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Error estimates for the numerical approximation of a quaslinear Neumann problem under minimal regularity of the data. (English) Zbl 1209.65129

The paper is concerned with the finite element approximation of the following quasilinear Neumann problem
\[ \begin{cases} -\text{div}[a(x,u(x))\nabla u(x)]+f(x,u(x))=0\quad &\text{in}\quad \Omega, \\ a(x,u(x))\nabla u(x)\cdot v(x)=g(x)\quad&\text{on}\qquad \Gamma. \end{cases} \]
\(\Omega\) is a polygonal or polyhedral domain. Suppose now that we have got a family of triangulations \(\{\mathcal{T}_h\}_{h>0}\) of \(\overline{\Omega}.\) With each element \(T\in \mathcal{T}_h\) associate two parameters \(\rho (T); \delta(T)\), where \(\rho \) denotes the diameter and \(\delta\) the diameter of the largest ball contained in \(T\). Define the size of the mesh by \(h=\max_{T\in \mathcal{T}_h}\rho(T).\) The triangulation is supposed to be regular, if there are \(\rho >0,\delta(T)>0\) such that:
\[ \frac{\rho (T)}{\delta(T)}\leq\delta \quad \frac{h}{\rho(T)}\leq\rho, \] and associate \[ V_{h}:=\{u_{h}\in\;C(\overline{\Omega})\Bigr|\quad u_{h}\big|_{T}\in\mathcal{P}_k,\quad T\in\mathcal{T}_h\}. \]
Here \(\mathcal{P}_k\) stands for the polynomials of degree at most \(k\in\mathbb{N}.\) The discrete version of the quasilinear equation is given by:
\[ \begin{cases} \text{find} \quad u_{h}\in V_{h}\quad \text{such that, for all} \quad z_{h}\in V_{h},\\ \underset\Omega\int\{a(x,u_{h})\nabla\;u_{h}(x)\cdot\bigtriangledown z_{h}(x)+f(x,u_{h})z_{h}(x)\}dx= \underset\Gamma\int gz_{h}(x)d\sigma(x). \end{cases}. \]
Under some smoothness and regularity conditions conditions the authors have for a convex domain and \(n=2\) the following result: There exists \(h_{0}>0\) such that for any \(h<h_{0}\) the discrete equation has at least one solution that obeys
\[ \parallel u-u_{h}\parallel _{L^{2}(\Omega)}+h^{1/2}\parallel u-u_{h}\parallel _{H^{1}(\Omega)}\leq \varepsilon_{h}h. \]
Here \(\varepsilon_{h}\rightarrow 0\) when \(h\rightarrow 0.\) This result can be extended to nonconvex domains and to the case \(n=3\) too.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J62 Quasilinear elliptic equations
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