MSC classification: 65N30, 65N30, 65D07, 74S05

1. Introduction

The finite element method is a widely used method for solving both ordinary and partial differential equations. For a long time, the finite element method has been applied by many researchers to physics, solid and fluid mechanics, engineering, medicine, and so on[1]-[9]. The main idea behind the finite element method is to divide the whole region of the given problem into an equivalent system of finite elements with associated nodes and to choose the most appropriate element type to model most closely the actual physical behavior. Thus, by means of the finite element method, a huge problem is converted into many solvable small problems. Those elements must be made small enough to give usable results and yet large enough to reduce computational effort[3].

B-spline finite element method has been previously and widely used to obtain very accurate solutions for partial differential equations in one dimensional problem[2]-[8], [10]-[15]. Nevertheless, developments in computational techniques and computer technology have shown that the method can also be effectively used to obtain accurate approximate solutions for two-dimensional problems. In this paper, the bi-quintic B-splines on the boundary of the model problem have been modified and then they have been used to solve a typical two--dimensional problem. For two-dimensional problems, there are some works in the literature.[2, 7, 9, 16].

In this study, we have used bi-quintic B-splines which are only modified on the boundary of the problem as basis functions and rectangles as element shapes for a two-dimensional problem. The modified bi-quintic B-spline functions have been constructed for two dimensional problems and applied to a typical two-dimensional problem of the form. First of all, we try a bi-quintic B-spline function of the form

as an approximation solution to this type of equations. In order to find an approximate solution in the above form to the considered problem by the Galerkin method, first of all we have to redefine the B-spline basis functions into a new set of functions, namely modified bi-quintic B-spline functions. This redefining process was successfully applied to cubic B-spline functions to obtain modified bi-cubic B-spline functions[2]. The newly obtained set of modified functions is identically zero on the boundary of the given problem. It should be pointed out that this process is necessary, since B-spline basis functions in the -direction and the B-spline basis functions in the -direction are not zero on the boundary of the given problem. After the redefining process of the basis functions, we can now try the modified bi-quintic B-spline functions of the form

as its approximate solution, where and are quintic B-splines modified only on the boundary in the and - directions, respectively. Since all of the modified quintic B-splines are zero on the boundary of the given problem, non-homogenous boundary conditions are satisfied by the term .

Figure 1. The bi-quintic B-spline B₃₃ centred on (3,3) covering 36 finite elements of side 1

2. Derivation of the Modified Bi-Quintic B-splines

2.1. Bi-quintic B-spline Element

Suppose that a rectangular grid is subdivided into a number of uniform rectangular finite elements of sides and by the knots where

. An approximation with quintic B-spline functions to is of the form

can easily be found by replacing with and with Fig.1 depicts a region where , so that it is divided into finite elements by the integer knots , and a single bi-quintic B-spline which peaks on the point and also covers a total of square elements. When the entire set of bi-quintic splines , each of which peaks on a knot , where are added to this figure, a total of splines cover each finite element[8].

(1)

where are the amplitudes of bi-quintic B-splines given by

(2)

and is defined as[11]

2.2. A Modified Two-dimensional Quintic B-spline Element

To show how to modify bi-quintic splines on the boundary, we consider the two-dimensional general linear problem given in the form

(3)

subject to boundary conditions

(4)

(5)

(6)

(7)

where and , are given functions, is a rectangular region in with boundary .

Now, it is supposed that both the -space variable domain and -space variable domain of the system (3)-(7) are divided into and sub-intervals, respectively, by the set of distinct grid points and distinct grid points such that

.

Since we use quintic B-splines, a quintic B-spline covers six consecutive elements, we add ten additional grid points in the -direction and ten additional grid points , in the -direction such that

To obtain an approximate solution in the form of Eq. (1) to the model problem given by Eqs.(3)-(7) with the Galerkin method, we do need to redefine the basis functions into a new set of basis functions which all vanish on . The redefining process of the basis functions is done in the following three steps.

