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New Approach for Numerical Solution of Poisson’s Equation by Cubic Spline

M. Yousefi*, J. Rashidinia, M. Yousefi, N.S BahrololoumiMofrad, Mehdi Moudi

School of Mathematics, Iran University of Science & Technology, Narmak, Tehran 16844-13114, Iran

Abstract: We consider the solution of various boundary value problems for Poisson’s equation in the unit square using a nodal cubic spline collocation method and modifications of it which produce optimal fourth order approximations. Uniform partition of the square with cost O(N²log N) using a direct fast Fourier transform method. The numerical results exhibit super convergence phenomena.

Keywords: Nodal collocation, Poisson’s equation, cubic spline, convergence analysis

Received December 23, 2014; Revised December 29, 2014; Accepted December 31, 2014

Citation: M. Yousefi, J. Rashidinia, M. Yousefi, N.S BahrololoumiMofrad, Mehdi Moudi, New Approach for Numerical Solution of Poisson’s Equation by Cubic Spline. Journal of Mathematical Sciences and Applications, 2 (3). 39-42. January 2014. doi:10.12691/jmsa-2-3-3

1. Introduction

The first spline collocation method proposed for the solution of second-order two-point boundary value problems was a nodal cubic spline collocation(NCSC) method [1]. In the basic nodal cubic spline collocation (NCSC) method, an approximate solutionis sought in the space of bicubic splines and is determined by satisfying the differential equation and the boundary conditions at the partition nodes of and , respectively.This method is well known to be suboptimal; in fact, it is no better than second order, whereas fourth order is expected since the approximate solution is piecewise bicubic [2]. In [3], two optimal order NCSC methods, a two step method (TSM) and a one step method (OSM), were presented for the solution of the Dirichlet BVP. These methods are based on optimal NCSC methods for solving second-order two-point BVPs.De Boor [4] proved that classical nodal cubic spline collocation for solving two-point boundary value problems. Archer [5] and, independly, Daniel and Swartz [6] developed a modified nodal cubic spline collocation (MNCSC) scheme which is fourth order accurate. Houstis,Vavalis, and Rice [7]. Derived a fourth order MNCSC scheme for solving elliptic boundary value problems on rectangles. In [5], for the Helmholtz equation, matrix decomposition algorithms (MDA) with fast Fourier transformswere formulated and implemented to solve the two step method(TSM) collocation equations for each the boundary value problem (BVP), but, for the one step method, it was possible to formulate an MDA only for the Dirichlet BVP.

In this paper, we consider Poisson’s equation with a constant coefficient in the unit square subject to the Dirichlet boundary value condition. We develop a new algorithm for the solution of the linear systems arising in the NCSC method for the Dirichlet problem. Our scheme involves perturbations of both the left- and right-hand sides. Numerical results show that our scheme exhibits super convergence phenomena while that [7] does not.

A brief outline of this paper is as follows. We give preliminaries in section 2. In section 3, the matrix vector of our sheme, and a direct fast Fourier transforms algorithm for solving the scheme are presented Section 4.) Includes numerical results obtained using our scheme.

2. Cubic Spline Function Properties and Cubic Spline Interpolation

We consider the Dirichlet boundary value problemfor Poisson’s equation

[picture] | (2.1)

Where is the Laplacian, and =, and is the boundary of .

Let with be a uniform partition of in the x-direction(For simplicity, in the remainder of the paper, a uniform partition of in the y-direction is such that .)

Let denote the space of cubic splines

Where and is the set of polynomials of degree ≤ 3 and let

To introduce basis functions for, we extend the partition using . As a basis for , we choose the functions , where

[picture] | (2.2)

And

[picture] | (2.3)

These basis functions are such that, for

[picture] | (2.4)

As a basis for,we choose the cubic splines defined in terms of splines by

[picture] | (2.5)

Cf.[8,Section 2].

It follows from (2.3) that

[picture] | (2.6)

[picture] | (2.7)

It also follows from (2.5),(2.6),(2.4), and (2.3) that, for i=1,…,N,

[picture] | (2.8)

And the matrix-vector form of

[picture] | (2.9)

Is

[picture] | (2.10)

Where . And

3. The Approximate Solution of Poisson’s Equation

In this section, we consider equation (2.1). we introduce

Our NCSC scheme for solving (2.1) is formulated as follows: seek such that

[picture] | (3.1)

(3.1) is equivalent to

[picture] | (3.2)

[picture] | (3.3)

[picture] | (3.4)

[picture] | (3.5)

The scheme (3.1)is motivated by the fourth order finite difference method for (2.1).

