%%% -*-BibTeX-*- %%% ==================================================================== %%% BibTeX-file{ %%% author = "Ulrich Ruede", %%% version = "0.08", %%% date = "11 May 2013", %%% time = "12:10:40 MDT", %%% filename = "ruede-ulrich.bib", %%% address = "Fakultaet fuer Mathematik %%% Technische Universitaet Chemnitz-Zwickau %%% D-09009 Chemnitz %%% Germany", %%% telephone = "+49 - 0371 - 561 -2159", %%% FAX = "?n/a?", %%% checksum = "28698 854 3895 39230", %%% email = "ruede at mathematik.tu-chemnitz.de (Internet)", %%% codetable = "ISO/ASCII", %%% keywords = "Scientific Computing, Multilevel Adaptive Methods", %%% license = "public domain", %%% supported = "yes", %%% docstring = "This is a bibliography of publications of %%% Ulrich Ruede. %%% %%% At version 0.08, the year coverage looked %%% like this: %%% %%% 1985 ( 1) 1989 ( 2) 1993 ( 8) %%% 1986 ( 3) 1990 ( 3) 1994 ( 3) %%% 1987 ( 2) 1991 ( 3) %%% 1988 ( 5) 1992 ( 9) %%% %%% Article: 6 %%% Book: 1 %%% InProceedings: 11 %%% TechReport: 20 %%% Unpublished: 1 %%% %%% Total entries: 39 %%% %%% The checksum field above contains a CRC-16 %%% checksum as the first value, followed by the %%% equivalent of the standard UNIX wc (word %%% count) utility output of lines, words, and %%% characters. This is produced by Robert %%% Solovay's checksum utility.", %%% } %%% ==================================================================== @Preamble{"\input path.sty"} %%% ==================================================================== %%% Journal abbreviations: @String{j-INT-J-HSC= "International Journal of High Speed Computing"} @String{j-KERNTECHNIK= "Kerntechnik"} @String{j-SIAM-J-NUMER-ANAL= "SIAM Journal on Numerical Analysis"} @String{j-SIAM-J-SCI-STAT-COMP= "SIAM J. Sci.~Stat.~Comput."} %%% ==================================================================== %%% Publisher abbreviations: @String{pub-AMS= "Amer. Math. Soc."} @String{pub-AMS:adr= "Providence, RI, USA"} @String{pub-BIRKHAUSER= "Birkh{\"{a}}user"} @String{pub-BIRKHAUSER:adr= "Cambridge, MA, USA; Berlin, Germany; Basel, Switzerland"} @String{pub-DEKKER= "Marcel Dekker"} @String{pub-DEKKER:adr= "New York, NY, USA"} @String{pub-INST-ANG-ANA-STOCH= "Institut f{\"u}r Angewandte Analysis und Stochastik"} @String{pub-INST-ANG-ANA-STOCH:adr= "Berlin, Germany"} @String{pub-NASA= "NASA"} @String{pub-SIAM= "SIAM"} @String{pub-SIAM:adr= "Philadelphia, PA, USA"} @String{pub-SV= "Springer-Verlag"} @String{pub-SV:adr= "Berlin, Germany~/ Heidelberg, Germany~/ London, UK~/ etc."} %%% ==================================================================== %%% Bibliography entries: @TechReport{Ruede:1985:AMB, author = "U. R{\"u}de", title = "{Anwendung der Mehrgittermethode zur Berechnung von digitalen H{\"o}henmodellen in der Photogrammetrie}", type = "Bericht", number = "I-8525", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = nov, year = "1985", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", } @InProceedings{Ruede:1986:DMM, author = "U. R{\"u}de", editor = "W. Hackbusch and U. Trottenberg", booktitle = "Multigrid Methods: Special Topics and Applications, Papers presented at the 2nd European Conference on Multigrid Methods, October 1-4, 1985", title = "Discretizations for Multigrid Methods", volume = "110", address = "Cologne", pages = "??--?? (of 178)", month = may, year = "1986", ISBN = "3-88457-110-9", ISBN-13 = "978-3-88457-110-1", LCCN = "QA377.E87 1985", bibdate = "Mon Jul 11 13:00:25 1994", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", note = "Also available as TU-Bericht I-8519", series = "GMD Studien", } @InProceedings{Ruede:1986:TSM, author = "U. R{\"u}de and C. Zenger", editor = "W. Hackbusch and U. Trottenberg", booktitle = "Lecture Notes in Mathematics 1228: Multigrid Methods II, Proceedings of the Conference Held at Cologne, October 1-4, 1985", title = "On the Treatment of Singularities in the Multigrid Method", publisher = pub-SV, address = pub-SV:adr, pages = "??