Multiscale finite element modeling of diffusion-reaction equation using bubble functions with bilinear and triangular elements

A multiscale finite element method for the solution of transport equations

“…Residual free bubble functions are also derived by solving the above equation within the element [12].…”

“…In multidimensional problems, the analytical solution of governing equations, which are partial differential equations (PDE), can cause major difficulties. To overcome this difficulty a semi-discrete method (SD method) is proposed by Parvazinia and Nassehi [12] in which the solution of a PDE is replaced by the analytical solution of ordinary differential equations (ODE). The analytical solution of ODE is then approximated by Taylor series expansion which helps us to have multidimensional bubble function using tensor product.…”

Finite volume element method for analysis of unsteady reaction–diffusion problems

“…Some examples of such difficulty are encountered in the cases where the reaction-diffusion equation cannot be satisfied by a globally smooth function. In particular, if the unknown scalar variables vary rapidly within a thin layer, many standard numerical schemes may lead to inaccurate and unstable solutions [8][9][10]. In particular, a robust numerical method, in the sense that the error should not deteriorate as the singular perturbation parameter tends to zero, is required.…”

“…and heat transfer has led to physically interesting and mathematically challenging problems related to the nontrivial numerical solution of the singularly perturbed initial/boundary value problem [1][2][3][4][5][6][7][8][9]. In general, solution of the singularly perturbed problem contains high-gradient boundary layers along the boundary of the domain [2].…”

“…During the past decade, many techniques of the finite element method for solving the singularly perturbed problems have been proposed and investigated [2,4,7,8,11,12]. The conventional Galerkin finite element method usually fails to analyze such unsteady reaction-diffusion problems when either the diffusion coefficient becomes small or a small time step is employed in the time discretization.…”