Abstract
We introduce an iterative method for computing the first eigenpair (λ p ,e p ) for the p-Laplacian operator with homogeneous Dirichlet data as the limit of (μ q, u q ) as q→p −, where u q is the positive solution of the sublinear Lane-Emden equation \(-\Delta_{p}u_{q}=\mu_{q}u_{q}^{q-1}\) with the same boundary data. The method is shown to work for any smooth, bounded domain. Solutions to the Lane-Emden problem are obtained through inverse iteration of a super-solution which is derived from the solution to the torsional creep problem. Convergence of u q to e p is in the C 1-norm and the rate of convergence of μ q to λ p is at least O(p−q). Numerical evidence is presented.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adimurthi, R., Yadava, S.L.: An elementary proof of the uniqueness of positive radial solutions of a quasilinear Dirichlet problem. Arch. Ration. Mech. Anal. 127, 219–229 (1994)
Ainsworth, M., Kay, D.: The approximation theory for the p-version finite element method and application to non-linear elliptic PDEs. Numer. Math. 82(3), 351–388 (1999)
Allegretto, W., Huang, Y.X.: A Picone’s identity for the p-Laplacian and applications. Nonlinear Anal. 32, 819–830 (1998)
Andreianov, B., Boyer, F., Hubert, F.: On the finite-volume approximation of regular solutions of the p-Laplacian. IMA J. Numer. Anal. 26(3), 472–502 (2006)
Antontsev, S.N., Díaz, J.I., de Oliveira, H.B.: Mathematical models in dynamics of non-Newtonian fluids and in glaciology. In: Proceedings of the CMNE/CILAMCE Congress. Universidade do Porto, Porto (2007), 20 pp.
Atkinson, C., Champion, C.R.: Some boundary value problems for the equation ∇⋅(|∇ϕ|N). Q. J. Mech. Appl. Math. 37, 401–419 (1984)
Azorero, J.G., Alonso, J.P.: On limits of solutions of elliptic problems with nearly critical exponent. Commun. Partial Differ. Equ. 17, 2113–2126 (1992)
Barrett, J.W., Liu, W.B.: Finite element approximation of the p-Laplacian. Math. Comput. 61(204), 523–537 (1993)
Bermejo, R., Infante, J.A.: A multigrid algorithm for the p-Laplacian. SIAM J. Sci. Comput. 21(5), 1774–1789 (2000)
Biezuner, R.J., Ercole, G., Martins, E.M.: Computing the first eigenvalue of the p-Laplacian via the inverse power method. J. Funct. Anal. 257, 243–270 (2009)
Biezuner, R.J., Ercole, G., Martins, E.M.: Computing the sin p function via the inverse power method. Comput. Methods Appl. Math. 11(2), 129–140 (2011)
Biezuner, R.J., Ercole, G., Martins, E.M.: Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift (submitted)
Blacker, T., Bohnhoff, W., Edwards, T., Hipp, J., Lober, R., Mitchell, S., Sjaardema, G., Tautges, T., Wilson, T., Oakes, W. et al.: CUBIT mesh generation environment. Technical Report, Sandia National Labs., Albuquerque, NM. Cubit Development Team (1994)
Bognár, G., Szabó, T.: Solving nonlinear eigenvalue problems by using p-version of FEM. Comput. Math. Appl. 43, 57–68 (2003)
Bognár, G.: Estimation on the first eigenvalue for some nonlinear Dirichlet eigenvalue problems. Nonlinear Anal. 71(12), e2242–e2448 (2009)
Bognár, G., Rontó, M.: Numerical-analytic investigation of the radially symmetric solutions for some nonlinear PDEs. Comput. Math. Appl. 50, 983–991 (2005)
Bueno, H., Ercole, G., Zumpano, A.: Positive solutions for the p-Laplacian and bounds for its first eigenvalue. Adv. Nonlinear Stud. 9, 313–338 (2009)
Brown, J.: Efficient nonlinear solvers for nodal high-order finite elements in 3D. J. Sci. Comput. 45(1), 48–63 (2010)
Damascelli, L.: Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Ann. Inst. Henry Poincaré 15, 493–516 (1998)
Descloux, J., Tolley, M.: An accurate algorithm for computing the eigenvalues of a polygonal membrane. Comput. Methods Appl. Mech. Eng. 39(1), 37–53 (1983)
