FEM discretization and a-priori error estimates for power-law diffusion problems

Dr. Svetoslav Nakov

Oct. 8, 2019, 3:30 p.m. S2 054

We start by considering power-law diffusion problems of the form
\begin{align*}
-\div\left({\bf A}(\nabla u)\right)=f,
\end{align*}
where
\begin{align*}
{\bf A}(\nabla u):=\left(k+\abs{\nabla u}\right)^{p-2}\nabla u \, \text{ for some }\, 1\end{align*}
We recall the ideas in \cite{Diening_Ruzicka_2007} by introducing a so-called quasi-norm and the respective near-best approximation result. From here, it is not hard to derive as a byproduct a near-best approximation result in the natural $W^{1,p}$ norm which coincides with the error bounds derived in \cite{Chow_FE_Error_Estimates_for_NonLinear_Elliptic_Eq_of_Monotone_Type_1989,Tyukhtin_1982}:
\begin{align*}
\abs{u-u_h}_{W^{1,p}(\Omega)}&\lesssim \inf\limits_{v_h\in V_h}{\abs{u-v_h}_{W^{1,p}(\Omega)}^{\frac{p}{2}}} \,\text{ for } 1\abs{u-u_h}_{W^{1,p}(\Omega)}&\lesssim \inf\limits_{v_h\in V_h}{\abs{u-v_h}_{W^{1,p}(\Omega)}^{\frac{2}{p}}} \,\text{ for } p>2.
\end{align*}

We extend the ideas from the elliptic case to the corresponding parabolic problem in the context of a space-time finite element discretization and present some numerical results.



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{Tyukhtin, V. B.}
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