Variational multiscale finite element methods for a nonlinear convection-diffusion-reaction equation - Zhelnin, Maksim Sergeyevich; Kostina, Anastasiya Andreyevna; Plekhov, Oleg Anatol’yevich (Computational Continuum Mechanics, 2019)
This paper focuses on the development of finite element methods for solving a two-dimensional boundary value problem for a singularly perturbed time-dependent convection-diffusion-reaction equation. Solution to the problem can vary rapidly in thin layers. As a result, spurious oscillations in the solution occur if the standard Galerkin method is used. In multiscale finite element methods, the initial problem is split into grid-scale and subgrid-scale problems, which allows one to capture the features of the problem at a scale smaller than an element mesh size. In the study two methods are considered: VMM-ASA (Variational Multiscale Method with Algebraic Sub-scale Approximation) and RFB (Residual-Free Bubbles). In the first method, the subgrid problem is modeled by the residual of the grid equation and intrinsic time scales. In the second method, the subgrid problem is approximated by special functions. The grid and subgrid problems are formulated through a linearization procedure on the subgrid component applied to the initial problem. The computer implementation of the methods has been carried out using a commercial finite element package. The efficiency of the developed methods has been studied by solving a test boundary value problem for the nonlinear equation. Several values of the diffusion coefficient of the equation have been analyzed. On the basis of the numerical study, it has been shown that the multiscale methods allow one to increase the stability of a numerical solution and to decrease the quantity and 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.