--?? (of vi + 335)", year = "1986", ISBN = "0-387-16491-X", ISBN-13 = "978-0-387-16491-5", LCCN = "QA3 .L35 v.1228", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", price = "DM50.00", } @TechReport{Ruede:1986:WMM, author = "U. R{\"u}de and C. Zenger", title = "A Workbench for Multigrid Methods", type = "Bericht", number = "I-8607", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = may, year = "1986", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", } @TechReport{Foessmeier:1987:OSS, author = "R. F{\"o}{\ss}meier and U. R{\"u}de", title = "Operating System Support for Parallel Numerical Software Development", type = "Bericht", number = "I-8712", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = oct, year = "1987", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", annote = "Foessmeier Ruede", } @TechReport{Ruede:1987:MTE, author = "U. R{\"u}de", title = "Multiple tau-Extrapolation for Multigrid Methods", type = "Bericht", number = "I-8701", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = jan, year = "1987", bibdate = "Mon May 23 09:36:47 MDT 1994", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", } @TechReport{Jaensch:1988:MET, author = "C. R. J{\"a}nsch and U. R{\"u}de and K. Schnepper", title = "Macro Expansion, a Tool for the Systematic Development of Scientific Software", type = "Bericht", number = "I-8814", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = nov, year = "1988", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", } @TechReport{Muszynski:1988:AAM, author = "P. Muszynski and U. R{\"u}de and C. Zenger", title = "Application of Algebraic Multigrid {(AMG)} to Constrained Quadratic Optimization", type = "Bericht", number = "I-8801", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = jan, year = "1988", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", } @InProceedings{Ruede:1988:ACS, author = "U. R{\"u}de", editor = "S. F. McCormick", booktitle = "Multigrid Methods: Theory, Applications, Supercomputing: Proceedings of the Third Copper Mountain Conference on Multigrid Methods, April 5-10, 1987", title = "On the Accurate Computation of Singular Solutions of {Laplace's and Poisson's} Equation", publisher = pub-DEKKER, address = pub-DEKKER:adr, pages = "??--?? (of xiv + 644)", year = "1988", ISBN = "0-8247-7979-7", ISBN-13 = "978-0-8247-7979-5", LCCN = "QA377 .M9431 1988", bibdate = "Mon May 23 09:36:47 MDT 1994", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", } @TechReport{Ruede:1988:NBS, author = "U. R{\"u}de", title = "{Zur numerischen Behandlung von Singularit{\"a}ten in elliptischen partiellen Differentialgleichungen}", type = "Bericht", number = "I-8810", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = aug, year = "1988", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", } @Article{Zenger:1988:BSE, author = "C. Zenger and R. F{\"o}{\ss}meier and U. R{\"u}de", title = "{Betriebssystem- und Software-Engineering-Aspekte bei parallelen Algorithmen}", journal = j-KERNTECHNIK, volume = "52", number = "2", pages = "120--125", year = "1988", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", annote = "Foessmeier Ruede", } @TechReport{McCormick:1989:FVC, author = "S. McCormick and U. R{\"u}de", title = "A Finite Volume Convergence Theory for the Fast Adaptive Composite Grid Method", institution = "University of Colorado at Denver", year = "1989", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", note = "To be published in Applied Numerical Mathematics (14) 1994, Elsevier", annote = "Ruede", } @InProceedings{Ruede:1989:LCE, author = "U. R{\"u}de", editor = "J. Mandel", booktitle = "Proceedings of the Fourth Copper Mountain Conference on Multigrid Methods, April 9-14, 1989", title = "Local Corrections for Eliminating the Pollution Effect of Reentrant Corners", publisher = pub-SIAM, address = pub-SIAM:adr, pages = "365--382", year = "1989", bibdate = "Mon May 23 09:36:47 MDT 1994", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", } @Article{Foessmeier:1990:KWP, author = "R. F{\"o}{\ss}meier and U. R{\"u}de", title = "{Konzepte und Werkzeuge zur parallelen Programmierung auf der Unix-Kommando-Ebene}", journal = "unix/mail", volume = "8", number = "2", pages = "66--73", year = "1990", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", } @Article{McCormick:1990:LRH, author = "S. McCormick and U. R{\"u}de", title = "On Local Refinement Higher Order Methods for Elliptic Partial Differential Equations", journal = j-INT-J-HSC, volume = "2", number = "4", pages = "311--334", year = "1990", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", note = "Also available as TU-Bericht I-9034", annote = "Ruede", } @TechReport{Slavkovsky:1990:SBK, author = "P. Slavkovsky and U. R{\"u}de", title = "{Schnellere Berechnung klassischer Matrix-Multiplikationen}", type = "SFB Bericht", number = "342/17/90", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = sep, year = "1990", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", annote = "Ruede", } @TechReport{Arbesmeier:1991:TMM, author = "M. Arbesmeier and U. R{\"u}de", title = "A Toolbox for Multigrid Methods", type = "Bericht", number = "I-9136", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = sep, year = "1991", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", URL = "file://www.tu-chemnitz.de/pub/Local/mathematik/Ruede/mgwb.ps.Z", abstract = "The multigrid workbench project is an ongoing effort to design an interactive environment for multigrid algorithms. Central components are a graphical user interface and visualization modules. The workbench in its present form is based on a specialized high level toolset that provides the basic multigrid components for two-dimensional stationary diffusion equations. Because of its functional and stream-oriented programming style the workbench can be used to solve large systems with up to several million unknowns on standard workstations. This report discusses the related software engineering and efficiency issues. A prototype of the interactive user environment and basic visualization components is available.", } @InProceedings{Ruede:1991:AHO, author = "U. R{\"u}de", editor = "W. Hackbusch and U. Trottenberg", booktitle = "Proceedings of the Third European Conference on Multigrid Methods, October 1-4, 1990", title = "Adaptive Higher Order Multigrid Methods", publisher = pub-BIRKHAUSER, address = pub-BIRKHAUSER:adr, pages = "339--351", year = "1991", bibdate = "Mon May 23 09:36:47 MDT 1994", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", note = "International Series of Numerical Mathematics, Vol.~98", } @TechReport{Ruede:1991:ERT, author = "U. R{\"u}de", title = "Extrapolation and Related Techniques for Solving Elliptic Equations", type = "Bericht", number = "I-9135", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = sep, year = "1991", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", URL = "file://www.tu-chemnitz.de/pub/Local/mathematik/Ruede/extra_rel.ps.Z", abstract = "Extrapolation is a well-known numerical technique for raising the approximation order. Several variants of extrapolation can be used for elliptic partial differential equations. The basic algorithmic variants are Richardson extrapolation, truncation error extrapolation and extrapolation of the functional. In multi-dimensional problems the error can be expanded into multivariate polynomials with respect to mesh parameters for the different coordinate directions. This can be exploited by multivariate extrapolation and the combination and sparse grid techniques. In this paper these methods are introduced and discussed in detail. The features and effectiveness are illustrated in numerical experiments for model problems.", } @TechReport{Bonk:1992:PAO, author = "T. Bonk and U. R{\"u}de", title = "Performance Analysis and Optimization of Numerically Intensive Programs", type = "SFB Bericht", number = "342/26/92 A", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = nov, year = "1992", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", URL = "file://www.tu-chemnitz.de/pub/Local/mathematik/Ruede/performance.ps.Z", abstract = "In this paper we explore the characteristics of numerically intensive programs and explore their efficient implementation on a variety of machine architectures. It is demonstrated that different architectures need different optimization techniques. The emphasis of the paper is on modern RISC-CPUs on the one side and advanced, recursive algorithms on the other side.", annote = "Ruede", keywords = "Numerically intensive computing, supercomputing, performance measurement, optimizing compilers, basic linear algebra operations, recursion, matrix multiplication, high dimensional