Keywords: Nodal collocation, Poisson’s equation, cubic spline, convergence analysis
Journal of Mathematical Sciences and Applications, 2014 2 (3),
pp 3942.
DOI: 10.12691/jmsa233
Received December 23, 2014; Revised December 29, 2014; Accepted December 31, 2014
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction
The first spline collocation method proposed for the solution of secondorder twopoint 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 secondorder twopoint BVPs.De Boor ^{[4]} proved that classical nodal cubic spline collocation for solving twopoint 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 righthand 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
 (2.1) 
Where is the Laplacian, and =, and is the boundary of ^{.}
Let with be a uniform partition of in the xdirection(For simplicity, in the remainder of the paper, a uniform partition of in the ydirection 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
 (2.2) 
And
 (2.3) 
These basis functions are such that, for
 (2.4) 
As a basis for,we choose the cubic splines defined in terms of splines by
 (2.5) 
Cf.[8,Section 2].
It follows from (2.3) that
 (2.6) 
 (2.7) 
It also follows from (2.5),(2.6),(2.4), and (2.3) that, for i=1,…,N,
 (2.8) 
And the matrixvector form of
 (2.9) 
Is
 (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
 (3.1) 
(3.1) is equivalent to
 (3.2) 
 (3.3) 
 (3.4) 
 (3.5) 
The scheme (3.1)is motivated by the fourth order finite difference method for (2.1).
For and , and
 (3.6) 
Where l in (3.6)is eather 1 or 2.
Since
 (3.7) 
And involve equations in the unknown coefficients
It follows from (2.4) and (2.5) that, for
 (3.8) 
Substituting (3.7) into (3.2), we obtain
 (3.9) 
Substituting (3.7) into, we obtain (3.3)
 (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
 (3.11) 
Where, and for , we introduce matrices and defined by
 (3.12) 
It follows from (2.4)(2.7) that
 (3.13) 
Where is the identity matrix and the matrix is given by
 (3.14) 
And (3.11) simplifies to
 (3.15) 
We have
 (3.16) 
Where the matrices and are given by
 (3.17) 
 (3.18) 
Cf., [1, section 2].
Using (3.18), we see (3.15) is equivalent to
 (3.19) 
If,and using (3.16) and (3.19) we obtain
 (3.20) 
The system (3.20) reduces to the N independent systems
 (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,u_{x},u_{y}_{}
Table 2. Global errors and convergence rates for u,u_{x},u_{y} and u_{xy}_{}
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
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