Step1. The approximate solution given by Eq. (1) can also be written as[2]

(8)

where

(9)

Allowing the approximate solution given by Eq. (8) to satisfy the boundary conditions (6) and (7) and eliminating and from the resulting equations, we obtain

(10)

where

(11)

(12)

(13)

Step 2. By evaluating the expression given by Eq. (9) at and and eliminating and from the resulting equations, we obtain the following expression for :

(14)

where

(15)

(16)

(17)

Step 3. Finally, substituting in Eq. (14) into Eq. (10) and allowing the resulting equation to satisfy the boundary conditions (4) and (5), we obtain

(18)

where

(19)

Applying exactly the same above steps to the approximate solution (1) to the problem given by Eqs. (3)-(7) again, but now writing the approximate solution as[2]

(20)

where

(21)

we obtain

(22)

Where

(23)

By taking the average of Eq. (19) and Eq. (23), we finally obtain the general approximation of the form

(24)

where the new set of basis functions are for , which all vanish on and given in Eq. (24) satisfies the boundary conditions given by Eqs. (4) - (7). For simplicity, the modified quintic B-splines are shown in Fig.2 for only 4 elements.

Figure 2. Modified B-spline functions

3. Numerical Example

There are some researches about two-dimensional Poisson equation in the literature. Wu and Zhang[7] have obtained numerical solutions of Dirichlet problem for two-dimensional Poisson equation using the Galerkin quartic B-spline finite element method. Schaerer et al.[17] have presented Multilevel Schwarz Shooting method for the numerical solution of the Poisson equation using a five point finite difference molecule, and subject to Dirichlet boundary conditions, which arises in two dimensional incompressible flow simulations. Ahmed and Monaquel[18] have developed multi-grid method with compact finite difference method of 9-point sixth order for two-dimensional Poisson equation and showed the efficiency of the method with numerical solutions.

In this section, we have applied the Galerkin method using the modified bi-quintic B-spline basis functions to obtain the numerical solutions of the two dimensional Poisson problem given as

where .

The exact solution of this problem is

To apply the Galerkin method with bi-quintic B-splines, we first need to find the weak formulation of the problem. Its weak form is given by the integral equation

(25)

where is the weight function taken as modified bi-quintic B-spline functions. Using the Green Theorem and the values of the modified bi-quintic B-spline functions on the boundary conditions, we obtain the weak form of the problem as

(26)

For the sake of simplicity, we divide the region into elements and show the procedure in detail. Thus, we have 16 squares with sides of , and . We use the following numeration of the region with elements in the following processes.

Since each part of a quintic B-spline should be multiplied by the same coefficient to satisfy the continuity condition, the modified quintic B-splines and are multiplied by local coefficients and respectively. Therefore, we have total global coefficients to be computed. The global coefficients and their corresponding local coefficients are determined as

Thus, we obtain the approximations for each element by writing the linear combination of and lying on the related element in Eq. (26).

By applying the Galerkin method with and as basis functions and with an approximation given by Eq.(26) to the model problem, in a similar way given in Ref.[2], we get the following algebraic system:

where

and is the coefficient vector to be computed.

Assembling all contributions from all elements, we reach the global combined matrix. Since the modified bi-quintic B-splines have been altered so as to meet the boundary conditions beforehand, the resulting equations are solved without imposing the boundary conditions at the end of the element assembly process. By solving the system of equations obtained in the global combined matrix, we evaluate the values of global variables .

Table 1 displays the obtained numerical and exact solutions of the problem with those given in Ref.[7]. It is clearly seen that the obtained numerical solutions are in very good agreement with exact ones. They are also much better than those given in Ref.[7] using quartic B-spline finite element and finite difference methods. To show how good and accurate the numerical solutions of the problem, a comparison of the obtained numerical and exact solutions at some different points are displayed in Table 2.

