Abstract
In the present paper we develop a representation of a norm frequently used in the analysis of multilevel methods. This allows us to examine the convergence of additive Schwarz schemes also in the case of non-nested subspaces. We demonstrate the usefulness of the given norm representation by studying in detail the stability of sparse grid splittings due to Griebel and Oswald, which turns out to be a special case of our unified theory. Further applications concerning approximation spaces and non-nested finite element spaces are given.
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References
R. Balder and C. Zenger, The solution of multidimensional real Helmholtz equations on sparse grids, SIAM J. Sci. Comput. 17 (1996) 631–646.
P.E. Bjørstad and J. Mandel, On the spectra of sums of orthogonal projections with applications to parallel computing, BIT 31 (1991) 76–88.
F.A. Bornemann and H. Yserentant, A basic norm equivalence for the theory of multilevel methods, Numer. Math. 64 (1993) 455–476.
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer, Berlin, 1994).
H.-J. Bungartz, Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung, Dissertation, TU München (1992).
A. Cohen, I. Daubechies and J. Feauveau, Bi-orthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992) 485–560.
A. Cohen, I. Daubechis and P. Vial, Wavelets on the interval and fast wavelet transforms, Appl. Comput. Harmon. Anal. 1(1) (1993) 54–81.
W. Dahmen and A. Kunoth, Multilevel preconditioning, Numer. Math. 63(2) (1992) 315–344.
I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988) 909–996.
R.A. DeVore and G.G. Lorentz, Constructive Approximation, Grundlehren der Mathematischen Wissenschaften 303 (Springer, Berlin, 1993).
T. Dornseifer and C. Pflaum, Elliptic differential equations on curvilinear bounded domains with sparse grids, Computing 56 (1996) 197–213.
M. Griebel and P. Oswald, On additive Schwarz preconditioners for sparse grid discretizations, Numer. Math. 66 (1994) 449–463.
M. Griebel and P. Oswald, On the abstract theory of additive and multiplicative Schwarz algorithms, Numer. Math. 70 (1995) 163–180.
M. Griebel and P. Oswald, Tensor product type subspace splittings and multilevel iterative methods for anistropic problems, Adv. Comput. Math. 4 (1995) 171–206.
P. Oswald, On function spaces related to finite element approximation theory, Z. Anal. Anwendungen 9 (1990) 43–64.
P. Oswald, Stable splittings of Sobolev spaces and fast solution of variational problems, Forschungsergebnisse Math/92/5, FSU Jena (1992).
P. Oswald, Multilevel Finite Element Approximation (Teubner, 1994).
S.A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR 148 (1963) 1042–1045 (English translation Soviet Math. Dokl. 4 240–243.)
V.N. Temlyakov, Approximation of functions with bounded mixed derivative, Trudy Math. Inst. Steklov 178 (1986). (English translation Proc. Steklov Inst. Math. 1 (1989).)
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (North-Holland, Amsterdam, 1978).
C. Zenger, Sparse grids, in: Parallel Algorithms for PDE, Proceedings 6th GAMM Seminar, Kiel, ed. W. Hackbusch (Vieweg, Braunschweig, 1991) pp. 241–251.
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Nießen, G. An explicit norm representation for the analysis of multilevel methods. Advances in Computational Mathematics 9, 311–335 (1998). https://doi.org/10.1023/A:1018949809444
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DOI: https://doi.org/10.1023/A:1018949809444