Two-grid mixed virtual element method for a nonlinear fourth-order time-fractional reaction-diffusion equation

Abstract

In this article, we developed a two-grid mixed virtual element method (MVEM) for solving a time-dependent fourth-order reaction-diffusion equation with a Riemann-Liouville fractional derivative and a nonlinear reaction term. By introducing an auxiliary variable σ = ∆u, we reformulated the problem into a coupled system of second-order equations. Then we discretized the equations at t_k−α by a second-order backward differentiation formula (BDF2) with a fractional parameter α. The method employed a two-grid strategy: First, we solved a nonlinear problem via Newton iterations on a coarse ploygonal grid, and subsequently, leveraging the coarse-grid solutions, we solved a linearized problem on a fine virtual element space. We established the stability of this method and derived a priori L²-norm error estimates with the optimal convergence result O(∆t² + h^k+1 + H^2k+2), where ∆t denotes the time step size, h and H are the fine and coarse spatial mesh sizes, respectively, and k is the polynomial degree of the virtual element space. Numerical experiments validated the theoretical convergence rates, and it was demonstrated that applying the two-grid MVEM is computationally more efficient than solving the nonlinear system directly on the fine grid.

Keywords

two-grid mixed virtual element method / fourth-order time-fractional reaction-diffusion equation / a priori error analysis in L²-norm / unstructured meshes

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