numerical quadrature", } @TechReport{Bungartz:1992:ECS, author = "H. Bungartz and M. Griebel and U. R{\"u}de", title = "Extrapolation, Combination and Sparse Grid Techniques for Elliptic Boundary Value Problems", type = "SFB Bericht", number = "342/10/92 A", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = may, year = "1992", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", note = "to be published in {\em Computer Methods in Applied Mechanics and Engineering} (1994)", annote = "Ruede", } @TechReport{Griebel:1992:CTPa, author = "M. Griebel and W. Huber and U. R{\"u}de and T. St{\"o}rtkuhl", title = "The Combination Technique for Parallel Sparse-Grid-Preconditioning and -Solution of {PDE}s on Multiprocessor Machines and Workstation Networks", type = "SFB Bericht", number = "342/11/92 A", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = may, year = "1992", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", annote = "Ruede", } @InProceedings{Griebel:1992:CTPb, author = "M. Griebel and W. Huber and U. R{\"u}de and T. St{\"o}rtkuhl", editor = "L. Boug{\'e} and M. Cosnard and Y. Robert and D. Trystram", booktitle = "Parallel Processing: {CONPAR 92 -- VAPP V}", title = "The Combination Technique for Parallel Sparse-Grid-Preconditioning or -Solution of {PDE}s on Workstation Networks", volume = "634", publisher = pub-SV, address = pub-SV:adr, pages = "217--228", year = "1992", bibdate = "Mon May 23 09:36:47 MDT 1994", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", note = "Proceedings of the Second Joint International Conference on Vector and Parallel Processing, Lyon, France, September 1--4, 1992", series = "Lecture Notes in Computer Science", annote = "Ruede", } @TechReport{Ruede:1992:DSM, author = "U. R{\"u}de", title = "Data Structures for Multilevel Adaptive Methods and Iterative Solvers", type = "Bericht", number = "I-9217", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = may, year = "1992", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", URL = "file://www.tu-chemnitz.de/pub/Local/mathematik/Ruede/data_structures.ps.Z", abstract = "The adaptive solution of partial differential equations by finite elements must be supported by suitable data structures. Besides an overview of existing adaptive mesh techniques, our analysis provides an abstract treatment discussing the necessary functionality independent of concrete representations. The different aspects can be organized in topological, geometric and the algebraic components. Further considerations are necessary to support self-adaptivity in a multilevel context. We distinguish between element based and node based data structures and will discuss their implementation in an object oriented language.", } @Article{Ruede:1992:HBE, author = "U. R{\"u}de", editor = "T. Manteuffel", title = "The Hierarchical Basis Extrapolation Method", journal = j-SIAM-J-SCI-STAT-COMP, volume = "13", number = "1", pages = "307--318", month = jan, year = "1992", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", note = "Proceedings of the First Copper Mountain Conference on Iterative Methods, April 1-5, 1990, T. Manteuffel ed.", } @InProceedings{Ruede:1992:REF, author = "U. R{\"u}de", editor = "A. Quarteroni", booktitle = "Proceedings of the Sixth International Conference on Domain Decomposition in Science and Engineering, Como, Italy, June 15-19, 1992", title = "On the robustness and efficiency of the fully adaptive multigrid method", publisher = pub-AMS, address = pub-AMS:adr, pages = "??--?? (of xxii + 484)", year = "1992", ISBN = "0-8218-5158-6", ISBN-13 = "978-0-8218-5158-6", LCCN = "QA402.2 .I55 1992", bibdate = "Mon May 23 09:36:47 MDT 1994", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", abstract = "The fully adaptive multigrid method (FAMe) is a finite element based elliptic solver integrating self-adaptivity, error estimation and efficient iterative solution. Refined elements are not restricted to predetermined regions and need not be grouped in patches. Instead, whether an element is refined, is decided individually for each element using an integrated error indicator. The refinement process induces a multilevel structure and therefore a natural decomposition of