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A finite element approximation for a class of
degenerate elliptic equations

Authors: Bruno Franchi and Maria Carla Tesi
Journal: Math. Comp. 69 (2000), 41-63
MSC (1991): Primary 46E30, 49N60
Published electronically: February 19, 1999
MathSciNet review: 1642821
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we exhibit a finite element method fitting a suitable geometry naturally associated with a class of degenerate elliptic equations (usually called Grushin type equations) in a plane region, and we discuss the related error estimates.

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  • [BDY] R.E. Bank, T.F. Dupont and H. Yserentant, The hierarchical basis multigrid method, Num. Math. 52 (1988), 427-458. MR 89b:65247
  • [CDG1] L. Capogna, D. Danielli and N. Garofalo, The geometric Sobolev embedding for vector fields and the isoperimetric inequality, Comm. in Analysis and Geometry 2 (1994), 203-215. MR 96d:46032
  • [CDG2] -, Subelliptic mollifiers and a basic pointwise estimate of Poincaré type, Math. Z. 226 (1997), 147-154. MR 98i:35025
  • [CF] C. Cancelier and B. Franchi, Subelliptic estimates for a class of degenerate elliptic integro-differential operators, Math. Nachr. 183 (1997), 19-41. MR 98d:35086
  • [DG] E. De Giorgi, Sulla differenziabilità e l'analiticità degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 3 (1957), 25-43. MR 20:172
  • [F] B. Franchi, Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations, Trans. Amer. Math. Soc. 327 (1991), 125-158. MR 91m:35095
  • [Fr] K.O. Friedrichs, The identity of weak and strong estension of differential operators, Trans. Amer. Math. Soc. 55 (1944), 132-151. MR 5:188b
  • [FGaW] B. Franchi, S. Gallot and R.L. Wheeden, Sobolev and isoperimetric inequalities for degenerate metrics, Math. Ann. 300 (1994), 557-571. MR 96a:46066
  • [FGuW1] B. Franchi, C. Gutierrez and R.L. Wheeden, Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. Partial Differential Equations 19 (1994), 523-604. MR 96h:26019
  • [FGuW2] B. Franchi, C. Gutierrez and R.L. Wheeden, Two-weight Sobolev-Poincaré inequalities and Harnak inequality for a class of degenerate elliptic operators, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur 5 (9) (1994), 167-175. MR 95i:35115
  • [FL] B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa 10 (4) (1983), 523-541. MR 85k:35094
  • [FLW] B. Franchi, G. Lu and R.L. Wheeden, Representation formulas and weighted Poincaré inequalities for Hörmander vector fields, Ann. Inst. Fourier, Grenoble 45 (1995), 577-604. MR 96i:46037
  • [FP] C. Fefferman and D.H. Phong, Subelliptic eigenvalue problems, Conference on Harmonic Analysis, Chicago, 1980, (W. Beckner et al., eds.), Wadsworth, 1981, pp. 590-606. MR 86c:35112
  • [FS] B. Franchi and R. Serapioni, Pointwise estimates for a class of strongly degenerate elliptic operators: a geometrical approach, Ann. Scuola Norm. Sup. Pisa 14 (4) (1987), 527-568. MR 90e:35076
  • [FSSC] B. Franchi, R. Serapioni and F. Serra Cassano, Champs de vecteurs, théorème d'approximation de Meyers-Serrin et phénomème de Lavrentev pour des fonctionnelles dégénérés, C.R. Acad. Sci. Paris Sér. I Math. 320 (1995), 695-698. MR 95m:46044
  • [FW] B. Franchi and R.L. Wheeden, Compensation couples and isoperimetric estimates for vector fields, Colloq. Math 74 (1997), 9-27. MR 98g:46042
  • [G] P. Grisvard, Behaviour of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, Numerical Solutions of Partial Differential Equations, III (B. Hubbard, ed.), Academic Press, New York, 1976, pp. 207-274. MR 57:6786
  • [Gr] M. Gromov, Carnot-Carathéodory spaces seen from within, Sub-Riemannian Geometry, Birkhäuser, 1996, pp. 79-323. CMP 97:04
  • [GN] N. Garofalo and D.M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), 1081-1144. MR 97i:58032
  • [GT] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1977. MR 57:13109
  • [J] V. Jurdjevic, Geometric Control Theory, Cambridge Studies in Advanced Mathematics, 1997. MR 98a:93002
  • [Mo] J. Moser, A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1961), 457-468. MR 30:332
  • [M1] W.F. Mitchell, Unified multilevel adaptive finite element method for elliptic problems, Ph.D. thesis, Report No. UIUCDSC-R-88-1436, Department of Computer Science, University of Illinois, Urbana, IL, 1988.
  • [M2] W.F. Mitchell, Optimal multilevel iterative methods for adaptive grids, SIAM J. Sci. Stat. Comput. 13 (1992), 146-167. MR 92j:65187
  • [MP] C. Mogavero and S. Polidoro, A finite difference method for a boundary value problem related to the Kolmogorov equation, Calcolo 32 (1995), 193-206. CMP 98:04
  • [NSW] A. Nagel, E.M. Stein and S. Wainger, Balls and metrics defined by vector fields I: basic properties, Acta Math. 155 (1985), 103-147. MR 86k:46049
  • [QV] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, Springer, Berlin, 1994. MR 95i:65006
  • [RS] L.P. Rothschild and E.M.Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247-320. MR 55:9171
  • [X] C.-J. Xu, The Harnack's inequality for second order degenerate elliptic operators, Chinese Ann. Math. Ser A 10 (1989), 359-365, in Chinese. MR 90m:35039

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Additional Information

Bruno Franchi
Affiliation: Dipartimento Matematico dell’Università, Piazza di Porta S. Donato, 5, 40127 Bologna, Italy

Maria Carla Tesi
Affiliation: Université de Paris-Sud, Mathématiques, Bât. 425, 91405 Orsay Cedex, France

Received by editor(s): June 28, 1996
Received by editor(s) in revised form: September 8, 1997, and March 31, 1998
Published electronically: February 19, 1999
Additional Notes: The first author is partially supported by M.U.R.S.T., Italy (40%) and by G.N.A.F.A. of C.N.R., Italy (60%).
The authors are indebted to A. Valli for many fruitful discussions.
Article copyright: © Copyright 1999 American Mathematical Society

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