A Two-Point Boundary Value Problem by Using a Mixed Finite Element Method ()
1. Introduction
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 [1] . We also have to mention to [2] and [3] which introduced of the fundamental ideas for the analysis of mixed finite elements. We also refer to [4] and [5] where general results are obtained.
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 [6] . We generalize the results to mixed methods using rectangular elements and we use the barycentric quadrature rule to compute the stiffness matrix, the load vector and the L2-norm. Finally, numerical experiments are given to illustrate the present theory [7] .
2. Error in the Finite Element Solution
Recall that in one dimension by Sobolev’s inequality, so that is defined for. We can prove that with and
and
In fact, by definition we have
(1)
where is the polynomial of degree 1 approximating v in, then we have
therefore, the error is of order because the fitting is until the second derivative, then
Now, we know that
So, we can write as
then we obtain
So
To write the norm in, fort take square
Next, integrate with respect to we have:
Taking the square root finally we obtain
3. Galerkin’s Method
Galerkin’s method: Let and satisfy the assumptions of the Lax-Milgram lemma
and be the solution of. Let be a finite-dimensional subspace and be determined by Galerkin’s method: for all. We want to prove that
(2)
and with symmetric,
(3)
(4)
In fact, let and with and. Thus,
Now, from the assumptions of the Lax-Milgram lemma we have
(5)
. And
Divide by both sides we have
Now, using (5), we get that
Finally, we can prove (4)
Now, for the symmetric we can apply Riesz representation theorem. Therefore the norm of the inner product can be written as
Similar to previous proof, we have
Therefore
As for the norm in V, we have
(6)
From the assumptions we obtain
With and we have
Using this inequality, (6) becomes
We know that
from Equation (4) which was proven in the previous section, therefore
4. FEM for Bilinear Form Non-Symmetric Problem
We consider the problem
in with on. A finite element method for this problem with an error bound in the -norm is as follows. First we need to find the variational formulation for this problem. In fact, multiply by a function on:
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 is dense in and by Lax Milgram theorem, there is a weak solution in. is coercive in, therefore
The bilinear form is also bounded:
Now, we would like to minimize the residual
Also we have (see [2] )
where. Therefore, we can apply the previously proven statement (2) in order to estimate a bound for the error
which for our case become
5. BVP by Finite Element Method
We consider the boundary value problem
Figure 1. Mesh used to solve the problem (7) by using (8).
Figure 2. (a) Solution of the system (7) using; (b) Error of the approximation in (a), compared to the real solution; (c) Solution of the system (7) using; (d) Error of the approximation in (b), compared to the real solution.
Figure 3. Logarithmic plot of the L2-norm of the error vs. the choice of h.
(7)
We want to solve it by the finite element method
(8)
based on piecewise linear approximating functions on the partition, divided into triangles by inserting a diagonal with positive slope into each mesh-square with. We will use the barycentric quadrature rule to compute the stiffness matrix, the load vector and the L2-norm. The Figure 1 shows the mesh used to solve this problem (system (7)). With this mesh, the stiffness matrix A was computed considering each node, from a total of interior nodes (, which h is the step size). The basis function is a set of pyramidal functions. At each node, there are two triangles coming at a straight angle, and four others coming with an acute angle. The basis functions are therefore
Therefore we obtain
・ . There are two triangles common to these neighboring nodes, therefore this inner product is. It is the same for the neighbours on the left (2 common triangles). Similarly for the neighbours on the rows above and below. This inner product was used in the stiffness matrix A.
・ 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 vertices.
・ Figure 2 shows the solution of (7) for (panel A) and (panel C). Compared to the correct solution
the errors are shown in (panel B) and (panel C) respectively.
・ norm of this error is shown Figure 3.
Acknowledgments
We thank the editor and the referee for their comments and group GEDNOL of the Universidad Tecnológica de Pereira-Colombia.