Numerical Solutions of Elliptic Partial Differential Equations by Using Finite Volume Method

Solution of Partial Differential Equations (PDEs) in some region R of the space of independent variables is a function, which has all the derivatives that appear on the equation, and satisfies the equation everywhere in the region R. Some linear and most nonlinear differential equations are virtually impossible to solve using exact solutions, so it is often possible to find numerical or approximate solutions for such type of problems. Therefore, numerical methods are used to approximate the solution of such type of partial differential equation to the exact solution of partial differential equation. The finite-volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages.


Introduction
Differential equations are mathematical expressions that how the variables and their derivatives with respect to one or more independent variables affect each other in dynamic way. A partial differential equation is a differential equation in which the unknown function F is a function of multiple independent variables and of their partial derivatives or Equations involving one or more partial derivatives of a function of two or more independent variables is called partial differential equations (PDEs). The highest derivative in the partial differential equation is the order of the partial differential equation. A PDE is linear if the dependent variable and its functions are all of first order. A PDE is homogeneous if each term in the equation contains either the dependent variable or one of its derivatives. Otherwise, the equation is said to be non-homogeneous in the given partial differential equation . The equation of the form   ,  +  ,  +  ,  +  ,   + , This is the general second order, linear and nonhomogeneous partial differential equation. The partial differential equation is classified as parabolic, hyperbolic and elliptic depending on the values of A, B and C in the above equation (1). That is if the discriminate defined by ∆= − 4 > 0, then the above equation (1) is said to be hyperbolic partial differential equation. If the discriminate ∆< 0, then the equation is said to be elliptic partial differential equation and if the discriminate ∆= 0,then the equation is also said to be parabolic partial differential equation.
PDEs are mathematical models of continuous physical phenomenon in which a dependent variable, say u, is a function of more than one independent variable, say y (time), and x (eg. spatial position). PDEs derived by applying a physical principle such as conservation of mass, momentum or energy. These equations, governing the kinematic and mechanical behavior of general bodies are referred to as Conservation Laws. These laws can be written in either the strong of differential form or an integral form.

Discretization of Partial Differential Equation by Using FVM
The starting point for a finite-volume discretization is a decomposition of the problem domain Ω into a finite number of sub domains = 1,2, … , called control volumes (CVs), and related nodes where the unknown variables are to be computed. The union of all CVs should cover the whole problem domain. In general, the CVs also may overlap, but since these results in unnecessary complications we consider here the non-overlapping case only. Since finally each CV gives one equation for computing the nodal values, their final number (i.e., after the incorporation of boundary conditions) should be equal to the number of CVs. Usually, the CVs and the nodes are defined on the basis of a numerical grid. For one-dimensional problems the CVs are subintervals of the problem interval and the nodes can be the midpoints or the edges of the sub-intervals.  In the two-dimensional case the CVs can be arbitrary polygons. For quadrilateral grids the CVs usually are chosen identically with the grid cells. The nodes can be defined as the vertices or the centers of the CVs often called edge or cell-centered approaches, respectively. For triangular grids, in principle, one could do it similarly, i.e., the triangles define the CVs and the nodes can be the vertices or the centers of the triangles. Here, the nodes are chosen as the vertices of the triangles and the CVs are defined as the polygons formed by the perpendicular bisectors of the sides of the surrounding the triangles. Definition This partition of the domain into smaller sub-domains is referred to as a mesh or grid. Using finite volume method, the solution domain is subdivided into a finite number of small control volumes by a grid that grid defines the boundaries of the control volumes while the computational node lies at the center of the control volume.
Nodal points are used within these control volumes for interpolating the field variable and usually, single node at the center of the control volume is used for each control volume.
The finite volume method is a discretization of the governing equation in integral form, in contrast to the finite difference method, which is unusually applied to the governing equation in differential form. In order to obtain a finite volume discretization, the domain Ω will be Sub divided into M sub-domains % such that the collection of all those sub domains forms a partition of Ω, that is: ). These sub domains i Ω are called control volumes or control domains. In the cell-centered methods, the unknowns are associated with the control volumes, for example, any control volume corresponds to a function value at some interior point. In the cell-vertex methods, the unknowns are locating at the vertices of the control volumes.
The following figure shows that Problem variables and control volumes in a cell-centered finite volume method.
. Approximating the integrals on ijk Γ by means of the midpoint rule and replacing the derivatives by difference quotients, we have and we can approximate the right hand side by using the midpoint rule. If Ω ∂ ∈ i a , then parts of the boundary i Ω ∂ lie on ∂Ω. At these nodes, the Dirichlet boundary conditions already prescribe values of the unknown function, and so there is no need to include the boundary control volumes into the balance equations.
In order to approximate the value of the solution of partial differential equation by finite volume method we use the following steps.
Step 1: Grid Generation: The first step in the finite volume method is grid generation by dividing the domain in to discrete control volumes. Let us place a number of nodal points in the space between the points. The boundaries of control volumes are positioned mid-way between adjacent nodes. Thus each node is surrounded by a control volume or cell. It is common practice to set up control volumes near the edge of the domain in such a way that the physical boundaries coincide with the control volume boundaries. A general nodal point defined by and its neighbors in a onedimensional geometry.
Step 2: Discretization: The most important features of finite volume method are the integration of the governing equation over a control volume to yield a discretized equation at its nodal points Step 3: Solution: After discretization over each volume method, we are finding a system of algebraic equation which is easily solved by numerical methods.

