, } @book {MR93k:35025, AUTHOR = {Ole{\u\i}nik, O. A. and Shamaev, A. S. and Yosifian, G. A.}, TITLE = {Mathematical problems in elasticity and homogenization}, SERIES = {Studies in Mathematics and its Applications}, VOLUME = {26}, PUBLISHER = {North-Holland Publishing Co.}, ADDRESS = {Amsterdam}, YEAR = {1992}, PAGES = {xiv+398}, ISBN = {0-444-88441-6}, } @book {MR1482803, AUTHOR = {Pankov, Alexander}, TITLE = {${G}$-convergence and homogenization of nonlinear partial differential operators}, SERIES = {Mathematics and its Applications}, VOLUME = {422}, PUBLISHER = {Kluwer Academic Publishers}, ADDRESS = {Dordrecht}, YEAR = {1997}, PAGES = {xiv+249}, ISBN = {0-7923-4720-X}, } @incollection {MR22:12710, AUTHOR = {Abramow, A. and Neuhaus, M.}, TITLE = {Bemerkungen \"uber {E}igenwertprobleme von {M}atrizen h\"oherer {O}rdnung}, YEAR = {1958}, BOOKTITLE = {Les math\'ematiques de l'ing\'enieur}, PAGES = {176--179}, PUBLISHER = {M\'em. Publ. Soc. Sci. Arts Lett. Hainaut, vol. hors S\'erie}, } @ARTICLE{s61, AUTHOR = {V. A. Shishov}, TITLE = {A method for partitioning a high order matrix into blocks in order to find its eigenvalues}, JOURNAL="{USSR} Computational Math. and Math. Physics", NUMBER="1", VOLUME="1", YEAR="1961", PAGES="186--190" } @ARTICLE{fg94, AUTHOR = {C. Farhat and M. Geradin}, TITLE = {On a component mode synthesis method and its application to incompatible substructures}, JOURNAL="Computers & Structures", NUMBER="5", VOLUME="51", YEAR="1994", PAGES="459--473" } @book {MR92i:35013, AUTHOR = {Maslov, V. P.}, TITLE = {{{A}simptoticheskie metody i teoriya vozmushcheni\u\i} [Asymptotic methods and perturbation theory] }, PUBLISHER = {``Nauka''}, ADDRESS = {Moscow}, YEAR = {1988}, PAGES = {311}, NOTE = {(In {R}ussian)}, ISBN = {5-02-013784-7}, } @ARTICLE{c01b, AUTHOR="Cosserat, E. and Cosserat, F. ", TITLE="Sur une application des fonctions potentielles de la th\'{e}orie de l'\'{e}lasticit\'{e} ", JOURNAL="C. R. Acad. Sci. (Paris) ", YEAR="1901", VOLUME="133", PAGES="210-213" } @ARTICLE{c01f, AUTHOR="Cosserat, E. and Cosserat, F. ", TITLE="Sur une point critique particulier de la solution des \'{e}quations de l'\'{e}lasticit\'{e} dans le cas o\`{u} les efforts sur la fronti\`{e}re sont donn\'{e}es ", JOURNAL="C. R. Acad. Sci. (Paris) ", YEAR="1901", VOLUME="133", PAGES="382-384" } @ARTICLE{m73, AUTHOR="Mikhlin, S. G. ", TITLE="The Spectrum of a Family of Operators in the Theory of Elasticity ", JOURNAL="Russian Math Surveys ", YEAR="1973", VOLUME="28", PAGES="45-88" } @BOOK{lad, AUTHOR="O. A. Ladyzhenskaya", TITLE="The mathematical theory of viscous incompressible flow", YEAR="1969", PUBLISHER="{G}ordon and {B}reach", ADDRESS="New York" } @book {gr86, AUTHOR = {Girault, Vivette and Raviart, Pierre-Arnaud}, TITLE = {Finite element methods for {N}avier--{S}tokes equations}, NOTE = {Theory and algorithms}, SERIES = {Springer Series in Computational Mathematics}, VOLUME = {5}, PUBLISHER = {Springer-Verlag}, ADDRESS = {Berlin}, YEAR = {1986}, PAGES = {x+374}, ISBN = {3-540-15796-4}, } @book {lit82, AUTHOR = {Litvinov, V. G.}, TITLE = {Motion of a nonlinearly viscous fluid}, NOTE = {(In {R}ussian)}, PUBLISHER = {``Nauka''}, ADDRESS = {Moscow}, YEAR = {1982}, PAGES = {375}, } @book {lit87, AUTHOR = {Litvinov, V. G.}, TITLE = { Optimization in elliptic boundary value problems with applications to mechanics },PUBLISHER = {``Nauka''}, ADDRESS = {Moscow}, YEAR = {1987}, PAGES = {367}, NOTE = {(In {R}ussian)},} @book {MR85b:73031, AUTHOR = {Banichuk, N. V.}, TITLE = {Problems and methods of optimal structural design}, NOTE = {Translated from the Russian by Vadim Komkov, Translation edited by Edward J. Haug, With a preface by Haug and Komkov}, SERIES = {Mathematical Concepts and Methods in Science and Engineering}, VOLUME = {26},PUBLISHER = {Plenum Press}, ADDRESS = {New York}, YEAR = {1983}, PAGES = {xxi+313}, ISBN = {0-306-41284-5},} @ARTICLE{ak89, AUTHOR="Aarden, J. M. C. and Karlsson, K.