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This paper describes a numerical solution for a two-point boundary value problem. It includes an algorithm for discretization by mixed finite element method. The discrete scheme allows the utilization a finite element method based on piecewise linear approximating functions and we also use the barycentric quadrature rule to compute the stiffness matrix and the
*L*
_{2}-norm.

Finite element methods in which two spaces are used received the domination of mixed finite element method. Sometimes a second variable is introduced in the formulation of the problem by its physical study, for example in the case of elasticity equations and also the Stokes equations where the mixed formulation is the natural one. The mathematical analyses of mixed finite element have been widely developed in the seventies. A general analysis was first developed by [

An outline of the paper is as follows. We derive the mixed variational formulation for bilinear form non- symmentric problem and we define the related discrete elements and the error analysis of the associated finite element method is made [_{2}-norm. Finally, numerical experiments are given to illustrate the present theory [

Recall that

and

In fact, by definition we have

where

therefore, the error is of order

Now, we know that

So, we can write

then we obtain

So

To write the norm in

Next, integrate with respect to

Taking the square root finally we obtain

Galerkin’s method: Let

and

and with

In fact, let

Now, from the assumptions of the Lax-Milgram lemma we have

Divide by

Now, using (5), we get that

Finally, we can prove (4)

Now, for the symmetric

Similar to previous proof, we have

Therefore

As for the norm in V, we have

From the assumptions we obtain

With

Using this inequality, (6) becomes

We know that

from Equation (4) which was proven in the previous section, therefore

We consider the problem

in

Next, integrate over the domain

Now, the left hand side can be written using integrating by parts:

Therefore we have the bilinear form

and the linear functional

The space

The bilinear form is also bounded:

Now, we would like to minimize the residual

Also we have (see [

where

which for our case become

We consider the boundary value problem

We want to solve it by the finite element method

based on piecewise linear approximating functions on the partition_{2}-norm. The

Therefore we obtain

・

・ The Barycentric Quadrature Rule were used to evaluate the integral on the right hand side,

where

where K is each triangle in the mesh,

・

the errors are shown in (panel B) and (panel C) respectively.

・

We thank the editor and the referee for their comments and group GEDNOL of the Universidad Tecnológica de Pereira-Colombia.

Pedro Pablo Cárdenas Alzate,José Rodrigo González Granada, (2015) A Two-Point Boundary Value Problem by Using a Mixed Finite Element Method. Applied Mathematics,06,1996-2003. doi: 10.4236/am.2015.612177