Weighted Poincaré inequalities

IMA Journal of Numerical Analysis

Volumn 33, Issue 2, 2013, Pages 652-686

  Pechstein, Clemens ^a   Scheichl, Robert ^b  

^a JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

^b University of Bath   (United Kingdom)

Author keywords

coefficient robustness; discrete Sobolev inequalities; geometric dependencies; Poincare Friedrichs inequalities; quasi monotone heterogeneous coefficients

Indexed keywords

ITERATIVE METHODS; NUMBER THEORY;

COEFFICIENT VALUES; CONDITION NUMBERS; GEOMETRIC DEPENDENCIES; MULTIGRID METHODS; POINCARE; QUASI-MONOTONE HETEROGENEOUS COEFFICIENTS; SECOND-ORDER ELLIPTIC PROBLEM; SOBOLEV INEQUALITIES;

DOMAIN DECOMPOSITION METHODS;

EID: 84876231083     PISSN: 02724979     EISSN: 14643642     Source Type: Journal    
DOI: 10.1093/imanum/drs017     Document Type: Article

References (37)

1. Bebendorf, M. (2003) A note on the Poincare inequality for convex domains. Z. Anal. Anwendungen, 22, 751-756.
2. Bernardi, C. & Verfurth, R. (2000) Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math., 85, 579-608.
3. Bramble, J. H. & Xu, J. (1991) Some estimates for a weighted L2 projection. Math. Comp., 56, 463-476.
4. Brenner, S. C. & Scott, L. R. (2002) The Mathematical Theory of Finite Element Methods, 2nd edn. Texts in Applied Mathematics 15. New York: Springer.
5. Chua, S.-K. (1993) Weighted Sobolev inequalities on domains satisfying the chain condition. Proc. Amer. Math. Soc., 117, 449-457.
6. Chua, S.-K. & Wheeden, R. L. (2006) Estimates of best constants for weighted Poincare inequalities on convex domains. Proc. London Math. Soc. (3), 93, 197-226.
7. Ciarlet, P. G. (2002) The Finite Element Method for Elliptic Problems. Philadelphia: SIAM.
8. Dohrmann, C. R., Klawonn, A. & Widlund, O. B. (2008) Domain decomposition for less regular subdomains: overlapping Schwarz in two dimensions. SIAM J. Numer. Anal., 46, 2153-2168.
9. Dryja, M., Sarkis, M. V. & Widlund, O. B. (1996) Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math., 72, 313-348.
10. Efendiev, Y. & Galvis, J. (2010) A domain decomposition preconditioner for multiscale high-contrast problems. Domain Decomposition Methods in Science and Engineering XIX (Y. Huang, R. Kornhuber, O. Widlund & J. Xu eds). Berlin: Springer, pp. 189-196.
11. Fabes, E. B., Kenig, C. E. & Serapioni, R. P. (1982) The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations, 7, 77-116.
12. Galvis, J. & Efendiev, Y. (2010a) Domain decomposition preconditioners for multiscale flows in high contrast media. Multiscale Model. Simul., 8, 1461-1483.
13. Galvis, J. & Efendiev, Y. (2010b) Domain decomposition preconditioners for multiscale flows in high contrast media: reduced dimension coarse spaces. Multiscale Model. Simul., 8, 1621-1644.
14. Graham, I. G., Lechner, P. & Scheichl, R. (2007) Domain decomposition for multiscale PDEs. Numer. Math., 106, 589-626.
15. Griebel, M., Scherer, K. & Schweitzer, M. A. (2007) Robust norm equivalencies for diffusion problems. Math. Comp., 76, 1141-1161.
16. Iwaniec, T. & Nolder, C. A. (1985) Hardy-Littlewood inequality for quasiregular mappings in certain domains in Rn. Ann. Acad. Sci. Fenn. Ser. A I Math., 10, 267-282.
17. Klawonn,A., Rheinbach, O. & Widlund, O. B. (2008) An analysis of a FETI-DP algorithm on irregular subdomains in the plane. SIAM J. Numer. Anal., 46, 2484-2504.
18. Klawonn,A. & Widlund, O. B. (2001) FETI and Neumann-Neumann iterative substructuring methods: connections and new results. Comm. Pure Appl. Math., 54, 57-90.
19. Knyazev, A. V. (2001) Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput., 23, 517-541.
20. Maz'ja, V. G. (1960) Classes of domains and imbedding theorems for functions spaces. Soviet Math. Dokl., 1, 882-885.
21. Maz'ja, V. G. (1985) Sobolev Spaces. Berlin: Springer
22. McLean,W. (2000) Strongly Elliptic Systems and Boundary Integral Equations. Cambridge, UK: Cambridge University Press.
23. Muckenhoupt, B. (1972) Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc., 165, 207-226.
24. Oswald, P. (1999) On the robustness of the BPX-preconditioner with respect to jumps in the coefficients. Math. Comp., 68, 633-650.
25. Payne, L. E. & Weinberger, H. F. (1960) An optimal Poincare inequality for convex domains. Arch. Ration. Mech. Anal., 5, 286-292.
26. Pechstein, C. & Scheichl, R. (2008) Analysis of FETI methods for multiscale PDEs. Numer. Math., 111, 293-333.
27. Pechstein, C. & Scheichl, R. (2009) Scaling up through domain decomposition. Appl. Anal., 88, 1589-1608.
28. Pechstein, C. & Scheichl, R. (2010) Weighted Poincare inequalities and applications in domain decomposition. Domain Decomposition Methods in Science and Engineering XIX (Y. Huang, R. Kornhuber, O. Widlund & J. Xu eds). Berlin: Springer, pp. 197-204.
29. Pechstein, C. & Scheichl, R. (2011) Analysis of FETI methods for multiscale PDEs-part II: interface variation. Numer. Math., 118, 485-529.
30. Sarkis, M. (1997) Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements. Numer. Math., 77, 383-406.
31. Scheichl, R. & Vainikko, E. (2007) Additive Schwarz and aggregation-based coarsening for elliptic problems with highly variable coefficients. Computing, 80, 319-343.
32. Scheichl, R., Vassilevski, P. S. & Zikatanov, L. (2012) Multilevel methods for elliptic problems with highly varying coefficients on non-aligned coarse grids. SIAM J. Numer. Anal., 50, 1675-1694.
33. Schenk, O. & Gartner, K. (2004) Solving unsymmetric sparse systems of linear equations with PARDISO. J. Future Gener. Comput. Syst., 20, 475-487.
34. Schenk, O. & Gartner, K. (2006) On fast factorization pivoting methods for symmetric indefinite systems. Electron. Trans. Numer. Anal., 23, 158-179.
35. Scott, L. R. & Zhang, S. (1990) Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp., 54, 483-493.
36. Toselli, A. & Widlund, O. (2005) Domain Decomposition Methods-Algorithms and Theory. Berlin: Springer.
37. Veeser, A. & Verfurth, R. (2009) Explicit upper bounds for dual norms of residuals. SIAM J. Numer. Anal., 47, 2387-2405.