STABILIZED FEM FOR CONVECTION-DIFFUSION PROBLEMS ON LAYER-ADAPTED MESHES

The application of a standard Galerkin finite element method for convection-diffusion problems leads to oscillations in the discrete solution, therefore stabilization seems to be necessary. We discuss several recent stabilization methods, especially its combination with a Galerkin method on layer-adapted meshes. Supercloseness results obtained allow an improvement of the discrete solution using recovery techniques.     

A variational multiscale method based on bubble functions for convection-dominated convection-diffusion equation

This work presents a variational multiscale method based on polynomial bubble functions as subgrid scale and a numerical implementation based on two local Gauss integrations. This method can be implemented easily and efficiently for the convection-dominated problem. Static condensation of the bubbles suggests the stability of the method and we establish its global convergence. Representative numerical tests are presented.     

UNIFORM SUPERCONVERGENCE OF A FINITE ELEMENT METHOD WITH EDGE STABILIZATION FOR CONVECTION-DIFFUSION PROBLEMS

In the present paper the edge stabilization technique is applied to a convection-diffusion problem with exponential boundary layers on the unit square, using a Shishkin mesh with bilinear finite elements in the layer regions and linear elements on the coarse part of the mesh. An error bound is proved for ‖πu-u^h‖Е, where πu is some interpolant of the solution u and uh the discrete solution. This supercloseness result implies an optimal error estimate with respect to the L2 norm and opens the door to the application of postprocessing for improving the discrete solution.     

A nonlocal convection-diffusion equation

In this paper we study a nonlocal equation that takes into account convective and diffusive effects, ut=J∗u−u+G∗(f(u))−f(u) in Rd, with J radially symmetric and G not necessarily symmetric. First, we prove existence, uniqueness and continuous dependence with respect to the initial condition of solutions. This problem is the nonlocal analogous to the usual local convection-diffusion equation ut=Δu+b⋅∇(f(u)). In fact, we prove that solutions of the nonlocal equation converge to the solution of the usual convection-diffusion equation when we rescale the convolution kernels J and G appropriately. Finally we study the asymptotic behaviour of solutions as t→∞ when f(u)=|u|^q−1u with q>1. We find the decay rate and the first-order term in the asymptotic regime.     

Regularity and derivative bounds for a convection-diffusion problem with a Neumann outflow condition

A convection-diffusion problem is considered on the unit square. The convective direction is parallel to two of the square's sides. A Neumann condition is imposed on the outflow boundary, with Dirichlet conditions on the other three sides. The precise relationship between the regularity of the solution and the global smoothness and corner compatibility of the data is elucidated. Pointwise bounds on derivatives of the solution are obtained; their dependence on the data regularity and compatibility and on the small diffusion parameter is made explicit. The analysis uses Fourier transforms and Mikhlin multipliers to sharpen regularity results previously published for certain subproblems in a decomposition of the solution.     

DISCONTINUOUS FINITE ELEMENT METHOD FOR CONVECTION-DIFFUSION EQUATIONS

1. IlltroductionThe finite element approximation of the convection--diffusin equations has been investigated using several different approaches (see e.g. [3] [4] and the references therein).Previous analysis in primal formulation of these problems was done for two types ofapproximation schemes: one which produces a continuous piecewise polynomial approximation and one which produces a piecewise polynomial approximation which arecontinuous for certain number of moments accross interelement edge…     

A high-order exponential scheme for solving 1D unsteady convection-diffusion equations

In this paper, a high-order exponential (HOE) scheme is developed for the solution of the unsteady one-dimensional convection-diffusion equation. The present scheme uses the fourth-order compact exponential difference formula for the spatial discretization and the (2,2) Padé approximation for the temporal discretization. The proposed scheme achieves fourth-order accuracy in temporal and spatial variables and is unconditionally stable. Numerical experiments are carried out to demonstrate its accuracy and to compare it with analytic solutions and numerical results established by other methods in the literature. The results show that the present scheme gives highly accurate solutions for all test examples and can get excellent solutions for convection dominated problems.     

A low order anisotropic nonconforming characteristic finite element method for a convection-dominated transport problem

A low order anisotropic nonconforming rectangular finite element method for the convection-diffusion problem with a modified characteristic finite element scheme is studied in this paper. The O(h²) order error estimate in L²-norm with respect to the space, one order higher than the expanded characteristic-mixed finite element scheme with order O(h), and the same as the conforming case for a modified characteristic finite element scheme under regular meshes, is obtained by use of some distinct properties of the interpolation operator and the mean value technique, instead of the so-called elliptic projection, which is an indispensable tool in the convergence analysis of the previous literature. Lastly, some numerical results of the element are provided to verify our theoretical analysis.     

A finite-volume ELLAM for non-linear flux convection-diffusion problems

In this paper, a modified finite-volume Eulerian-Lagrangian localized adjoint method (FVELLAM) extended for convection-diffusion problems with a non-linear flux function is introduced. Tracking schemes are discussed using viscous Burgers’ equation. It is shown that in order to have smooth results, only the new time level values should be used in tracking process. Then, the proposed method is employed to study immiscible incompressible two-phase flows in porous media. Various one- and two-dimensional test cases involving internal sources and sinks are solved and accuracy of solution and performance of the method are investigated by comparing the results obtained using FVELLAM with those of fine grid solutions. Finally, it is concluded that although proposed FVELLAM produces satisfactory results even on coarse grids and allows fairly large time-steps, its major advantage is in solving convection-dominated problems in heterogeneous media.