the solution space into a nested sequence. This can be exploited to define an efficient solver and error estimator.", annote = "Contemporary Mathematics", } @TechReport{Ruede:1992:TSF, author = "U. R{\"u}de and C. Zenger", title = "On the Treatment of Singularities in the Finite Element Method", type = "Bericht", number = "I-9220", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = aug, year = "1992", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", } @TechReport{Ruede:1992:VCF, author = "U. R{\"u}de", title = "On the {V}-cycle of the fully adaptive multigrid method", type = "Bericht", number = "I-9215", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = may, year = "1992", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", note = "to be published in the proceedings of the 9th GAMM Seminar, Kiel, January 22--24, 1993", URL = "file://www.tu-chemnitz.de/pub/Local/mathematik/Ruede/kiel93.ps.Z", abstract = "The Fully Adaptive Multigrid Method (FAMe) is a concept for the effective solution of elliptic problems including robust and efficient iterative solution, error estimation, and self-adaptive refinement. In this paper we introduce a variant of the FAMe similar in structure to a multigrid V-cycle and a multiplicative multilevel Schwarz method. This variant permits a convergence analysis showing that the FAMe provides optimal convergence rates when the classical methods do. The FAMe, however, will be more efficient in a local refinement context by exploiting the locality of the computations and will be more robust, because it naturally provides diagnostic information that can serve as rigorous error bounds.", } @TechReport{Pflaum:1993:GAR, author = "C. Pflaum and U. R{\"u}de", title = "{Gau\ss}' adaptive relaxation for the multilevel solution of partial differential equations on sparse grids", type = "SFB-Bericht", number = "342/13/93 A", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = sep, year = "1993", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", note = "to appear in the proceedings of the 2nd Gau{\ss} Symposium, Munich, Aug.~2--7, 1993", URL = "file://www.tu-chemnitz.de/pub/Local/mathematik/Ruede/gauss.ps.Z", abstract = "In combination with the multilevel principle, relaxation methods are among the most efficient numerical solution techniques for elliptic partial differential equations. Typical methods used today are derivations of the Gauss -Seidel or Gauss -Jacobi method. Recently it has been recognized that in the context of multilevel algorithms, the original method suggested by Gauss has specific advantages. For this method the iteration is concentrated on unknowns where fast convergence can be obtained by intelligently monitoring the residuals. We will present this algorithm in the context of a sparse grid multigrid algorithm. Using sparse grids the dimension of the discrete approximation space can be reduced additionally.", annote = "Ruede", } @InProceedings{Regler:1993:LOA, author = "H. Regler and U. R{\"u}de", booktitle = "Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, Copper Mountain, April 4-9, 1993", title = "Layout optimization with Algebraic Multigrid Methods {(AMG)}", publisher = pub-NASA, year = "1993", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", series = "Conference Publication", URL = "file://www.tu-chemnitz.de/pub/Local/mathematik/Ruede/amg.ps.Z", annote = "Ruede", } @InProceedings{Ruede:1993:DAT, author = "U. R{\"u}de", booktitle = "Proceedings of the GAMM-Seminar on Multigrid Methods, Sept. 21 -- 25, 1992 in Gosen, Germany", title = "Data abstraction techniques for multilevel algorithms", publisher = pub-INST-ANG-ANA-STOCH, address = pub-INST-ANG-ANA-STOCH:adr, year = "1993", bibdate = "Mon May 23 09:36:47 MDT 1994", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", note = "Report 5, ISSN 0942--9077", URL = "file://www.tu-chemnitz.de/pub/Local/mathematik/Ruede/gosen92.ps.Z", abstract = "Multilevel methods are fast and efficient solvers for a wide range of technical and scientific applications. Their structural complexity makes the construction of powerful multilevel based software difficult. Conventional software engineering concepts do not provide a sufficient basis for the