For and , and

[picture] | (3.6)

Where l in (3.6)is eather 1 or 2.

Since

[picture] | (3.7)

And involve equations in the unknown coefficients

It follows from (2.4) and (2.5) that, for

[picture] | (3.8)

Substituting (3.7) into (3.2), we obtain

[picture] | (3.9)

Substituting (3.7) into, we obtain (3.3)

[picture] | (3.10)

Where

Each of the methods gives rise to a linear systemof the form

Where the matrices , are with , where the matrices , are , and where denotes the matrix tensor product (cf.,[8]), thus we can (3.10) as

[picture] | (3.11)

Where, and for , we introduce matrices and defined by

[picture] | (3.12)

It follows from (2.4)-(2.7) that

[picture] | (3.13)

Where is the identity matrix and the matrix is given by

[picture] | (3.14)

And (3.11) simplifies to

[picture] | (3.15)

We have

[picture] | (3.16)

Where the matrices and are given by

[picture] | (3.17)

[picture] | (3.18)

Cf., [1, section 2].

Using (3.18), we see (3.15) is equivalent to

[picture] | (3.19)

If,and using (3.16) and (3.19) we obtain

[picture] | (3.20)

The system (3.20) reduces to the N independent systems

[picture] | (3.21)

We thus have the following algorithm for solving (3.15) :

Step 1. Compute

Step 2. Solve the N systems in (3.21)

Step 3. Compute

It follows from (3.18) that the cost of steps 1 and 3 is each (cf.,[8]).In step 2, the systems are tridiagonal, so this step is performed at a cost . Consequently, the total cost of the algorithm is .

4. Numerical Experiment

In our numerical study, we used the following testproblem. Thecomputations were carried out in double precision. We determined the nodal and global errors using the formulas. test problem from [7] were considered.

We determined the nodal and global errors using the formulas

Where

Convergence rates were determines using the formula

Where is the error corresponding to partition.

Problem D–1: Poisson’s equation with exact solution

Table 1. Nodal errors and convergence rates for u,ux,uy

Table 2. Global errors and convergence rates for u,ux,uy and uxy

5. Conclusion

We see from the results in Table 1 and Table 2 that Scheme produces fourth order accuracy for u in both the discrete and the continuous maximum norms. We also observe super convergence phenomena since the derivative approximation at the partition nodes are of order four.

References

[1]   | B. Bialecki, G. Fairweather, and A. Karageorghis, Matrix decomposition algorithms for modified spline collocation for Helmholtz problems, SIAM J. Sci. Comput., 24 (2003), pp. 1733-1753.
[2]   | B. Bialecki, G. Fairweather, and A. Karageorghis, Optimal superconvergent one step nodal cubic spline collocation methods, SIAM J. Sci. Comput., 27 (2005), pp. 575-598.
[3]   | E. N. Houstis, E. A. Vavalis, and J. R. Rice, Convergence of O(h⁴) cubic spline collocation methods for elliptic partial differential equations, SIAM J. Numer. Anal., 25 (1988), pp. 54-74.
[4]   | C. de Boor, The Method of Projections as Applied to the Numerical Solution of Two Point Boundary Value Problems Using Cubic Splines, Ph.D. thesis, University of Michigan, Ann Arbor, MI, 1966.
[5]   | D. Archer, An O(h⁴) cubic spline collocation method for quasilinear parabolic equations, SIAM J. Number. Anal., 14 (1977), pp. 620-637.
[6]   | J. W. Daniel and B. K. Swartz, Extrapolated collocation for two-point boundary-value problems using cubic splines, J. Inst. Math Appl., 16 (1975), pp. 161-174.
[7]   | E. N. Houstis, E. A. Vavalis, and J. R. Rice, Convergence of O(h⁴) cubic spline collocation methods for elliptic partial differential equations, SIAM J. Numer. Anal., 25 (1988), pp. 54-74.
[8]   | B. Bialecki and G. Fairweather, Matrix decomposition methods for separable elliptic boundary value problems in two dimensions, J.Comput. Appl. Math., 46 (1993), pp. 369-386.