A Two-Point Boundary Value Problem by Using a Mixed Finite Element Method

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 L2-norm.


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 nonsymmentric 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 L 2 -norm.Finally, numerical experiments are given to illustrate the present theory [7].

Error in the Finite Element Solution
Recall that 1 0 H in one dimension by Sobolev's inequality, so that h I v is defined for 1 0 v H ∈ .We can prove that with ( ) In fact, by definition we have ( ) where 1 Q v is the polynomial of degree 1 approximating v in 1 , ∫ So, we can write v′′ as ( ) ( ) ( ) then we obtain ( ) ( ) To write the norm in 2 L , fort take square Next, integrate with respect to dx we have: Taking the square root finally we obtain

Galerkin's Method
Galerkin's method: Let ( ) , a ⋅ ⋅ and ( ) , , , and u V ∈ be the solution of ( ) ( ) and with ( ) In fact, let ( ) ( ) Now, from the assumptions of the Lax-Milgram lemma we have Divide by 2 C both sides we have Now, using (5), we get that ( ) Finally, we can prove (4) Now, for the symmetric ( ) , a ⋅ ⋅ we can apply Riesz representation theorem.Therefore the norm of the inner product can be written as Similar to previous proof, we have ( ) As for the norm in V, we have From the assumptions we obtain Using this inequality, ( ( ) 4) which was proven in the previous section, therefore

FEM for Bilinear Form Non-Symmetric Problem
We consider the problem ( ) in Ω with 0 u = on Γ .A finite element method for this problem with an error bound in the 1 H -norm is as follows.First we need to find the variational formulation for this problem.In fact, multiply by a function Now, the left hand side can be written using integrating by parts: Therefore we have the bilinear form and the linear functional The space 1 0 C is dense in 1 H and by Lax Milgram theorem, there is a weak solution in 1 The bilinear form is also bounded: ( ) Now, we would like to minimize the residual Also we have (see [2]) . Therefore, we can apply the previously proven statement (2) in order to estimate a bound for the error

BVP by Finite Element Method
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 ( ) ( ) , divided into triangles by inserting a diagonal with positive slope into each mesh-square with 1 10,1 20 h = .We will use the barycentric quadrature rule to compute the stiffness matrix, the load vector and the L 2 -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 ( ) ( ) , 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 ( ) , , , φ φ φ φ .The basis functions are therefore . There are two triangles common to these neighboring nodes, therefore this inner product is 1 − .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, ( ) ( )

Figure 2 .
Figure 2. (a) Solution of the system (7) using 1 10 h = ; (b) Error of the approximation in (a), compared to the real solution; (c) Solution of the system (7) using 1 20 h = ; (d) Error of the approximation in (b), compared to the real solution.

Figure 3 .
Figure 3. Logarithmic plot of the L 2 -norm of the error vs. the choice of h.

(
shown in (panel B) and (panel C) respectively.• 2 h L × norm of this error is shown Figure 3.