FEM for quasilinear obstacle problems in bad domains

Capitanelli, Raffaela; Vivaldi, Maria Agostina

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2465-2485.

We study obstacle problems involving the $p$-Laplace operator in domains with fractal boundary and the corresponding pre-fractals problems. We obtain error estimates for FEM solutions based on smoothness properties.

Y. Achdou and N. Tchou, Neumann conditions on fractal boundaries. Asymptot. Anal. 53 (2007) 61–82. | MR | Zbl

Y. Achdou, C. Sabot and N. Tchou, A multiscale numerical method for Poisson problems in some ramified domains with a fractal boundary. Multiscale Model. Simul. 5 (2006) 828–860. | DOI | MR | Zbl

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

P. Bagnerini, P. Buffa and E. Vacca, Mesh generation and numerical analysis of a Galerkin method for highly conductive prefractal layers. Appl. Numer. Math. 65 (2013) 63–78. | DOI | MR | Zbl

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). | Numdam | MR | Zbl

J.W. Barrett and W.B. Liu, 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. | Numdam | MR | Zbl

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

R. Capitanelli and M.A. Vivaldi, Uniform weighted estimates on pre-fractal domains. Discrete Contin. dyn. Syst. Ser. B 19 (2014) 1969–1985. | MR | Zbl

R. Capitanelli and M.A. Vivaldi, Weighted estimates on fractal domains. Mathematika 61 (2015) 370–384. | DOI | MR | Zbl

R. Capitanelli and M.A. Vivaldi, Reinforcement problems for variational inequalities on fractal sets. Calc. Var. Partial Differ. Equ. 54 (2015) 2751–2783. | DOI | MR | Zbl

R. Capitanelli and M.A. Vivaldi, Dynamical Quasi-Filling Fractal Layers. SIAM J. Math. Anal. 48 (2016) 3931–3961. | DOI | MR | Zbl

R. Capitanelli and M.A. Vivaldi, Regularity results for p-Laplacian in pre-fractal domains. In preparation (2017). | MR

M. Cefalo, M.R. Lancia and H. Liang, 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

M. Dobrowolski, On quasilinear elliptic equations in domains with conical boundary points. J. Reine Angew. Math. 394 (1989) 186–195. | MR | Zbl

C. Ebmeyer and W.B. Liu, Quasi-norm interpolation error estimates for the piecewise linear finite element approximation of p-Laplacian problems. Numer. Math. 100 (2005) 233–258. | DOI | MR | Zbl

C. Ebmeyer, W.B. Liu and N. Yan, A posteriori FE error control for p-Laplacian by gradient recovery in quasi-norm. Math. Comput. 75 (2006) 1599–1616. | DOI | MR | Zbl

R.S. Falk, Approximation of an elliptic boundary value problem with unilateral constraints. RAIRO Anal. Numér. 9 (1975) R-2, 5–12. | Numdam | MR | Zbl

D.S. Grebenkov, M. Filoche and B. Sapoval, Mathematical basis for a general theory of Laplacian transport towards irregular interfaces. Phys. Rev. E 73 (2006) 021103. | DOI | MR

J. Gwinner, Cea’s error estimate for strongly monotone variational inequalities. Appl. Anal. 45 (1992) 179–192. | DOI | MR | Zbl

I. Holopainen, 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

J.E. Hutchinson, Fractals and selfsimilarity. Indiana Univ. Math. J. 30 (1981) 713–747. | DOI | MR | Zbl

D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130 (1995) 161–219. | DOI | MR | Zbl

T. Kuusi and G. Mingione, Guide to nonlinear potential estimates. Bull. Math. Sci. 4 (2014) 1–82. | DOI | MR | Zbl

M.R. Lancia, M. Cefalo and G. Dell’Acqua, Numerical approximation of transmission problems across Koch-type highly conductive layers. Appl. Math. Comput. 218 (2012) 5453–5473. | MR | Zbl

M.R. Lancia and M. Cefalo, An optimal mesh generation algorithm for domains with Koch type boundaries. Math. Comput. Simul. 106 (2014) 133–162. | DOI | MR | Zbl

H. Lewy and G. Stampacchia, On the smoothness of superharmonics which solve a minimum problem. J. Anal. Math. 23 (1970) 227–236. | DOI | MR | Zbl

J.L. Lions and E. Magenes, Non Homogeneous Boundary Value Problems and ApplicationsNo. 1, 2. Springer Verlag, Berlin (1972). | MR | Zbl

W.B. Liu and J.W. Barrett, Quasi-norm error bounds for the finite element approximation of some degenerate quasilinear elliptic equations and variational inequalities. RAIRO: M2AN 28 (1994) 752–744. | Numdam | MR | Zbl

W.B. Liu and N. Yan, Quasi-norm local error estimators for p-Laplacian. SIAM J. Numer. Anal. 39 (2001) 100–127. | DOI | MR | Zbl

W.B. Liu and N. Yan, Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian. Numer. Math. 89 (2001) 341–378. | DOI | MR | Zbl

W.B. Liu and N. Yan, Some a posteriori error estimators for p-Laplacian based on residual estimation or gradient recovery. J. Sci. Comput. 16 (2002) 435–477. | DOI | MR | Zbl

W.B. Liu and N. Yan, On quasi-norm interpolation error estimation and a posteriori error estimates for p-Laplacian. SIAM J. Numer. Anal. 40 (2002) 1870-1895. | DOI | MR | Zbl

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

C. Mercuri, G. Riey and B. Sciunzi, A regularity result for the p-Laplacian near uniform ellipticity. SIAM J. Math. Anal. 48 (2016) 2059–2075. | DOI | MR | Zbl

U. Mosco and G.M. Troianiello, On the smoothness of solutions of unilateral Dirichlet problem. Boll. Un. Mat. Ital. 8 (1973) 57–67. | MR | Zbl

U. Mosco, Convergence of convex sets and of solutions of variational inequalities. Advances Math. 3 (1969) 510–585. | DOI | MR | Zbl

U. Mosco, Implicit Variational Problems and Quasi Variational Inequalities. Vol. 543 of Lect. Notes Math. Springer Verlag (1976). | MR | Zbl

U. Mosco, Gauged Sobolev Inequalities. Appl. Anal. 86 (2007) 367–402. | DOI | MR | Zbl

U. Mosco and M.A. Vivaldi, Layered fractal fibers and potentials. J. Math. Pures Appl. 103 (2015) 1198–1227. | DOI | MR | Zbl

K. Nyström, Smoothness properties of Dirichlet problems in domains with a fractal boundary. Ph.D. Dissertation, Umeä, (1994).

J. F. Rodrigues and R. Teymurazyan, On the two obstacles problem in Orlicz-Sobolev spaces and applications. Complex Var. Elliptic Equ. 56 (2011) 769–787. | DOI | MR | Zbl

G. Savaré, Regularity results for elliptic equations in Lipschitz domains. J. Funct. Anal. 152 (1998) 176–201. | DOI | MR | Zbl

V.A. Solonnikov and M.A. Vivaldi, Mixed type, nonlinear systems in polygonal domains. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 24 (2013) 39–81. | DOI | MR | Zbl

G.M. Troianiello, Elliptic partial differential equations and obstacle problems. The University Series in Mathematics. Plenum Press, New York (1987). | MR | Zbl

K. Uhlenbeck, Regularity for a class of non-linear elliptic systems. Acta. Math. 138 (1977) 219–240. | DOI | MR | Zbl