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A weighted least squares finite element method for elliptic problems with degenerate and singular coefficients

Authors: S. Bidwell, M. E. Hassell and C. R. Westphal
Journal: Math. Comp. 82 (2013), 673-688
MSC (2010): Primary 65N30, 65N15, 35J70
Published electronically: December 4, 2012
MathSciNet review: 3008834
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Abstract: We consider second order elliptic partial differential equations with coefficients that are singular or degenerate at an interior point of the domain. This paper presents formulation and analysis of a novel weighted-norm least squares finite element method for this class of problems. We propose a weighting scheme that eliminates the pollution effect and recovers optimal convergence rates. Theoretical results are carried out in appropriately weighted Sobolev spaces and include ellipticity bounds on the weighted homogeneous least squares functional, regularity bounds on the elliptic operator, and error estimates. Numerical experiments confirm the predicted error bounds.

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  • 1. G. Acosta, T. Apel, R.G. Durán, and A.L. Lombardi, Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra, Math. Comp. 80 (2011), No. 273, 141-163. MR 2728975 (2011m:65262)
  • 2. R. Adams and J. Fournier, Sobolev spaces, second ed., Academic Press, 2003. MR 2424078 (2009e:46025)
  • 3. D. Arroyo, A. Bespalov, and N. Heuer, On the finite element method for elliptic problems with degenerate and singular coefficients, Math. Comp. 76 (2007), no. 258, 509-537. MR 2291826 (2008e:65336)
  • 4. C. Bacuta, V. Nistor, and L. Zikatanov, Improving the rate of convergence of high-order finite elements on polyhedra i: a priori estimates, Numer. Funct. Anal. Optim. 26 (2005), no. 6, 613-639. MR 2187917 (2006i:35036)
  • 5. P.B. Bochev and M.D. Gunzburger, Finite element methods of least-squares type, SIAM Rev. 40 (1998), 789-837. MR 1659689 (99k:65104)
  • 6. D. Boffi, F. Brezzi, L.F. Demkowicz, R.G. Duran, R.S. Falk, and M. Fortin, Mixed finite elements, compatibility conditions, and applications, Lecture Notes in Mathematics, vol. 1939, Springer-Verlag, 2008, Edited by D. Boffi and L. Gastaldi. MR 2459075 (2010h:65219)
  • 7. D. Braess, Finite elements: Theory, fast solvers and applications in solid mechanics, Cambridge, 2001. MR 1827293 (2001k:65002)
  • 8. J. Bramble and J. Pasciak, New estimates for multilevel algorithms including the V-cycle, Math. Comp. 60 (1993), 447-471. MR 1176705 (94a:65064)
  • 9. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, Springer-Verlag, 1994. MR 1278258 (95f:65001)
  • 10. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer-Verlag, 1991. MR 1115205 (92d:65187)
  • 11. Z. Cai, R. Lazarov, T.A. Manteuffel, and S.F. McCormick, First-order system least squares for second-order partial differential equations: Part I, SIAM J. Numer. Anal. 31 (1994), no. 6, 1785-1799. MR 1302685 (95i:65133)
  • 12. Z. Cai, T.A. Manteuffel, and S.F. McCormick, First-order system least squares for second-order partial differential equations: Part II, SIAM J. Numer. Anal. 34 (1997), no. 2, 425-454. MR 1442921 (98m:65039)
  • 13. Z. Cai and C.R. Westphal, A weighted H(div) least-squares method for second-order elliptic problems, SIAM J. Numer. Anal. 46 (2008), no. 3, 1640-1651. MR 2391010 (2008m:65314)
  • 14. M.D. Gunzburger and P.B. Bochev, Least-squares finite element methods, Springer, 2009. MR 2490235 (2010b:65004)
  • 15. Dennis Jespersen, Ritz-Galerkin methods for singular boundary value problems, SIAM J. Numer. Anal. 15 (1978), no. 4, 813-834. MR 0488786 (58:8296)
  • 16. R.C. Kirby, From functional analysis to iterative methods, SIAM Review 52 (2010), no. 2, 269-293. MR 2646804 (2011j:65279)
  • 17. V.A. Kozlov, V.G. Maz'ya, and J. Rossmann, Elliptic boundary vaule problems in domains with point singularities, vol. 52, American Mathematical Society, 1997. MR 1469972 (98f:35038)
  • 18. L.D. Landau and E.M. Lifshitz, Quantum mechanics: Non-relativistic theory, Pergamon Press, 1958. MR 0400931 (53:4761)
  • 19. E. Lee, T.A. Manteuffel, and C.R. Westphal, Weighted-norm first-order system least squares (FOSLS) for problems with corner singularities, SIAM J. Numer. Anal. 44 (2006), no. 5, 1974-1996. MR 2263037 (2008a:65221)
  • 20. -, Weighted-norm first-order system least squares (FOSLS) for div/curl systems with three dimensional edge singularities, SIAM J. Numer. Anal. 46 (2008), no. 3, 1619-1639. MR 2391009 (2009c:65316)
  • 21. H. Li, A-priori analysis and the finite element method for a class of degenerate elliptic equations, Math. Comp. 78 (2009), no. 266, 713-37. MR 2476557 (2010b:35163)
  • 22. -, Finite element analysis for the axisymmetric laplace operator on polygonal domains, J. Comput. Appl. Math. 235 (2011), 5155-76. MR 2817318
  • 23. A. Lunardi, G. Metafune, and D. Pallara, Dirichlet boundary conditions for elliptic operators with unbounded drift, Journal: Proc. Amer. Math. Soc. (2005), no. 133, 2625-2635. MR 2146208 (2006e:35140)
  • 24. G. Metafune, D. Pallara, J. Pruss, and R. Schnaubelt, $ {L}^p$-theory for elliptic operators on $ {R}^d$ with singular coefficients, Z. Anal. Anwendungen 24 (2005), no. 3, 497-521. MR 2208037 (2006k:35163)
  • 25. P.J. Rabier, Elliptic problems on $ R^N$ with unbounded coefficients in classical Sobolev spaces, Math. Z. 249 (2005), no. 1, 1-30. MR 2106968 (2005h:35054)
  • 26. J. Ruge and K. Stüben, Efficient solution of finite difference and finite element equations, Multigrid methods for integral and differential equations (Bristol, 1983), Inst. Math. Appl. Conf. Ser. New Ser., vol. 3, Oxford Univ. Press, New York, 1985, pp. 169-212. MR 849374 (87i:65047)
  • 27. U. Trottenberg, C. Oosterlee, and A. Schüller, Multigrid, Academic Press, 2001. MR 1807961 (2002b:65002)
  • 28. H. Wu and D.W.L. Sprung, Inverse-square potential and the quantum vortex, Phy. Rev. A 49 (1994), no. 6, 4305-11.

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Additional Information

S. Bidwell
Affiliation: Department of Mathematics, Tufts University, Medford, Massachusetts 02155

M. E. Hassell
Affiliation: Department of Mathematics, Binghamton University, Binghamton, New York 13902-6000

C. R. Westphal
Affiliation: Department of Mathematics and Computer Science, Wabash College, P.O. Box 352, Crawfordsville, Indiana 47933

Received by editor(s): August 31, 2010
Received by editor(s) in revised form: May 27, 2011, and October 5, 2011
Published electronically: December 4, 2012
Additional Notes: The research in this paper was supported by National Science Foundation Grant DMS-0755260.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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