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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
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