THE WELL-POSEDNESS OF AN ELLIPTIC PROBLEM AND ITS SOLUTION USING THE FINITE ELEMENT METHOD

In this paper, we deal with an elliptic problem with the Dirichlet boundary condition. We operate in Sobolev spaces and the main analytic tool we use is the Lax-Milgram lemma. First, we present the variational approach of the problem which allows us to apply different functional analysis techniques. Then we study thoroughly the wellposedness of the problem. We conclude our work with a solution of the problem using numerical analysis techniques and the free software freefem++.


INTRODUCTION
Partial differential equations are one of the most efficient tools in the study of physics phenomena since many physics problems are modeled mathematically. The theory of modern PDE (partial differential equations) is closely related to functional analysis. The use of functional analysis tools is essential in the study of partial differential problems, especially the elliptic ones. Finding the classical solution for PDE problems with boundary conditions is a real challenge and to ease the burden of continuity and differentiability in classic PDE we deal with Sobolev spaces and weak 3392 BESIANA COBANI, ERISELDA GOGA solutions. We study the existence, the unicity, and the stability of the problem, its well-posedness with the variational approach. We use the celebrated theorem of Lax-Milgram that is formulated as follows: Theorem 1.1 [2] (Lax -Milgram lemma). Let be a Hilbert space and let: → ℝ be a bilinear form. The inverse operator satisfies the inequality ‖ −1 ‖ ≤ 1 .

MATERIALS AND METHODS
Homogeneous second-order elliptic operators. [ This proves that the operator is strictly positive definite for = ( +1) .
Proof. Let 1 = + + be the linear, second-order differential operator. The partial differential equation can be written as = -where 3395 THE WELL-POSEDNESS OF AN ELLIPTIC PROBLEM AND ITS SOLUTION = -((ux)x + (xux)y + (uy)y), − = 1 2 (Ω) and Ω ⊂ ℝ 2 is a bounded open set. First, we prove that the operator is uniformly elliptic.
) is the symmetric matrix of the quadratic form and the determinant | 1 2 is called discriminant of Q. It is easy to see that We use the notation D1 = a =1 and D2 = ac then the form is of 2 + 2 type, so is positive definite for ∀ ( , ) ≠ 0.
We derive the variational formulation of the elliptic problem in the unit disc.
The set 0 ∞ (Ω) denotes the space of test functions , infinitely differentiable in Ω that vanish in some neighborhood of Ω [4]. Now we multiply the equation and integrate over Ω. Then, Green's first identity).
, the same integral identity remains valid for every 1. The bilinear form is continuous. Indeed, 2. is strictly positive definite.
There exists > 0 such that Since Ω is bounded, from the Poincaŕ inequality there exists a constant such that Combining the two above inequalities we find This proves is strictly positive definite with = ( +1) .
By the Lax-Milgram lemma, for every ̃ The solution depends continuously on the initial data. The space of bounded linear functionals on 0 1 (Ω) is given by ( 0 1 (Ω)) * = −1 (Ω) [6] and from the definition of the norm in −1 (Ω) we find .
Using the inequality above we have Now let us show that for 1 ( 0 1 (Ω)) * the problem is well-posed. By the definition of the well-posedness [7], the elliptic problem (1.4)-(1.5) is well-posed, more specifically it has a uniquely determined solution that depends continuously on its initial data.

THE FINITE ELEMENT METHOD
In this section, we focus on the numerical resolution of the elliptic problem by the Finite Element Method (FEM) and the construction of it. Later on, we shall illustrate the power of this method by solving this problem with the free software freefem++.
The first step in the construction of a finite element method for the elliptic boundary value problem is done, we have already converted the problem into its weak formulation: The second step in the construction of the FEM is to replace (which is infinite-dimensional) [8][10] by a finitedimensional subspace ℎ ⊂ which consists of continuous piecewise polynomial functions of a fixed degree associated with a subdivision of the computational domain [9]. In ℎ we can solve the variational problem and hence define a finite element approximation ℎ .

Construction of a triangulation
Here Ω ⊂ ℝ 2 is a bounded domain with a smooth boundary Ω, thus Ω can be exactly covered by a finite number of triangles [9]. Let ℎ be a triangulation of Ω. ℎ is a set , =1,2,…..
On each triangle , =1,2,….. , a function ℎ is simply required to belong to 1 ( ). Requiring also continuity of ℎ between neighboring triangles, we obtain the space of all continuous piecewise linear polynomials ℎ which is a finite-dimensional subspace of V defined by With this choice of approximation space, the finite element method takes the form: Find ℎ ℎ such that which is a linear system for the unknowns .

Derivation of a Linear System of Equation
Solving the linear system (1.11) we obtain the unknowns , and thus ℎ .

PRACTICAL IMPLEMENTATION
Using the software program freefem++ to solve the problem numerically The generation of a good mesh and the definition of the corresponding finite element space may be, in practice, a very difficult task [10], so an easy way to do it is by using the free software freefem++, that is based on the finite element method and executes all the usual steps required by this method we described on section 4. We note that part of the material of this section has been adapted from [11,12].

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BESIANA COBANI, ERISELDA GOGA The standart process in freefem++ for solving the problem (1.8) is the following: Step 1 (Define the geometry). For defining the geometry of the given domain, below is used the parametric method. The boundary = Ω is and is defined by the analytic description such as We define the boundary of Ω in freefem++ by using the keyword border (line 1 below).
Step 2 (Mesh Generation) The triangulation ℎ of Ω is automatically generated by using the keyword buildmesh.
ℎ Th = ℎ(C(50)); The parameter 50 dictates the number of uniform discretization points taken on the curve C as in Fig.1.1. Refinement of the mesh are done by increasing the number of points on C [11].
Once the mesh is built we use the command below to visualize and save it : (Th, = "malla. eps").
The name ℎ referes to the family of triangles shows in Fig. 1.1.
Step 3 (Construct and solve the problem). First, we define a finite element space (where we want to solve the problem) on the constructed mesh by using the command Vh(Th, 1).
1 means that we use the 1 finite elements (continuous piecewise linear on ℎ ).
Once we have a finite element space we can define variables in this space.

Vh uh, vh;
That means that the unknown function ℎ and test functions ℎ belongs to ℎ.
We next define the given function 1 by using the command func 1 = x*y; The function 1 is defined analytically by using the keyword func.

CONCLUSION
In this paper, we presented an elliptic problem in the unit disc domain with Dirichlet boundary condition. We used the variational method to prove that the solution exists and depends continuously on the initial data. The proof was given in detail to be as helpful as possible for the new researchers in this field. We concluded the theoretical study proving the well-posedness of the given elliptic problem with Dirichlet boundary conditions in the unit disc. 3405 THE WELL-POSEDNESS OF AN ELLIPTIC PROBLEM AND ITS SOLUTION Then we moved forward in our research presenting an approximate solution using numerical analysis methods, more specifically the finite element method but we didn't intend to solve the problem using FEM. Emphasis was placed on the numerical resolution of it by using the free software freefem++, which executes all the steps presented on this method and allows us to obtain quickly and in an easy way the numerical result of the elliptic problem.