Variational multiscale finite element methods for a nonlinear convection-diffusion-reaction equation

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...