Higher order finite element approximation of a quasilinear elliptic boundary value problem of a non-monotone type. (English) Zbl 0870.65096

The problem \(-\operatorname {div}(A(x,u) \operatorname {grad}u)=f\) in \(\Omega \), \(u=0\) on \(\partial \Omega \), in the weak formulation is examined in a polyhedral domain \(\Omega \subset \mathbb{R}^d\), \(d\in \{1,2,3\}\). Among others, the problem describes a stationary heat conduction in nonlinear inhomogeneous and anisotropic media.
The equation is approximated via the finite elements of degree \(k\geq 1\). Both the original and approximated problem have a solution, the reader is referred to the paper by I. Hlaváček, M. Křížek and J. Malý [J. Math. Anal. Appl. 184, No. 1, 168-189 (1994; Zbl 0802.65113)] for proofs. Employing the Aubin-Nitsche trick and assuming some regularity of the solution of an auxiliary adjoint problem, the authors prove the optimal rate of convergence \(\mathcal O(h^k)\) in the \(H^1\)-norm and \(\mathcal O(h^{k+1})\) in the \(L^2\)-norm. Numerical integration is not taken into account.
Reviewer: J.Chleboun (Praha)


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74A15 Thermodynamics in solid mechanics
35J65 Nonlinear boundary value problems for linear elliptic equations


Zbl 0802.65113
Full Text: EuDML


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