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This paper examines the numerical solution of the convection-diffusion equation in 2-D. The solution of this equation possesses singularities in the form of boundary or interior layers due to non-smooth boundary conditions. To overcome such singularities arising from these critical regions, the adaptive finite element method is employed. This scheme is based on the streamline diffusion method combined with Neumann-type posteriori estimator. The effectiveness of this approach is illustrated by different examples with several numerical experiments.

This paper deals with the scalar convection-diffusion equation. This equation describes the transport of scalar quantity, e.g., temperature or concentration. We are interested in the convection dominated case. In this case, the solution of a convection-diffusion equation frequently has boundary or interior layers. It is well known that the standard Galerkin finite element discretization on uniform grids produces inaccurate oscillatory solutions to convection diffusion problems. Therefore several stabilized finite element methods have been developed, e.g., the streamline-upwind Petrov-Galerkin (SUPG) method [

Another related problem is to obtain reliable estimates of the accuracy of the computed numerical solution. A priori estimate are often insufficient and can’t be used to estimate the exact error. Therefore, it is natural to acquire a posteriori error estimators to pinpoint where the error is large and, at the same time, properly bound the exact error on the whole domain. The error estimator should be local and should yield reliable upper and lower bounds for the true error in a user-specified norm. Global upper bounds are sufficient to obtain a numerical solution with accuracy below a prescribed tolerance. Local lower bounds are necessary to ensure that the grid is correctly refined so that one obtains a numerical solution with a prescribed tolerance using a nearly minimal number of grid-points.

For two-dimensional problems, several estimators have been shown to be asymptotically exact when used on uniform meshes provided the solution of the problem is smooth enough [

In this paper we introduce and analyze from theoretical and experimental points of view an adaptive scheme to efficiently solve the convection-diffusion equation. This scheme is based on the streamline-diffusion finite element method (SDFEM) introduced in [

The paper is organized as follows. In Section 2 we recall the convection-diffusion problem under consideration and the Streamline Diffusion Finite Element Method. In Section 3 we define a posteriori error estimator with the energy norm of the finite element approximation error. Finally, in Section 4, we introduce the adaptive scheme and report the results of the numerical tests.

We consider the following steady linear convection-diffusion equation

where

(A.1)

(A.2)

(A.3)

(A.4)

The

norm,

To define weak form of Equation (2.1), we need two classes of functions: the trial functions

The standard variational formulation of Equation (2.1) is given by: Find

where

Let

We need to make the following geometrical assumptions on the family of triangulations

1) Admissibility: whenever

1)

3)

4) Shape regularity: the ratio of

which means for any

We define the finite element spaces

for triangular elements, where

In the case of convection-dominated problem, the standard Galerkin approximation of Equation (2.6) may produce unphysical behavior, oscillation, if the mesh is too coarse in critical regions. To circumvent these difficulties, stability of the discretization has to be increased by introducing artificial diffusion along streamlines. The Streamline-Diffusion Finite Element Method (SDFEM) [

The SDFEM yields the following discrete problem obtained: Find

where

In Equation (2.11), a constant

fined by,

following choice of

where

In this topic, we introduce the analysis of a Neumann-type error estimator proposed in [

We now introduce some definitions and notations that will be needed for the error estimates.

We denote by

and the subsets relating to internal, Dirichlet and Neumann edges respectively as

We denote

do not lie on the Dirichlet boundary

For the lowest order

and the internal residual is approximated by

where

For any edge

where

across

The approximation space is denoted by

consisting of edge and interior bubble functions respectively:

where each member of the space is a quadratic (or biquadratic) edge bubble function

where each function is associated with an element K, and is zero on all edges of K, nonzero on the interior of K, and

The upshot is that the local problems are always well posed and that for each triangular element a 4 × 4 system of equations must be solved to compute

For an element

where

In the following, we use the short-hand notation

Theorem 1. If the variational Equation (2.6) solved with a grid of linear triangular elements, and if the triangle aspect ratio condition is satisfied with

where C is independent of

Proof. See the details in [

Theorem 2. If the variational Equation (2.6) with

where c is independent of

Proof. See the details in [

In this section we report three series of numerical experiments with the Streamline Diffusion stabilization method described in Section (2) and an h-adaptive mesh-refinement strategy based on the error estimator analyzed in Section (3). In all the experiments we have used piecewise linear finite elements (i.e.,

The adaptive procedure consists in solving Equation (2.11) on a sequence of meshes up to finally attain a solution with an estimated error within a prescribed tolerance. To attain this purpose, we initiate the process with a quasi-uniform mesh and, at each step, a new mesh better adapted to the solution of Equation (2.6) must be created. This is done by computing the local error estimators

those elements K^{*} with

riments we have chosen

The implementation used in this paper is derived from iFEM [

Example 1 (Exponential boundary layer) The first test problem contains an exponential boundary layer. This

problem corresponds to the case of

the exact solution is given by

We report the results obtained for

and

Example 2 (Interior layers) We consider Equation (2.1) with

Example 3 (Interior and boundary layers) We consider Equation (2.1) with

Discontinuities at

In the case of

For the case of

An adaptive finite element scheme for the convection-diffusion equation has been introduced and analyzed. This scheme is based on the Streamline Diffusion Finite element method combined with a Neumann-type error estimator.

Several numerical experiments are reported. For

quite evident that our error estimator provides an effective refinement indicator even in the presence of internal layers.

Galeage Kaelo,Brothers Wilright Malema,Gelaw Temesgen Mekuria,Jakkula Anand Rao, (2016) Adaptive Finite Element Method for Steady Convection-Diffusion Equation. American Journal of Computational Mathematics,06,275-285. doi: 10.4236/ajcm.2016.63029