Linear convergence of an adaptive finite element method for the \(p\)-Laplacian equation.

*(English)*Zbl 1168.65060The nonlinear Dirichlet problem with the so called \(\varphi\) -Laplacian operator of the form
\[
-\text{div} ({\mathbf A}(\nabla u )) =f \;\;\text{in } \Omega \;\;\;u=0 \;\;\text{ on }\partial \Omega,
\]
with
\[
{\mathbf A} (\nabla u) =\varphi^\prime(|\nabla u|)\frac{\nabla u}{|\nabla u|}
\]
where \(\varphi \) is called \(N\) -function is studied.

The weak formulation of the \(\varphi\) Laplacian problem and corresponding minimizing problem is derived. Then the finite element discretization with piecewise linear continuous functions, and an adaptive method for refining the mesh is constructed. The convergence analysis is presented for this method . For measuring errors the quasi norm of J. W. Barrett and W. B. Liu [Math. Comput. 61, No. 204, 523–537 (1993; Zbl 0791.65084)] is used. An essential tool in calculations is using of so-called shifted \(N\) -functions which provide to handle with more complex problem than \(p\)-Laplacian and moreover they simplify and clarify the calculations for \(p\)-Laplacian problems too. Residual based error estimators without a gap between the upper and lower bound are proposed. Results are obtained without extra marking for the oscillation. The linear convergence of the algorithm similar to the one derived by P. Morin, R. H. Nochetto and K. G. Siebert [SIAM J. Numer. Anal. 38, No. 2 466–488 (2000; Zbl 0970.65113); SIAM Rev. 44, No. 4, 631–658 (2002; Zbl 1016.65074)] is proved.

The weak formulation of the \(\varphi\) Laplacian problem and corresponding minimizing problem is derived. Then the finite element discretization with piecewise linear continuous functions, and an adaptive method for refining the mesh is constructed. The convergence analysis is presented for this method . For measuring errors the quasi norm of J. W. Barrett and W. B. Liu [Math. Comput. 61, No. 204, 523–537 (1993; Zbl 0791.65084)] is used. An essential tool in calculations is using of so-called shifted \(N\) -functions which provide to handle with more complex problem than \(p\)-Laplacian and moreover they simplify and clarify the calculations for \(p\)-Laplacian problems too. Residual based error estimators without a gap between the upper and lower bound are proposed. Results are obtained without extra marking for the oscillation. The linear convergence of the algorithm similar to the one derived by P. Morin, R. H. Nochetto and K. G. Siebert [SIAM J. Numer. Anal. 38, No. 2 466–488 (2000; Zbl 0970.65113); SIAM Rev. 44, No. 4, 631–658 (2002; Zbl 1016.65074)] is proved.

Reviewer: Angela Handlovičová (Bratislava)

##### MSC:

65N15 | Error bounds for boundary value problems involving PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

35J60 | Nonlinear elliptic equations |

35J70 | Degenerate elliptic equations |