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Konvergenzraten finiter Elemente für die Poröse-Medien-Gleichung im $$\mathbb{R}^ n$$. (Convergence rate of finite elements for the porous-media equation in $$\mathbb{R}^ n$$.) (German) Zbl 0889.76035
Bonner Mathematische Schriften. 287. Bonn: Univ. Bonn, Mathematisch-Naturwissenschaftliche Fakultät, 69 S. (1996).
The author investigates the initial-boundary value problem for degenerate nonlinear parabolic equation which describes gas flows in porous media in the so-called “slow diffusion approximation”: $$u_t= \Delta (u|u|^{m-1})$$ in $$\Omega \times (0,T]$$, $$u(x,t)=0$$ on $$\partial\Omega \times(0,T]$$, $$u(x,0)=u_0(x)$$ for $$x\in\Omega$$, where $$\Omega \subset \mathbb{R}^n$$, $$n\geq 2$$, and $$m>1$$; it is assumed that $$u_0\geq 0$$ and $$\text{supp} u_0 \Subset\Omega$$. First, the author shows that the Lebesgue measure $$|\Omega_0|$$ of the set $$\Omega_0: =\{(x,t)\in \Omega\times [0,T]:0< |u|< \varepsilon\}$$ satisfies the inequality $$|\Omega_0 |<\text{const} \cdot \varepsilon^{m-1 \over 2}$$. Then he defines finite element approximations $$U_{h,\tau}$$ of the solution $$u$$, with mesh size $$h$$ in $$x$$ and piecewise linear in $$x$$, and with mesh size $$\tau$$ in $$t$$ and piecewise constant in $$t$$, and, by using the above inequality, establishes the main estimate $$(\int^T_0|u-U_{h,\tau} |_{L^2(\Omega)} dt)^{1/2} \leq \text{const} \cdot h^{\alpha (m)}$$. A special feature of this estimate is that the exponent $$\alpha(m)$$ does not become arbitrarily small for $$m\to\infty$$, as in earlier works (more precisely, always $$\alpha(m) \geq{1\over 5})$$. In some particular cases, this estimate can be essentially improved. The analysis is based on the Green operator technique and duality arguments.
Reviewer: O.Titow (Berlin)

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76S05 Flows in porous media; filtration; seepage 76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics 35K65 Degenerate parabolic equations