Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Finite element approximation of the $p$-Laplacian
HTML articles powered by AMS MathViewer

by John W. Barrett and W. B. Liu PDF
Math. Comp. 61 (1993), 523-537 Request permission

Abstract:

In this paper we consider the continuous piecewise linear finite element approximation of the following problem: Given $p \in (1,\infty )$, f, and g, find u such that \[ - \nabla \cdot (|\nabla u{|^{p - 2}}\nabla u) = f\quad {\text {in}}\;\Omega \subset {\mathbb {R}^2},\quad u = g\quad {\text {on}}\;\partial \Omega .\] The finite element approximation is defined over ${\Omega ^h}$, a union of regular triangles, yielding a polygonal approximation to $\Omega$. For sufficiently regular solutions u, achievable for a subclass of data f, g, and $\Omega$, we prove optimal error bounds for this approximation in the norm ${W^{1,q}}({\Omega ^h}),q = p$ for $p < 2$ and $q \in [1,2]$ for $p > 2$, under the additional assumption that ${\Omega ^h} \subseteq \Omega$. Numerical results demonstrating these bounds are also presented.
References
  • C. Atkinson and C. R. Champion, Some boundary-value problems for the equation $\nabla \cdot (\mid \nabla \varphi \mid ^{N}\nabla \varphi )=0$, Quart. J. Mech. Appl. Math. 37 (1984), no. 3, 401–419. MR 760209, DOI 10.1093/qjmam/37.3.401
  • C. Atkinson and C. W. Jones, Similarity solutions in some non-linear diffusion problems and in boundary-layer flow of a pseudo plastic fluid, Quart. J. Mech. Appl. Math. 27 (1974), 193-211.
  • S.-S. Chow, Finite element error estimates for nonlinear elliptic equations of monotone type, Numer. Math. 54 (1989), no. 4, 373–393. MR 972416, DOI 10.1007/BF01396320
  • Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
  • R. Glowinski and A. Marrocco, Sur l’approximation, par Ă©lĂ©ments finis d’ordre un, et la rĂ©solution, par pĂ©nalisation-dualitĂ©, d’une classe de problĂšmes de Dirichlet non linĂ©aires, Rev. Française Automat. Informat. Recherche OpĂ©rationnelle SĂ©r. Rouge Anal. NumĂ©r. 9 (1975), no. R-2, 41–76 (French, with English summary). MR 388811, DOI 10.1051/m2an/197509R200411
  • A. Kufner, O. John, and S. Fučik, Function spaces, Noordhoff, Leyden, 1977.
  • Gary M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), no. 11, 1203–1219. MR 969499, DOI 10.1016/0362-546X(88)90053-3
  • J. R. Philip, $n$-diffusion, Austral. J. Phys. 14 (1961), 1–13. MR 140343, DOI 10.1071/PH610001
  • V. B. Tyukhtin, The rate of convergence of approximation methods for solving one-sided variational problems. I, Teoret. Mat. Fiz. 51 (1982), no. 2, 111–113, 121 (Russian, with English summary). MR 672607
  • Dongming Wei, Finite element approximations of solutions to $p$-harmonic equation with Dirichlet data, Numer. Funct. Anal. Optim. 10 (1989), no. 11-12, 1235–1251 (1990). MR 1050712, DOI 10.1080/01630568908816355
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC: 65N30
  • Retrieve articles in all journals with MSC: 65N30
Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Math. Comp. 61 (1993), 523-537
  • MSC: Primary 65N30
  • DOI: https://doi.org/10.1090/S0025-5718-1993-1192966-4
  • MathSciNet review: 1192966