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
-\div\left({\bf A}(\nabla u)\right)=f,
{\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}:
\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.

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, L. and R\r{u}\v{z}i\v{c}ka, M.}
\newblock Interpolation operators in {O}rlicz-{S}obolev spaces.
\newblock {\em Numer. Math.}, 107(1): 107--129, 2007.

{Chow, S.-S.}
\newblock Finite element error estimates for nonlinear elliptic
equations of monotone type.
\newblock {\em Numer. Math.}, 54(4): 373--393, 1989.

{Tyukhtin, V. B.}
\newblock The rate of convergence of approximation methods for solving one-sided variational problems. {I}.
\newblock {\em Teoret. Mat. Fiz.}, 51(2): 111--113, 1982.