Table 1. Comparison of the numerical solutions obtained by the present method with results from Ref. [7] and the exact solution at some points Present [7] Quintic Quartic Finite Difference Exact (x,y) (4×4) (10×10) (0.25,0.25) 0.533970 0.533803 0.523514 0.543869 0.533804 (0.25,0.50) 0.224352 0.224426 0.229845 0.220623 0.224400 (0.25,0.75) -0.107419 -0.107167 -0.098704 -0.117514 -0.107168 (0.50,0.25) 0.355980 0.355869 0.345257 0.364192 0.355869 (0.50,0.50) 0.149568 0.149618 0.152811 0.146201 0.149618 (0.50,0.75) -0.071613 -0.071445 -0.064679 -0.080613 -0.071445 (0.75,0.25) 0.177990 0.177935 0.173169 0.182276 0.177935 (0.75,0.50) 0.074784 0.074809 0.076689 0.072922 0.074809 (0.75,0.75) -0.035806 -0.035722 -0.032649 -0.040688 -0.035723

Table 2. Comparison of the numerical solutions obtained by the present method with a partition of 4×4=16 and 10×10=100 elements with exact ones at some points Present Exact (x,y) (4×4) (10×10) (0.1,0.1) 0.388262 0.388335 0.388335 (0.2,0.2) 0.538588 0.538542 0.538541 (0.3,0.3) 0.488891 0.488773 0.488773 (0.4,0.4) 0.327186 0.327347 0.327347 (0.5,0.5) 0.149568 0.149618 0.149618 (0.6,0.6) 0.022734 0.022579 0.022579 (0.7,0.7) -0.031608 -0.031571 -0.031571 (0.8,0.8) -0.030308 -0.030272 -0.030272 (0.9,0.9) -0.009757 -0.009775 -0.009775 (0.0,1.0) 0.000000 0.000000 0.000000 (0.1,0.9) -0.087821 -0.087978 -0.087978 (0.2,0.8) -0.121231 -0.121089 -0.121088 (0.3,0.7) -0.073754 -0.073665 -0.073665 (0.4,0.6) 0.034100 0.033868 0.033869 (0.6,0.4) 0.218123 0.218231 0.218231 (0.7,0.3) 0.209524 0.209474 0.209474 (0.8,0.2) 0.134648 0.134636 0.134635 (0.9,0.1) 0.043140 0.043148 0.043148 (1.0,0.0) 0.000000 0.000000 0.000000

Table 3. The error norms L₂ and L∞ of the model problem for various numbers of elements and B-spline base functions

In order to measure how good the numerical solutions obtained by the Galerkin Finite Element Method with the modified bi-quintic B-spline basis functions, the error norms and defined as

are computed and given in Table 3. Here is the number of inner points, and are the exact and approximate solutions at the point , respectively. The error norms and are computed by taking the values and at points obtained by dividing the region into equal elements in the directions and . As seen from Table 3, the approximate solution becomes better as the number of elements increase.

Moreover, in order to demonstrate how good the numerical solutions obtained by the present method exhibit the correct physical characteristic of the problem, we also give the graphs in Figs. 3, 4 and 5 which illustrate the exact and numerical solutions with and partitions of the model problem, respectively. It is clearly seen from the figures that both graphs are nearly indistinguishable since both solutions are in very good agreement with each other.

Figure 3. Exact solution

Figure 4. Numerical solution with 16 elements partitions

Figure 5. Numerical solution with 100 elements partitions

4. Conclusions

In this paper, the modified bi-quintic B-spline finite element method is proposed and successfully applied to two dimensional Poisson problem to obtain its numerical solutions. The agreement between our numerical results and the exact ones is satisfactorily good. The obtained numerical results showed that the modified bi-quintic B-spline finite element method is remarkably a successful numerical technique. In conclusion, this method with a different choice of B-spline basis functions and element numbers can also be applied to more general non-linear partial differential equations arising in physics and mathematics and resulting in both two dimensional and coupled two dimensional equations.