Formulation of Finite Volume Scheme of One Dimensional Elliptic PDEs
The principle of the finite volume method will be shown here on the academic Dirichlet problem, namely a second order differential operator without time dependent terms and with homogeneous Dirichilet boundary conditions. Let f be a given function from (0, 1) to IR, consider the following differential equation: consider the equation of the form & = ' , ( 0.1) with the boundary condition Let ' ( )0, 1*, +, there exists a unique solution ( -)0,1*, +, to the Problem (3.1). In the sequel, this exact solution will be denoted by and (3.1) can be written in the conservative form div (F) = f, with F = . In order to compute a numerical approximation to the solution of this equation, let us define a mesh or grid, denoted by T, of the interval (0, 1) consisting of N cells or control volumes, denoted by . , 1,2, … … and N points of (0,1), denoted by , 12,3 … … . , satisfying the following definitions: Definition: An admissible mesh of (0, 1), denoted by T, is given by a family * . After integration we get the following expression.

Formulation of Finite Volume Method for Linear Systems of PDEs
This section is concerned with the discretization of linear system of two dimensions of elliptic partial differential equation Let v Wℎ, wx q x, i = 0,1,2, , N "! 1 yy n z, j = 0,1,2, … … . , N + 1 such that x q# 1 2 < x q < x q! 1 2 for i = 1,2 ,……., " + 1 and 4 = 0 , 6 1 !" =1.Similarlly  Now using the data that are given in the three tables, we can draw the figure on the same coordinate axes as follow

The Formulation of Finite Volume Method of Nonlinear System of Two Dimensions of Elliptic Partial Differential Problems
Since the finite volume method (FVM) for nonlinear twodimensional elliptic problem of PDE is difficult to be used directly; thus the Newton's method used to overcome such computational difficulties. This method is used to find the nonlinear partial differential equation. In these case the functional iteration is written as: ‡ ˆ‚ ‡ ˆ‚#" ‰ ŠW OE• ‚ #"Ž #" • ‡ ˆ‚#" with boundary conditions 0, , 1, The exact solution of the problem is , , , .

With
After some computations, we have the following expression.

Error Estimates of Two Dimensional Elliptic Problem of Dirichlet Boundary Conditions
And Where V ,j = OE ,j Ž − ,j , '|} = 1,2, " , l = 1,2, … , . The application of the above theorem to estimate the error of two-dimensional elliptic problem followed in example 2. ,j ℎ . j ≤ 0.008

Conclusion
In order to obtain a finite volume descritization, the domain Ω will be sub-divided in to many sub domains such that the collection of all those sub-domains forms a partition of Ω. consider the second order one dimensional linear elliptic