## 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)

### MSC:

 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
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### References:

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