-E.", TITLE=" Preconditioned {CG}-type methods for solving the coupled systems of fundamental semiconductor equations", JOURNAL="BIT", VOLUME="29", NUMBER="4", YEAR="1989", PAGES=916 } @ARTICLE{a72, AUTHOR="V. B. Andreev", TITLE="Stability of finite difference schemes for elliptic equations with respect to {D}irichlet boundary conditions ", JOURNAL="{USSR} Computational Math. and Math. Physics", NUMBER="12", YEAR="1972", PAGES="35--52" } @ARTICLE{acl89, AUTHOR="Axelsson, O. and Carey, G. and Lindskog, G.", TITLE=" On a Class of Preconditioned Iterative Methods on Parallel Computers", JOURNAL="International J. for Numerical Methods in Engineering", VOLUME="27", NUMBER="3", YEAR="1989", PAGES={637-654} } @ARTICLE{av89, AUTHOR="Axelsson, O. and Vassilevski, P. S.", TITLE=" A survey of multilevel preconditioned iterative methods", JOURNAL="BIT", VOLUME="29", NUMBER="4", YEAR="1989", PAGES=769 } @ARTICLE{as91, AUTHOR="Ayari, M. L. and Saouma, V. E. ", TITLE="Static and Dynamic Contact/Impact Problems Using Fictitious Forces ", JOURNAL="International Journal for Numerical Methods in Engineering ", YEAR="1991", VOLUME="32", NUMBER="3", PAGES=623 } @article {MR95g:65146, AUTHOR = {Babu{\v{s}}ka, Ivo and Caloz, Gabriel and Osborn, John E.}, TITLE = {Special finite element methods for a class of second order elliptic problems with rough coefficients}, JOURNAL = {SIAM J. Numer. Anal.}, FJOURNAL = {SIAM Journal on Numerical Analysis}, VOLUME = {31}, YEAR = {1994}, NUMBER = {4}, PAGES = {945--981}, ISSN = {0036-1429}, CODEN = {SJNAAM}, review=" The authors consider the numerical solution of an elliptic problem of the form (1) $Lu(x,y)\equiv -(\partial /\partial x)(a\partial u/\partial x)-(\partial/\partial y)(a\partial u/\partial y)=f(x,y)$ for $(x,y)\in\Omega$, $u=0$ on $\partial \Omega$ with a nonsmooth coefficient $a=a(x,y)$ satisfying $0<\alpha\leq a\leq \beta$. In this review all spaces and norms refer to the domain $\Omega$ and results are stated only for $\Omega=\Omega\sb 0=(0,1)\times(0,1)$, and for unidirectional coefficients $a=a(x)$, a case of practical importance (unidirectional composite materials). The solution of (1) is sought in the space $H\sp L=\{u\in H\sp 1\colon a\partial u/\partial x,\partial u/\partial y\in H\sp 1\}$ with the norm $\Vert u\Vert \sp 2\sb L=\Vert u\Vert \sp 2\sb 1+\vert u\vert \sp 2\sb L$, where $$\vert u\vert \sp 2\sb L=\int\sb \Omega\bigg\{a\bigg\vert \frac{\partial}{\partial x}\bigg(a\frac{\partial u}{\partial x}\bigg)\bigg\vert \sp 2+a\bigg\vert \frac{\partial\sp 2 u}{\partial x\partial y}\bigg\vert \sp 2+\frac{1}{a}\bigg\vert \frac{\partial\sp 2 u}{\partial y\sp 2}\bigg\vert \sp 2\bigg\}dx\,dy.$$ First, a regularity result states the following. Suppose that $\Omega=\Omega\sb 0$ and $a=a(x)$; then, for $f\in L\sp 2$, the solution $u$ of (1) belongs to $H\sp 1\sb 0\cap H\sp L$. Furthermore, there is a constant $C=C(\alpha,\beta)$, independent of $f$, such that $\Vert u\Vert \sb L\leq C\Vert f\Vert \sb 0$. This theorem is extended to more general $\Omega$ and to the curvilinear unidirectional case $a(x,y)=a'(\xi)$, for some coordinates $\xi=\xi(x,y)$, $\eta=\eta(x,y)$. A first method of finite elements uses a mapping $\Omega\to\tilde\Omega$, $\tilde x(x)=\int\sp x\sb 0a\sp {-1}(s)ds$, $\tilde y(y)=y$, to construct a triangulation $\scr C\sb h$ as well as a corresponding conforming space of trial shape functions $S\sb h$. A curvilinear triangle $T$ of $\scr C\sb h$ is mapped into an ordinary triangle $\tilde T\subset\tilde\Omega$, and satisfies a minimal angle condition and diam $T\leq h$. The trial space is defined by the conditions $S\sb h=\{v\in L\sp 2\colon v\vert \sb T\in{\rm span}(1,\tilde x(x),y)$ for all $T\in \scr C\sb h\}$, $v$ is continuous at the nodes of $\scr C\sb h$ and vanishes at the boundary nodes. The finite element approximation $u\sb h$ to the solution $u$ of (1) is obtained from the weak formulation with the bilinear form $B(u,v)=\int\sb {\Omega}a\nabla uo\nabla vdx\,dy$ $(u,v\in H\sp 1\sb 0)$, namely, $u\sb h\in S\sb h$, $B(u\sb h,v)=\int\sb \Omega fvdx\,dy$, for all $v\in S\sb h$. In the case of a smooth coefficient $a$, with piecewise linear approximation, here too one has the estimate (2) $\Vert u-u\sb h\Vert \sb 1\leq{\rm Ch}\Vert f\Vert \sb 0$ $(0