Abstract

This paper is devoted to developing finite-element methods for solving a two-dimensional boundary value problem for a singularly perturbed time-dependent convection–diffusion–reaction equation. The solution to this problem may vary rapidly in thin layers. As a result, spurious oscillations occur in the solution if the standard Galerkin method is used. In the multiscale finite-element methods, the original problem is split into the grid-scale and subgrid-scale problems, which allows capturing the problem features at a scale smaller than the element mesh size. In this study two methods are considered: the variational multiscale method with algebraic sub-scale approximation (VMM-ASA) and the residual-free bubbles (RFB) method. In the first method the subgrid-scale problem is simulated by the residual of the grid-scale equation and intrinsic time scales. In the second method the subgrid-scale problem is approximated by special functions. The grid-scale and subgrid-scale problems are formulated via the linearization procedure on the subgrid component applied to the original problem. The computer implementation of the methods was carried out using a commercial finite-element package. The efficiency of the developed methods is evaluated by solving a test boundary value problem for the nonlinear equation. Cases with different values of the diffusion coefficient have been analyzed. Based on the numerical investigation, it is shown that the multiscale methods enable improving the stability of the numerical solution and decreasing the quantity and the amplitude of oscillations compared to the standard Galerkin method. In the case of a small diffusion coefficient, the developed methods can yield a satisfactory numerical solution on a sufficiently coarse mesh.

中文翻译：

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非线性对流扩散反应方程的变分多尺度有限元方法

摘要

本文致力于发展有限元方法来求解二维奇异摄动对流-扩散-反应方程的边值问题。该问题的解决方案在薄层中可能会迅速变化。结果，如果使用标准的Galerkin方法，则溶液中会产生杂散振荡。在多尺度有限元方法中，将原始问题分为网格尺度问题和子网格尺度问题，从而可以以小于元素网格大小的尺度捕获问题特征。在这项研究中，考虑了两种方法：带代数子尺度近似的变分多尺度方法（VMM-ASA）和无残差气泡（RFB）方法。在第一种方法中，子网格规模问题是通过网格规模方程和固有时间尺度的残差来模拟的。在第二种方法中，子网格规模问题是通过特殊函数来近似的。网格规模和子网格规模的问题是通过对应用到原始问题的子网格组件上的线性化过程来表述的。该方法的计算机实现是使用商业有限元软件包进行的。通过解决非线性方程的测试边界值问题来评估所开发方法的效率。分析了具有不同扩散系数值的情况。根据数值研究，结果表明，与标准的Galerkin方法相比，多尺度方法可以提高数值解的稳定性，并减少振荡的数量和幅度。在扩散系数小的情况下，所开发的方法可以在足够粗糙的网格上产生令人满意的数值解。