We study obstacle problems involving the -Laplace operator in domains with fractal boundary and the corresponding pre-fractals problems. We obtain error estimates for FEM solutions based on smoothness properties.
Accepté le :
DOI : 10.1051/m2an/2017033
Keywords: Fractals, degenerate elliptic equations, FEM
Capitanelli, Raffaela 1 ; Vivaldi, Maria Agostina 1
@article{M2AN_2017__51_6_2465_0,
author = {Capitanelli, Raffaela and Vivaldi, Maria Agostina},
title = {FEM for quasilinear obstacle problems in bad domains},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2465--2485},
year = {2017},
publisher = {EDP Sciences},
volume = {51},
number = {6},
doi = {10.1051/m2an/2017033},
mrnumber = {3745178},
zbl = {1381.28007},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2017033/}
}
TY - JOUR AU - Capitanelli, Raffaela AU - Vivaldi, Maria Agostina TI - FEM for quasilinear obstacle problems in bad domains JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 2465 EP - 2485 VL - 51 IS - 6 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2017033/ DO - 10.1051/m2an/2017033 LA - en ID - M2AN_2017__51_6_2465_0 ER -
%0 Journal Article %A Capitanelli, Raffaela %A Vivaldi, Maria Agostina %T FEM for quasilinear obstacle problems in bad domains %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 2465-2485 %V 51 %N 6 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2017033/ %R 10.1051/m2an/2017033 %G en %F M2AN_2017__51_6_2465_0
Capitanelli, Raffaela; Vivaldi, Maria Agostina. FEM for quasilinear obstacle problems in bad domains. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2465-2485. doi: 10.1051/m2an/2017033
and , Neumann conditions on fractal boundaries. Asymptot. Anal. 53 (2007) 61–82. | MR | Zbl
, and , A multiscale numerical method for Poisson problems in some ramified domains with a fractal boundary. Multiscale Model. Simul. 5 (2006) 828–860. | MR | Zbl | DOI
R.A. Adams, Sobolev spaces. Pure and Applied Mathematics, Vol. 65. Academic Press, New York London (1975). | MR | Zbl
H. Attouch, Variational Convergence for Functions and Operators. Pitman Advanced Publishing Program, London (1984). | MR | Zbl
, and , Mesh generation and numerical analysis of a Galerkin method for highly conductive prefractal layers. Appl. Numer. Math. 65 (2013) 63–78. | MR | Zbl | DOI
M.T. Barlow and B.M. Hambly, Transition density estimates for Brownian motion on scale irregular Sierpinski gaskets. Ann. Inst. Henri Poincaré Probab. Statist. 33(1997). | MR | Zbl | Numdam
and , Finite element approximation of the p-Laplacian. Math. Comput. 61 (1993) 523–537. | MR | Zbl
J. Bergh and J. Löfstrom, Interpolation Spaces. An Introduction. Springer Verlag, Berlin (1976). | MR | Zbl
M. Borsuk and V. Kondratiev, Elliptic Boundary value problems of Second Order in Piecewise Smooth Domains. Noth-Holland Mathematical Library Eds. (2006). | MR | Zbl
L. Brasco and F. Santambrogio, A sharp estimate à la Calderón−Zygmund for the p-Laplacian. To appear in Commun. Contemp. Math. (2017) | DOI | MR
F. Brezzi and G. Gilardi, FEM Mathematics, in Finite Element Handbook. edited by H. Kardestuncer, D.H. Norrie. McGraw-Hill Book Co., New York (1987). | MR | Zbl
J. Céa, Approximation variationnelle des problèmes aux limites. Vol. 4 of Studies in Mathematics and its Applications. Ann. Inst. Fourier 14 (1964) 345–444. | MR | Zbl | Numdam
P.G. Ciarlet, The finite element method for elliptic problems. Vol. 4 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam New York Oxford (1978). | MR | Zbl
and , Uniform weighted estimates on pre-fractal domains. Discrete Contin. dyn. Syst. Ser. B 19 (2014) 1969–1985. | MR | Zbl
and , Weighted estimates on fractal domains. Mathematika 61 (2015) 370–384. | MR | Zbl | DOI
and , Reinforcement problems for variational inequalities on fractal sets. Calc. Var. Partial Differ. Equ. 54 (2015) 2751–2783. | MR | Zbl | DOI
and , Dynamical Quasi-Filling Fractal Layers. SIAM J. Math. Anal. 48 (2016) 3931–3961. | MR | Zbl | DOI
R. Capitanelli and M.A. Vivaldi, Regularity results for p-Laplacian in pre-fractal domains. In preparation (2017). | MR
, and , Heat-flow problems across fractal mixtures: regularity results of the solutions and numerical approximation. Differ. Integral Equ. 26 (2013) 1027–1054. | MR | Zbl
J.I. Diaz, Nonlinear partial differential equations and free boundaries. I. Elliptic equations. Vol. 106 of Res. Notes Math. Pitman, Boston, MA (1985). | MR | Zbl
