Virtual element method for the parabolic $p$-Laplacian equation

We propose a virtual element method (VEM) for a nonlinear parabolic equation, specically the $p$-Laplacian equation. We analyze a $W^{1,p}(Ω)$-conforming discretization by means of a VEM suited for general polytopal meshes. We prove that the methods are well-posed by using a novel stabilization term for the nonlinear discrete operator. We obtain space-time error estimates in the $L^\infty(0, T ; L^2(Ω))$ and $L^2(0, T ; W^{1,p}(Ω))$ norms. Finally, we consider two linearization strategies depending on p. More precisely, we use a fixed-point iteration for $p \in (1, 2]$, and we use the standard Newton iteration for $p \in (2, ∞)$. Next, numerical experiments are provided to validate the theoretical results.

Joint work with Verónica Anaya, Mostafa Bendahmane „and David Mora.