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.
翻译:在这项工作中,我们已经将一个基于时间的、非线性对流-扩散-反应方程式系统与空间域的虚拟元素法和时间间隔的 Euler 法分离。对于非线性全分解方案,我们用布鲁韦尔的固定点理论来证明解决办法的存在和独特性。为了克服解决非线性离散系统的复杂性,我们定义了等效的线性方程式系统。我们得出了一个先验错误估计,显示在$H$1$半诺姆方面最佳趋同顺序。此外,为了解决离散方程式系统,我们提出了一种迭代法和二格方法。在数字部分,实验结果证实了我们的理论估计,并指明了二格方法相对于迭代法的更好性能。