implementation of general, fast, and robust multilevel applications. In particular, there is a severe tradeoff between the generality of such software and its efficiency. These problems can be alleviated on the basis of a consequent data abstraction. To equally satisfy the demands for generality and efficiency it is necessary to introduce a two level software model based on a generation and an execution phase. Suitable implementation techniques are discussed.", editors = "S. Hengst", } @Unpublished{Ruede:1993:EEB, author = "U. R{\"u}de", title = "Error estimators based on stable splittings", year = "1993", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", note = "Submitted to the proceedings of the 7th International Conference On domain Decomposition, Penn State University", URL = "file://www.tu-chemnitz.de/pub/Local/mathematik/Ruede/ddm7.ps.Z", abstract = "Multilevel- and domain decomposition methods are generally based on a splitting of the solution space. The subspaces in such a splitting and their Hilbert space structure define elementary operations, called subspace corrections which can in turn be used to construct iterative methods and preconditioners. In this general setup many solution methods can be described, including classical relaxation schemes and domain decomposition algorithms. We are particularly interested in the special case of (variational) multigrid methods, based on a nested system of subspaces and the multiplicative combination of the subspace corrections, or the corresponding additive methods that lead to efficient preconditioners. The discussion of methods in this setup turns out to be useful, because the performance of solvers and preconditioners can be described by abstract features of the subspace system. The stability of the splitting has been introduced as the basic property that determines the efficiency of the corresponding methods. Furthermore, however, the splitting of the space can be used to derive error estimators. If the splitting is stable, the lower and upper bounds obtained for the error are uniformly bounded. Depending on the interpretation of the spaces, the bounds apply to the (algebraic) iteration error or the (continuous) discretization error. In any case, the estimate is based on subspace corrections, as they are computed in each iteration step. Therefore the error estimate does not create any additional cost. In the paper we review the theory of stable splittings and the derivation of the abstract error estimates. Furthermore we discuss their performance in realistic algorithms and their efficient integration with the multilevel adaptive iteration and the virtual global mesh refinement technique to a fully adaptive multilevel technique for the solution of partial differential equations.", } @TechReport{Ruede:1993:ETC, author = "U. R{\"u}de", title = "Extrapolation techniques for constructing higher order finite element methods", type = "Bericht", number = "I-9304", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", year = "1993", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", URL = "file://www.tu-chemnitz.de/pub/Local/mathematik/Ruede/extrapolation.ps.Z", abstract = "The p-version of the finite element methods requires the exact calculation of the stiffness matrix by a special form of numerical integration. As an alternative to classical techniques that are based on Gauss quadrature, we propose to use low order methods combined with extrapolation. To this purpose we derive asymptotic expansions for basic integration methods on triangles. In contrast to conventional extrapolation methods for elliptic equations these results use only a local analysis and can thus be used on unstructured meshes. We present a complete analysis and examples with practical suggestions for extrapolation-based high order finite element methods.", } @Article{Rude:1993:FAM, author = "Ulrich R{\"u}de", title = "Fully Adaptive Multigrid Methods", journal = j-SIAM-J-NUMER-ANAL, volume = "30", number = "1", pages = "230--248", month = feb, year = "1993", CODEN = "SJNAAM", ISSN = "0036-1429 (print), 1095-7170 (electronic)", MRclass = "65N55", MRnumber = "93i:65117", bibdate = "Mon Jan 20 15:27:00 MST 