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

## Dr. Svetoslav Nakov

Oct. 8, 2019, 1:30 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 + |{\nabla u}| \right )^{p-2}\nabla u , \text{ for some }, 1 \lt p \lt \infty , \text{ and some }, k\ge 0.
\end{align*}

We recall the ideas in [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 [Chow_FE_Error_Estimates_for_NonLinear_Elliptic_Eq_of_Monotone_Type_1989,Tyukhtin_1982]:

\begin{align*}
|{u-u_h}_{W^{1,p}(\Omega)}| & \lesssim \inf\limits_{v_h \in V_h}{|{u-v_h}_{W^{1,p}(\Omega)}^{\frac{p}{2}}|} , \text{ for } 1 \lt p \lt 2,\\
|{u-u_h}_{W^{1,p}(\Omega)}| &\lesssim \inf\limits_{v_h \in V_h}{|{u-v_h}_{W^{1,p}(\Omega)}^{\frac{2}{p}}|} , \text{ for } p \gt 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.

[Diening_Ruzicka_2007]
Diening, L. and Ruzicka, M., "Interpolation operators in {O}rlicz-{S}obolev spaces.", Numer. Math.}, 107(1): 107--129, 2007.

[Chow_FE_Error_Estimates_for_NonLinear_Elliptic_Eq_of_Monotone_Type_1989]
Chow, S.-S., "Finite element error estimates for nonlinear elliptic equations of monotone type.", Numer. Math., 54(4): 373--393, 1989.

[Tyukhtin_1982]
Tyukhtin, V. B., "The rate of convergence of approximation methods for solving one-sided variational problems.", Teoret. Mat. Fiz., 51(2): 111--113, 1982.