A weighted least squares finite element method for elliptic problems with degenerate and singular coefficients
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- by S. Bidwell, M. E. Hassell and C. R. Westphal PDF
- Math. Comp. 82 (2013), 673-688 Request permission
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.References
- Gabriel Acosta, Thomas Apel, Ricardo G. Durán, and Ariel L. Lombardi, Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra, Math. Comp. 80 (2011), no. 273, 141–163. MR 2728975, DOI 10.1090/S0025-5718-2010-02406-8
- Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
- Daniel Arroyo, Alexei Bespalov, and Norbert Heuer, On the finite element method for elliptic problems with degenerate and singular coefficients, Math. Comp. 76 (2007), no. 258, 509–537. MR 2291826, DOI 10.1090/S0025-5718-06-01910-7
- Constantin Bacuta, Victor Nistor, and Ludmil T. 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, DOI 10.1080/01630560500377295
- Pavel B. Bochev and Max D. Gunzburger, Finite element methods of least-squares type, SIAM Rev. 40 (1998), no. 4, 789–837. MR 1659689, DOI 10.1137/S0036144597321156
- Daniele Boffi, Franco Brezzi, Leszek F. Demkowicz, Ricardo G. Durán, Richard S. Falk, and Michel Fortin, Mixed finite elements, compatibility conditions, and applications, Lecture Notes in Mathematics, vol. 1939, Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence, 2008. Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26–July 1, 2006; Edited by Boffi and Lucia Gastaldi. MR 2459075, DOI 10.1007/978-3-540-78319-0
- Dietrich Braess, Finite elements, 2nd ed., Cambridge University Press, Cambridge, 2001. Theory, fast solvers, and applications in solid mechanics; Translated from the 1992 German edition by Larry L. Schumaker. MR 1827293
- James H. Bramble and Joseph E. Pasciak, New estimates for multilevel algorithms including the $V$-cycle, Math. Comp. 60 (1993), no. 202, 447–471. MR 1176705, DOI 10.1090/S0025-5718-1993-1176705-9
- Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
- Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
- Z. Cai, R. Lazarov, T. A. Manteuffel, and S. F. McCormick, First-order system least squares for second-order partial differential equations. I, SIAM J. Numer. Anal. 31 (1994), no. 6, 1785–1799. MR 1302685, DOI 10.1137/0731091
- Zhiqiang Cai, Thomas A. Manteuffel, and Stephen F. McCormick, First-order system least squares for second-order partial differential equations. II, SIAM J. Numer. Anal. 34 (1997), no. 2, 425–454. MR 1442921, DOI 10.1137/S0036142994266066
- Z. Cai and C. R. Westphal, A weighted $H(\textrm {div})$ least-squares method for second-order elliptic problems, SIAM J. Numer. Anal. 46 (2008), no. 3, 1640–1651. MR 2391010, DOI 10.1137/070698531
- Pavel B. Bochev and Max D. Gunzburger, Least-squares finite element methods, Applied Mathematical Sciences, vol. 166, Springer, New York, 2009. MR 2490235, DOI 10.1007/b13382
- Dennis Jespersen, Ritz-Galerkin methods for singular boundary value problems, SIAM J. Numer. Anal. 15 (1978), no. 4, 813–834. MR 488786, DOI 10.1137/0715054
- Robert C. Kirby, From functional analysis to iterative methods, SIAM Rev. 52 (2010), no. 2, 269–293. MR 2646804, DOI 10.1137/070706914
- V. A. Kozlov, V. G. Maz′ya, and J. Rossmann, Elliptic boundary value problems in domains with point singularities, Mathematical Surveys and Monographs, vol. 52, American Mathematical Society, Providence, RI, 1997. MR 1469972, DOI 10.1090/surv/052
- L. D. Landau and E. M. Lifshitz, A shorter course of theoretical physics. Vol. 2, Pergamon Press, Oxford-New York-Toronto, Ont., 1974. Quantum mechanics; Translated from the Russian by J. B. Sykes and J. S. Bell. MR 0400931
- 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, DOI 10.1137/050636279
- E. Lee, T. A. Manteuffel, and C. R. Westphal, 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, DOI 10.1137/06067345X
- Hengguang Li, A-priori analysis and the finite element method for a class of degenerate elliptic equations, Math. Comp. 78 (2009), no. 266, 713–737. MR 2476557, DOI 10.1090/S0025-5718-08-02179-0
- Hengguang Li, Finite element analysis for the axisymmetric Laplace operator on polygonal domains, J. Comput. Appl. Math. 235 (2011), no. 17, 5155–5176. MR 2817318, DOI 10.1016/j.cam.2011.05.003
- A. Lunardi, G. Metafune, and D. Pallara, Dirichlet boundary conditions for elliptic operators with unbounded drift, Proc. Amer. Math. Soc. 133 (2005), no. 9, 2625–2635. MR 2146208, DOI 10.1090/S0002-9939-05-08068-8
- Giorgio Metafune, Diego Pallara, Jan Prüss, and Roland Schnaubelt, $L^p$-theory for elliptic operators on $\Bbb R^d$ with singular coefficients, Z. Anal. Anwendungen 24 (2005), no. 3, 497–521. MR 2208037, DOI 10.4171/ZAA/1253
- Patrick J. Rabier, Elliptic problems on $\Bbb R^N$ with unbounded coefficients in classical Sobolev spaces, Math. Z. 249 (2005), no. 1, 1–30. MR 2106968, DOI 10.1007/s00209-004-0686-4
- 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
- U. Trottenberg, C. W. Oosterlee, and A. Schüller, Multigrid, Academic Press, Inc., San Diego, CA, 2001. With contributions by A. Brandt, P. Oswald and K. Stüben. MR 1807961
- H. Wu and D.W.L. Sprung, Inverse-square potential and the quantum vortex, Phy. Rev. A 49 (1994), no. 6, 4305–11.
Additional Information
- S. Bidwell
- Affiliation: Department of Mathematics, Tufts University, Medford, Massachusetts 02155
- Email: bidwell.stephen@gmail.com
- M. E. Hassell
- Affiliation: Department of Mathematics, Binghamton University, Binghamton, New York 13902-6000
- Email: hassell.matthew@gmail.com
- C. R. Westphal
- Affiliation: Department of Mathematics and Computer Science, Wabash College, P.O. Box 352, Crawfordsville, Indiana 47933
- Email: westphac@wabash.edu
- 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.
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 82 (2013), 673-688
- MSC (2010): Primary 65N30, 65N15, 35J70
- DOI: https://doi.org/10.1090/S0025-5718-2012-02659-7
- MathSciNet review: 3008834