1997", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", abstract = "Adaptivity is a key concept for the effective numerical solution of differential equations. The multilevel solution of elliptic partial differential equations can be combined with adaptive mesh refinement and an adaptive choice of the discretization order. Additionally, adaptivity may be built into the relaxation and the multilevel cycling strategy. The goal of these {\em fully adaptive} methods is to spend work only where it is most effective in the solution process. This approach includes concepts like {\em local relaxation} and not only leads to particularly fast convergence but also to additional robustness and generality. The efficient implementation of fully adaptive multilevel methods in a finite element framework will be discussed.", acknowledgement = ack-nhfb, amsmos = "65N22, 65N30, 65N50, 65N55", keywords = "adaptivity, multilevel techniques, elliptic PDE, finite elements", } @Book{Ruede:1993:MCT, author = "U. R{\"u}de", title = "Mathematical and computational techniques for multilevel adaptive methods", volume = "13", publisher = pub-SIAM, address = pub-SIAM:adr, pages = "xii + 140", year = "1993", ISBN = "0-89871-320-X", ISBN-13 = "978-0-89871-320-6", LCCN = "QA377 .R87 1993", bibdate = "Mon May 23 09:36:47 MDT 1994", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", series = "Frontiers in Applied Mathematics", } @TechReport{Ruede:1993:MES, author = "U. R{\"u}de", title = "Multilevel, Extrapolation, and Sparse Grid Methods", type = "SFB Bericht", number = "342/10/93 A/ I-9319", institution = "Institut f{\"u}r Informatik, TU M{\"u}nchen", month = jul, year = "1993", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", note = "to appear in the Proceedings of the \ European Conference on Multigrid Methods, Amsterdam, July 6--9, P. Hemker and P. Wesseling eds.", URL = "file://www.tu-chemnitz.de/pub/Local/mathematik/Ruede/emg93.ps.Z", abstract = "Multigrid Methods are asymptotically optimal solvers for discretized partial differential equations (PDE). For the optimal solution of PDEs, however, the quality of the discretization is of the same importance as the speed of the algebraic solution process. Especially for high accuracy requirements, high order discretizations become increasingly attractive. We describe higher order techniques, like extrapolation and sparse grid combination that are particularly interesting in the context of multilevel algorithms, because they are based on discretizing the problems on grids with different mesh sizes. Classical Richardson extrapolation can be extended and generalized in many ways. One generalization is to consider the mesh widths in the different coordinate directions as distinct parameters. This leads to the so-called multivariate extrapolation and the combination technique.", } @InProceedings{Balder:1994:SGE, author = "R. Balder and U. R{\"u}de and S. Schneider and C. Zenger", booktitle = "Proceedings of the 10th International Conference on Computational Methods in Water Resources, Heidelberg, 19.-22. Juli 1994", title = "Sparse Grid and Extrapolation Methods for Parabolic Problems", year = "1994", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", URL = "file://www.tu-chemnitz.de/pub/Local/mathematik/Ruede/heidel94.ps.Z", } @InProceedings{Jung:1994:IEM, author = "M. Jung and U. R{\"u}de", editor = "T. Manteuffel", booktitle = "Preliminary Proceedings of the Colorado Conference on Iterative Methods, Breckenridge, Colorado, April 4-10, 1994", title = "Implicit extrapolation methods for multilevel finite element computations", year = "1994", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", URL = "file://www.tu-chemnitz.de/pub/Local/mathematik/Ruede/impl_extrapolation.ps.Z", } @Article{Ruede:1994:MAI, author = "U. R{\"u}de", title = "On the Multilevel Adaptive Iterative Method", journal = j-SIAM-J-SCI-STAT-COMP, volume = "15", year = "1994", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/r/ruede-ulrich.bib", note = "also available as TU-Bericht I-9216, and published in the Preliminary Proceedings of the 2nd Copper Mountain Conference on Iterative Methods, April 9--14, 1992, ed. T. Manteuffel, University of Colorado at Denver.", }