Virtual element method for the system of time dependent nonlinear convection-diffusion-reaction equation

TL;DR

This paper develops a virtual element method combined with Euler time-stepping for nonlinear convection-diffusion-reaction equations, proving solution existence, deriving error estimates, and comparing iterative and two-grid solution approaches.

Contribution

It introduces a novel virtual element discretization for nonlinear time-dependent PDEs, with proven stability, convergence, and efficient solution strategies.

Findings

Optimal order of convergence in $H^1$ semi-norm.

Two-grid method outperforms simple iteration.

Numerical results confirm theoretical error estimates.

Abstract

In this work, we have discretized a system of time-dependent nonlinear convection-diffusion-reaction equations with the virtual element method over the spatial domain and the Euler method for the temporal interval. For the nonlinear fully-discrete scheme, we prove the existence and uniqueness of the solution with Brouwer's fixed point theorem. To overcome the complexity of solving a nonlinear discrete system, we define an equivalent linear system of equations. A priori error estimate showing optimal order of convergence with respect to $H^1$ semi-norm was derived. Further, to solve the discrete system of equations, we propose an iteration method and a two-grid method. In the numerical section, the experimental results validate our theoretical estimates and point out the better performance of the two-grid method over the iteration method.