Diaz, J.I., Hernandez, J.: On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in climatology. J. Math. Anal. Appl. 216, 593–613 (1997)
Diaz, J.I., de Thelin, F.: On a nonlinear parabolic problem arising in some models related to turbulent flows. SIAM J. Math. Anal. 25(4), 1085–1111 (1994)
Diening, L., Kreuzer, C.: Linear convergence of an adaptative finite element method for the p-Laplacian equation. SIAM J. Numer. Anal. 46(2), 614–638 (2008)
Drábek, P.: The uniqueness for a superlinear eigenvalue problem. Appl. Math. Lett. 12, 47–50 (1999)
Droniou, J.: Finite volume schemes for fully non-linear elliptic equations in divergence form. Modél. Math. Anal. Numér. 40(6), 1069–1100 (2006)
Glowinski, R., Rappaz, J.: Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid model in glaciology. Modél. Math. Anal. Numér. 37(1), 175–186 (2003)
Guidotti, P., Lambers, J.V.: Eigenvalue characterization and computation for the Laplacian on general 2-D domains. Numer. Funct. Anal. Optim. 29(5–6), 507–531 (2008)
Guan, M., Zheng, L.: The similarity solution to a generalized diffusion equation with convection. Adv. Dyn. Syst. Appl. 1(2), 183–189 (2006)
Heuveline, V.: On the computation of a very large number of eigenvalues for selfadjoint elliptic operators by means of multigrid methods. J. Comput. Phys. 184, 321–337 (2003)
Huang, Y.Q., Li, R., Liu, W.: Preconditioned descent algorithms for p-Laplacian. J. Sci. Comput. 32(2), 343–371 (2007)
Huang, Y.X.: A note on the asymptotic behavior of positive solutions for some elliptic equation. Nonlinear Anal. TMA 29, 533–537 (1997)
Juutine, J., Lindqvist, P., Manfredi, J.: The ∞-eigenvalue problem. Arch. Ration. Mech. Anal. 148, 89–105 (1999)
Kawohl, B.: On a family of torsional creep problems. J. Reine Angew. Math. 410, 1–22 (1990)
Kawohl, B., Fridman, V.: Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant. Comment. Math. Univ. Carol. 44, 659–667 (2003)
Kuttler, J.R., Sigillito, V.G.: Eigenvalues of the Laplacian in two dimensions. SIAM Rev. 26(2), 163–193 (1984)
Lefton, L., Wei, D.: Numerical approximation of the first eigenpair of the p-Laplacian using finite elements and the penalty method. Numer. Funct. Anal. Optim. 18(3–4), 389–399 (1997)
Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. TMA 12, 1203–1219 (1988)
Lindqvist, P.: Some remarkable sine and cosine functions. Ric. Mat. 2, 269–290 (1995)
Pélissier, M.-C., Reynaud, M.L.: Etude d’un modèle mathématique d’écoulement de glacier. C. R. Acad. Sci. Paris Ser. I Math. 279, 531–534 (1974)
Philip, J.R.: N-diffusion. Aust. J. Phys. 14, 1–13 (1961)
Sakaguchi, S.: Concavity properties of solutions to some degenerated quasilinear elliptic Dirichlet problems. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 14, 403–421 (1987)
Balay, S., Brown, J., Buschelman, K., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., Curfman McInnes, L., Smith, B.F., Zhang, H.: PETSc Users Manual, Technical Report ANL-95/11—Revision 3.1, Argonne National Laboratory (2010)
Showalter, R.E., Walkington, N.J.: Diffusion of fluid in a fissured medium with microstructure. SIAM J. Math. Anal. 22, 1702–1722 (1991)
Veeser, A.: Convergent adaptive finite elements for the nonlinear Laplacian. Numer. Math. 92(4), 743–770 (2002)
Yao, X., Zhou, J.: Numerical methods for computing nonlinear eigenpairs. I. Iso-homogeneous cases. SIAM J. Sci. Comput. 29(4), 1355–1374 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Biezuner, R.J., Brown, J., Ercole, G. et al. Computing the First Eigenpair of the p-Laplacian via Inverse Iteration of Sublinear Supersolutions. J Sci Comput 52, 180–201 (2012). https://doi.org/10.1007/s10915-011-9540-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-011-9540-0