Simple Positivity Preserving Nonlinear Finite Volume Scheme for Subdiffusion Equations on General Non-conforming Distorted Meshes

We propose a positivity preserving finite volume scheme on non-conforming quadrilateral distorted meshes with hanging nodes for subdiffusion equations, where the differential equations have a sum of time-fractional derivatives of different orders, and the typical solutions of the problem have a weak singularity at the initial time t =0 for given smooth data. In this paper, a positivity-preserving nonlinear method with centered unknowns is obtained by the two point flux technique, where a new method to handling vertex-unknown including hanging nodes is the highlight of our paper. For each time derivative, we apply the L1 scheme on a temporal graded mesh. Especially, the existence of a solution is strictly proved for the nonlinear system by applying the Brouwer’s fixed point theorem. Numerical results show that the proposed positivity-preserving method is effective for strongly anisotropic and heterogeneous full tensor subdiffusion coefficient problems.

A simple virtual element-based flux recovery on quadtree

<p style='text-indent:20px;'>In this paper, we introduce a simple local flux recovery for <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{Q}_k $\end{document}</tex-math></inline-formula> finite element of a scalar coefficient diffusion equation on quadtree meshes, with no restriction on the irregularities of hanging nodes. The construction requires no specific ad hoc tweaking for hanging nodes on <inline-formula><tex-math id="M2">\begin{document}$ l $\end{document}</tex-math></inline-formula>-irregular (<inline-formula><tex-math id="M3">\begin{document}$ l\geq 2 $\end{document}</tex-math></inline-formula>) meshes thanks to the adoption of virtual element families. The rectangular elements with hanging nodes are treated as polygons as in the flux recovery context. An efficient <i>a posteriori</i> error estimator is then constructed based on the recovered flux, and its reliability is proved under common assumptions, both of which are further verified in numerics.</p>