A finite element approximation for a class of degenerate elliptic equations

Franchi, Bruno; Tesi, Maria

doi:10.1090/S0025-5718-99-01075-3

A finite element approximation for a class of degenerate elliptic equations
HTML articles powered by AMS MathViewer

by Bruno Franchi and Maria Carla Tesi;

Math. Comp. 69 (2000), 41-63

DOI: https://doi.org/10.1090/S0025-5718-99-01075-3

Published electronically: February 19, 1999

PDF | Request permission

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.

References

Randolph E. Bank, Todd F. Dupont, and Harry Yserentant, The hierarchical basis multigrid method, Numer. Math. 52 (1988), no. 4, 427–458. MR 932709, DOI 10.1007/BF01462238
Luca Capogna, Donatella Danielli, and Nicola Garofalo, The geometric Sobolev embedding for vector fields and the isoperimetric inequality, Comm. Anal. Geom. 2 (1994), no. 2, 203–215. MR 1312686, DOI 10.4310/CAG.1994.v2.n2.a2
Luca Capogna, Donatella Danielli, and Nicola Garofalo, Subelliptic mollifiers and a basic pointwise estimate of Poincaré type, Math. Z. 226 (1997), no. 1, 147–154. MR 1472145, DOI 10.1007/PL00004330
Claudy Cancelier and Bruno Franchi, Subelliptic estimates for a class of degenerate elliptic integro-differential operators, Math. Nachr. 183 (1997), 19–41. MR 1434973, DOI 10.1002/mana.19971830103
Ennio De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43 (Italian). MR 93649
Bruno Franchi, Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations, Trans. Amer. Math. Soc. 327 (1991), no. 1, 125–158. MR 1040042, DOI 10.1090/S0002-9947-1991-1040042-8
Albert Eagle, Series for all the roots of a trinomial equation, Amer. Math. Monthly 46 (1939), 422–425. MR 5, DOI 10.2307/2303036
B. Franchi, S. Gallot, and R. L. Wheeden, Sobolev and isoperimetric inequalities for degenerate metrics, Math. Ann. 300 (1994), no. 4, 557–571. MR 1314734, DOI 10.1007/BF01450501
Bruno Franchi, Cristian E. Gutiérrez, and Richard L. Wheeden, Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. Partial Differential Equations 19 (1994), no. 3-4, 523–604. MR 1265808, DOI 10.1080/03605309408821025
Bruno Franchi, Cristian E. Gutiérrez, and Richard L. Wheeden, Two-weight Sobolev-Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 5 (1994), no. 2, 167–175 (English, with English and Italian summaries). MR 1292572
Bruno Franchi and Ermanno Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 4, 523–541. MR 753153
B. Franchi, G. Lu, and R. L. Wheeden, Representation formulas and weighted Poincaré inequalities for Hörmander vector fields, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 577–604 (English, with English and French summaries). MR 1343563, DOI 10.5802/aif.1466
C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 590–606. MR 730094
B. Franchi and R. Serapioni, Pointwise estimates for a class of strongly degenerate elliptic operators: a geometrical approach, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 4, 527–568 (1988). MR 963489
Bruno Franchi, Raul Serapioni, and Francesco Serra Cassano, Champs de vecteurs, théorème d’approximation de Meyers-Serrin et phénomène de Lavrent′ev pour des fonctionnelles dégénérées, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 6, 695–698 (French, with English and French summaries). MR 1323940
Bruno Franchi and Richard L. Wheeden, Compensation couples and isoperimetric estimates for vector fields, Colloq. Math. 74 (1997), no. 1, 9–27. MR 1455453, DOI 10.4064/cm-74-1-9-27
Pierre Grisvard, Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York-London, 1976, pp. 207–274. MR 466912
M. Gromov, Carnot-Carathéodory spaces seen from within, Sub–Riemannian Geometry, Birkhäuser, 1996, pp. 79–323.
Nicola Garofalo and Duy-Minh Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), no. 10, 1081–1144. MR 1404326, DOI 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 473443
Velimir Jurdjevic, Geometric control theory, Cambridge Studies in Advanced Mathematics, vol. 52, Cambridge University Press, Cambridge, 1997. MR 1425878
Jürgen Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457–468. MR 170091, DOI 10.1002/cpa.3160130308
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.
William F. Mitchell, Optimal multilevel iterative methods for adaptive grids, SIAM J. Sci. Statist. Comput. 13 (1992), no. 1, 146–167. MR 1145181, DOI 10.1137/0913009
C. Mogavero and S. Polidoro, A finite difference method for a boundary value problem related to the Kolmogorov equation, Calcolo 32 (1995), 193–206.
Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539
J. Stoer and R. Bulirsch, Introduction to numerical analysis, 2nd ed., Texts in Applied Mathematics, vol. 12, Springer-Verlag, New York, 1993. Translated from the German by R. Bartels, W. Gautschi and C. Witzgall. MR 1295246, DOI 10.1007/978-1-4757-2272-7
Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247–320. MR 436223, DOI 10.1007/BF02392419
Chao Jiang Xu, On Harnack’s inequality for second-order degenerate elliptic operators, Chinese Ann. Math. Ser. A 10 (1989), no. 3, 359–365 (Chinese). MR 1024922