, On quasilinear elliptic equations in domains with conical boundary points. J. Reine Angew. Math. 394 (1989) 186–195. | MR | Zbl
and , Quasi-norm interpolation error estimates for the piecewise linear finite element approximation of p-Laplacian problems. Numer. Math. 100 (2005) 233–258. | MR | Zbl | DOI
, and , A posteriori FE error control for p-Laplacian by gradient recovery in quasi-norm. Math. Comput. 75 (2006) 1599–1616. | MR | Zbl | DOI
, Approximation of an elliptic boundary value problem with unilateral constraints. RAIRO Anal. Numér. 9 (1975) R-2, 5–12. | MR | Zbl | Numdam
, and , Mathematical basis for a general theory of Laplacian transport towards irregular interfaces. Phys. Rev. E 73 (2006) 021103. | MR | DOI
, Cea’s error estimate for strongly monotone variational inequalities. Appl. Anal. 45 (1992) 179–192. | MR | Zbl | DOI
, Quasiregular mappings and the p -Laplace operator. Heat kernels and analysis on manifolds, graphs, and metric spaces. Contemp. Math. Amer. Math. Soc. 338 (2003) 219–239. | MR | Zbl
, Fractals and selfsimilarity. Indiana Univ. Math. J. 30 (1981) 713–747. | MR | Zbl | DOI
and , The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130 (1995) 161–219. | MR | Zbl | DOI
and , Guide to nonlinear potential estimates. Bull. Math. Sci. 4 (2014) 1–82. | MR | Zbl | DOI
, and , Numerical approximation of transmission problems across Koch-type highly conductive layers. Appl. Math. Comput. 218 (2012) 5453–5473. | MR | Zbl
and , An optimal mesh generation algorithm for domains with Koch type boundaries. Math. Comput. Simul. 106 (2014) 133–162. | MR | Zbl | DOI
and , On the smoothness of superharmonics which solve a minimum problem. J. Anal. Math. 23 (1970) 227–236. | MR | Zbl | DOI
J.L. Lions and E. Magenes, Non Homogeneous Boundary Value Problems and ApplicationsNo. 1, 2. Springer Verlag, Berlin (1972). | MR | Zbl
and , Quasi-norm error bounds for the finite element approximation of some degenerate quasilinear elliptic equations and variational inequalities. RAIRO: M2AN 28 (1994) 752–744. | MR | Zbl | Numdam
and , Quasi-norm local error estimators for p-Laplacian. SIAM J. Numer. Anal. 39 (2001) 100–127. | MR | Zbl | DOI
and , Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian. Numer. Math. 89 (2001) 341–378. | MR | Zbl | DOI
and , Some a posteriori error estimators for p-Laplacian based on residual estimation or gradient recovery. J. Sci. Comput. 16 (2002) 435–477. | MR | Zbl | DOI
and , On quasi-norm interpolation error estimation and a posteriori error estimates for p-Laplacian. SIAM J. Numer. Anal. 40 (2002) 1870-1895. | MR | Zbl | DOI
A. Lunardi, Interpolation theory. 2nd edition. Lecture Notes. Scuola Normale Superiore di Pisa (New Series) Edizioni della Normale, Pisa (2009). | MR
V. G. Maz’ya and S.V. Poborchi, Differentiable functions on bad domains. World Scientific Publishing Co., Inc., River Edge, NJ (1997). | MR | Zbl
, and , A regularity result for the p-Laplacian near uniform ellipticity. SIAM J. Math. Anal. 48 (2016) 2059–2075. | MR | Zbl | DOI
and , On the smoothness of solutions of unilateral Dirichlet problem. Boll. Un. Mat. Ital. 8 (1973) 57–67. | MR | Zbl
, Convergence of convex sets and of solutions of variational inequalities. Advances Math. 3 (1969) 510–585. | MR | Zbl | DOI
U. Mosco, Implicit Variational Problems and Quasi Variational Inequalities. Vol. 543 of Lect. Notes Math. Springer Verlag (1976). | MR | Zbl
, Gauged Sobolev Inequalities. Appl. Anal. 86 (2007) 367–402. | MR | Zbl | DOI
and , Layered fractal fibers and potentials. J. Math. Pures Appl. 103 (2015) 1198–1227. | MR | Zbl | DOI
K. Nyström, Smoothness properties of Dirichlet problems in domains with a fractal boundary. Ph.D. Dissertation, Umeä, (1994).
and , On the two obstacles problem in Orlicz-Sobolev spaces and applications. Complex Var. Elliptic Equ. 56 (2011) 769–787. | MR | Zbl | DOI
, Regularity results for elliptic equations in Lipschitz domains. J. Funct. Anal. 152 (1998) 176–201. | MR | Zbl | DOI
and , Mixed type, nonlinear systems in polygonal domains. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 24 (2013) 39–81. | MR | Zbl | DOI
G.M. Troianiello, Elliptic partial differential equations and obstacle problems. The University Series in Mathematics. Plenum Press, New York (1987). | MR | Zbl
, Regularity for a class of non-linear elliptic systems. Acta. Math. 138 (1977) 219–240. | MR | Zbl | DOI
Cité par Sources :
