Abstract
For saddle point problems stemming from appending essential boundary conditions in connection with Galerkin methods for elliptic boundary value problems, a class of multilevel preconditioners is developed. The estimates are based on the characterization of Sobolev spaces on the underlying domain and its boundary in terms of weighted sequence norms relative to corresponding multilevel expansions. The results indicate how the various ingredients of a typical multilevel framework affect the growth rate of the condition numbers. In particular, it is shown how to realize even condition numbers that are uniformly bounded independently of the discretization.
These investigations are motivated by the idea of employing nested refinable shift-invariant spaces as trial spaces covering various types of wavelets that are of advantage for the solution of boundary value problems from other points of view. Instead of incorporating the boundary conditions into the approximation spaces in the Galerkin formulation, they are appended by means of Lagrange multipliers leading to a saddle point problem.
Similar content being viewed by others
References
R.A. Adams,Sobolev Spaces (Academic Press, 1978).
R.E. Bank, B.D. Welfert and H. Yserentant, A class of iterative methods for solving saddle point problems, Numer. Math. 56 (1990) 645–666.
D. Braess,Finite Elemente (Springer, 1992).
J.H. Bramble, The Lagrange multiplier method for Dirichlet's problem, Math. Comp. 37 (1981) 1–11.
J.H. Bramble and J.E. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximations for elliptic problems, Math. Comp. 50 (1988) 1–17.
J.H. Bramble and J.E. Pasciak, A preconditioned algorithm for eigenvector/eigenvalue computation, Manuscript (June 1992).
J.H. Bramble, J.E. Pasciak and J. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990) 1–22.
F. Brezzi and M. Fortin,Mixed and Hybrid Finite Element Methods (Springer, 1991).
J.M. Carnicer, W. Dahmen and J.M. Peña, Local decomposition of refinable spaces, Manuscript (1994).
A. Cohen, W. Dahmen and R.A. DeVore, private communication.
S. Dahlke and A. Kunoth, Biorthogonal wavelets and multigrid, in:Adaptive Methods — Algorithms, Theory and Applications, eds. W. Hackbusch and G. Wittum, NNFM Series Vol. 46 (Vieweg, 1994) pp. 99–119.
M. Dæhlen and T. Lyche, Box splines and applications, in:Geometric Modelling: Methods and Applications, eds. H. Hagen and D. Roller (Springer, 1991) pp. 35–94.
W. Dahmen, Decomposition of refinable spaces and applications to operator equations, in:Algorithms for Approximation III, eds. M.G. Cox and J.C. Mason (Baltzer, Amsterdam, 1993).
W. Dahmen, Stability of multiscale transformations, Manuscript (1994).
W. Dahmen and A. Kunoth, Multilevel preconditioning, Numer. Math. 63 (1992) 315–344.
W. Dahmen and C.A. Micchelli, Using the refinement equation for evaluating integrals of wavelets, SIAM J. Numer. Anal. 30 (1993) 507–537.
W. Dahmen, S. Prößdorf and R. Schneider, Multiscale methods for pseudodifferential equations, in:Recent Advances in Wavelet Analysis, eds. L.L. Schumaker and G. Webb (Academic Press, 1994) pp. 191–235.
I. Daubechies,Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Maths. (SIAM, 1992).
R.A. DeVore, B. Jawerth and V.A. Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992) 737–785.
R.A. DeVore and B. Lucier, Wavelets, Acta Numerica 1 (1991) 1–56.
R.A. DeVore and V.A. Popov, Interpolation of Besov spaces, Trans. Amer. Math. Soc. 305 (1988) 397–414.
A. Jonsson and H. Wallin,Function Spaces on Subsets of ℝn (Harwood Academic Publishers, Mathematical Reports, Vol. 2, 1984).
A. Kunoth,Multilevel Preconditioning (Verlag Shaker, Aachen, Germany, 1994).
A. Kunoth, Computing Integrals of Refinable Functions — Documentation of the Program, Version 1.0, Techn. Rep. STF33A94045, SINTEF, Oslo, Norway (September 1994).
J.L. Lions and E. Magenes,Non-Homogeneous Boundary Value Problems and Applications, Vol. I (Springer, 1972).
T. Lyche and K. Mørken, Spline-wavelets of minimal support, in:Numerical Methods in Approximation Theory, vol. 9, eds. D. Braess and L.L. Schumaker (Birkhäuser, 1992) pp. 177–194.
S.M. Nikolskii,Approximation of Functions of Several Variables and Imbedding Theorems (Nauka, Moscow, 2nd ed., 1977).
P. Oswald, On function spaces related to finite element approximation theory, Z. Anal. Anwendungen 9 (1990) 43–64.
P. Oswald, Hierarchical conforming finite element methods for the biharmonic equation, SIAM J. Numer. Anal. 29 (1992) 1610–1625.
P. Oswald, On discrete norm estimates related to multilevel preconditioners in the finite element method, in:Constructive Theory of Functions, eds. K.G. Ivanov, P. Petrushev and B. Sendov (Bulg. Acad. Sci., Sofia, 1992) pp. 203–214.
P. Oswald, Stable splittings of Sobolev spaces and fast solution of variational problems, Preprint Nr. Math/92/5, Friedrich-Schiller-Universität Jena (May 1992).
S. Riemenschneider and Z. Shen, Wavelets and pre-wavelets in low dimensions, J. Approx. Theory 71 (1992) 18–38.
H. Triebel,Interpolation Theory, Function Spaces, Differential Operators (Dt. Verl. Wiss., Berlin, 1978).
R. Verfürth, A combined conjugate gradient-multigrid algorithm for the numerical solution of the Stokes problem, IMA J. Numer. Anal. 4 (1984) 441–455.
J. Xu, Theory of multilevel methods, Report AM 48, Department of Mathematics, Pennsylvania State University (1989).
H. Yserentant, On the multilevel splitting of finite element spaces, Numer. Math. 49 (1986) 379–412.
H. Yserentant, Two preconditioners based on the multilevel splitting of finite element spaces, Numer. Math. 58 (1990) 163–184.
Author information
Authors and Affiliations
Additional information
The work of the author is partially supported by the Deutsche Forschungsgemeinschaft under grant numbers Ku1028/1-1 and Pr336/4-1.
Rights and permissions
About this article
Cite this article
Kunoth, A. Multilevel preconditioning — Appending boundary conditions by Lagrange multipliers. Adv Comput Math 4, 145–170 (1995). https://doi.org/10.1007/BF02123477
Issue Date:
DOI: https://doi.org/10.1007/BF02123477
Keywords
- Galerkin methods
- condition numbers
- multilevel representations
- saddle point problems
- Lagrange multipliers
- Sobolev spaces
- refinable shift-invariant spaces
- wavelets
- boundary conditions