Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion-reaction problems

It is known that the enrichment of the polynomial finite element space of degree 1 by bubble functions results in a stabilized scheme of the SUPG-type for the convection-diffusion-reaction problems. In particular, the residual-free bubbles (RFB) can assure stabilized methods, but they are usually difficult to compute, unless the configuration is simple. Therefore it is important to devise numerical algorithms that provide cheap approximations to the RFB functions, contributing a good stabilizing effect to the numerical method overall. Here we propose a stabilization technique based on the RFB method and particularly designed to treat the most interesting case of small diffusion. We replace the RFB functions by their cheap, yet efficient approximations which retain the same qualitative behavior. The approximate bubbles are computed on a suitable sub-grid, the choice of whose nodes are critical and determined by minimizing the residual of a local problem with respect to L 1 norm. The resulting numerical method has similar stability features with the RFB method for the whole range of problem parameters. This fact is also confirmed by numerical experiments. We also note that the location of the sub-grid nodes suggested by the strategy herein coincides with the one in Brezzi et al. (Math. Models Methods Appl. Sci. 13:445-461, 2003). © 2011 Springer-Verlag

A fully discrete ε-uniform method for convection-diffusion problem on equidistant meshes

For a singularly-perturbed two-point boundary value problem, we propose an ε-uniform finite difference method on an equidistant mesh which requires no exact solution of a differential equation. We start with a full-fitted operator method reflecting the singular perturbation nature of the problem through a local boundary value problem. However, to solve the local boundary value problem, we employ an upwind method on a Shishkin mesh in local domain, instead of solving it exactly. We further study the convergence properties of the numerical method proposed and prove it nodally converges to the true solution for any ε

A Stabilizing subgrid for convection-diffusion problem

A stabilizing subgrid which consists of a single additional node in each triangular element is analyzed by solving the convection-diffusion problem, especially in the case of small diffusion. The choice of the location of the subgrid node is based on minimizing the residual of a local problem inside each element. We study convergence properties of the method under consideration and its connection with previously suggested stabilizing subgrids. We prove that the standard Galerkin finite element solution on augmented grid produces a discrete solution that satisfy the same a priori error estimates that are typically obtained with SUPG and RFB methods. Some numerical experiments that confirm the theoretical findings are also presented

On the choice of stabilizing sub-grid for convection-diffusion problem on rectangular grids

A stabilizing sub-grid which consists of a single additional node in each rectangular element is analyzed for solving the convection-diffusion problem, especially in the case of small diffusion. We provide a simple recipe for spotting the location of the additional node that contributes a very good stabilizing effect to the overall numerical method. We further study convergence properties of the method under consideration and prove that the standard Galerkin finite element solution on augmented grid produces a discrete solution that satisfies the same type of a priori error estimates that are typically obtained with the SUPG method. Some numerical experiments that confirm the theoretical findings are also presented. © 2010 Elsevier Ltd. All rights reserved

Bubble-based stabilized finite element methods for time-dependent convection–diffusion–reaction problems

In this paper, we propose a numerical algorithm for time-dependent convection–diffusion–reaction problems and compare its performance with the well-known numerical methods in the literature. Time discretization is performed by using fractional-step θ-scheme, while an economical form of the residual-free bubble method is used for the space discretization. We compare the proposed algorithm with the classical stabilized finite element methods over several benchmark problems for a wide range of problem configurations. The effect of the order in the sequence of discretization (in time and in space) to the quality of the approximation is also investigated. Numerical experiments show the improvement through the proposed algorithm over the classical methods in either cases

Pseudo-multi-scale functions for the stabilization of convection-diffusion equations on rectangular grids

We propose a finite element method of Petrov-Galerkin type for a singularly perturbed convection diffusion problem on a discretization consisting of rectangular elements. The method is based on enriching the finite-element space with a combination of multiscale and residual-free bubble functions. These functions require the solution of the original differential problem, which makes the method quite expensive, especially in two dimensions. Therefore, we instead employ their cheap, yet efficient approximations, using only a few nodes in each element. Several numerical tests confirm the good performance of the corresponding numerical method

The nearly-optimal Petrov-Galerkin method for convection-diffusion problems

The nearly-optimal Petrov-Galerkin (NOPG) method is employed to improve finite element computation of convection-dominated transport phenomena. The design of the NOPG method for convection-diffusion is based on consideration of the advective limit. Nonetheless, the resulting method is applicable to the entire admissible range of problem parameters. An investigation of the stability properties of this method leads to a coercivity inequality. The convergence features of the NOPG method for convection-diffusion are studied in an error analysis that is based on the stability estimates. The proposed method compares favorably to the performance of an established technique on several numerical tests.TÜBİTA

A fully discrete ε-uniform method for convection-diffusion problem on equidistant meshes

For a singularly-perturbed two-point boundary value problem, we propose an ε-uniform finite difference method on an equidistant mesh which requires no exact solution of a differential equation. We start with a full-fitted operator method reflecting the singular perturbation nature of the problem through a local boundary value problem. However, to solve the local boundary value problem, we employ an upwind method on a Shishkin mesh in local domain, instead of solving it exactly. We further study the convergence properties of the numerical method proposed and prove it nodally converges to the true solution for any ε

Applications of the pseudo residual-free bubbles to the stabilization of the convection-diffusion-reaction problems in 2D

A stabilized finite element method is studied herein for two-dimensional convection-diffusion-reaction problems. The method is based on the residual-free bubbles (RFB) method. However we replace the RFB functions by their cheap, yet efficient approximations computed on a specially chosen subgrid, which retain the same qualitative behavior. Since the correct spot of subgrid points plays a crucial role in the approximation, it is important to determine their optimal locations, which we do it through a minimization process with respect to the L1-norm. The resulting numerical method has similar stability features with the well-known stabilized methods in the literature for the whole range of problem parameters and this fact is also confirmed by numerical experiments

Solution of Navier-Stokes Equations Using FEM with Stabilizing Subgrid

The Galerkin finite element method (FEM) is used for solving the incompressible Navier Stokes equations in 2D. Regular triangular elements are used to discretize the domain and the finite-dimensional spaces employed consist of piece wise continuous linear interpolants enriched with the residual-free bubble (RFB) functions. To find the bubble part of the solution, a two-level FEM with a stabilizing subgrid of a single node is described in our previous paper [Int. J. Numer. Methods Fluids 58, 551-572 (2007)]. The results for backward facing step flow and flow through 2D channel with an obstruction on the lower wall show that the proper choice of the subgrid node is crucial to get stable and accurate solutions consistent with the physical configuration of the problems at a cheap computational cost