Finite Element Approximation of the Parabolic p -Laplacian
John W. Barrett, W. B. Liu · SIAM Journal on Numerical Analysis · 1994
In this paper the authors consider the continuous piecewise linear finite element approximation in space of the following problem: Given $p \in (1,\infty )$, f and $u_0 $; find u such that \[ \quad \begin{gathered} u_t = abla \cdot \left( {| abla u|^{p - 2} abla u} \right) + f\ {\text{in}}\ \Omega \times ( {0,T} ], \hfill \\ u = 0\quad {\text{on}}\quad \partial \Omega \times ( {0,T} ], \hfill \\ u(x,0) = u_0 (x)\forall x \in \Omega , \hfill \\ \end{gathered} \] where $\Omega \subset {\bf R}^d ,d = 1{\text{ or }}$. The authors analyse the semidiscrete approximation and a fully discrete approximation using the backward Euler time discretisation, obtaining error bounds which improve on those in the literature.