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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 859315, 11 pages
http://dx.doi.org/10.1155/2012/859315
Research Article

Bernstein-Polynomials-Based Highly Accurate Methods for One-Dimensional Interface Problems

School of Mathematics and Statistics, Central South University, Changsha 410083, China

Received 10 June 2012; Accepted 11 August 2012

Academic Editor: Francisco J. Marcellán

Copyright © 2012 Jiankang Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A new numerical method based on Bernstein polynomials expansion is proposed for solving one-dimensional elliptic interface problems. Both Galerkin formulation and collocation formulation are constructed to determine the expansion coefficients. In Galerkin formulation, the flux jump condition can be imposed by the weak formulation naturally. In collocation formulation, the results obtained by B-polynomials expansion are compared with that obtained by Lagrange basis expansion. Numerical experiments show that B-polynomials expansion is superior to Lagrange expansion in both condition number and accuracy. Both methods can yield high accuracy even with small value of N.

1. Introduction

In this paper, we consider the following two-point boundary value problem: 𝛽𝑢𝑥𝑥1+𝑢=𝑓+𝑣𝛿(𝑥𝛼)+2𝛽+𝛽+𝑤𝛿(𝑥𝛼),𝑥Ω=(𝑎,𝑏),(1.1) with boundary conditions 𝑢(𝑎)=𝑢𝑎,𝑢(𝑏)=𝑢𝑏,(1.2) where 𝑎<𝛼<𝑏,𝛿(𝑥) is the Dirac delta function and 𝛿(𝑥) is the dipole source term. The function 𝛽(𝑥) is allowed to be discontinuous at 𝑥=𝛼. For simplicity here we assume that 𝑓(𝑥) is smooth function. Due to the presence of singular source, the solution 𝑢(𝑥) possesses the following jump relations [1]: [𝑢]=𝑤,𝛽𝑢𝑥=𝑣,(1.3) where the jump notation [] is defined as [𝑢]=lim𝑥𝛼+𝑢(𝑥)lim𝑥𝛼𝑢(𝑥)=𝑢+𝑢.(1.4) This problem is referred to as the interface problem and is used in various applications of physics, engineering, and biological sciences, see [24] and the references therein.

For interface problems, since sharp interfaces or local jumps exist within the solution domain, any high-order method, such as the spectral method, suffers from the Gibbs phenomenon [5]. Here we evoke items we care most for solving interface problems. In 1993, the immersed interface method (IIM) was proposed for interface problems [1]. It is a two-order finite difference method based on Cartesian grids by incorporating the jump relations into difference schemes. The authors have constructed the IIM-based ADI finite difference scheme for 2D nonlinear convection diffusion interface problems [6]. In [7], a high-order method was developed for both discontinuous coefficients and singular source based on finite element method. To enhance the accuracy, the modified Hermite polynomials are used for the basic functions in each element. The matched interface and boundary method was proposed in [8] for elliptic interface problems with discontinuous coefficients and singular source. In [9], the coupling interface method was developed for elliptic interface problems. Recently, Shin and Jung [5] developed the spectral collocation method for one-dimensional interface problems (1.1) with 𝑤=0, in which Lagrange basis functions are chosen as the trial functions.

In this work, we also consider this problem and take Bernstein polynomials basis as the trial functions. Bernstein polynomials are useful polynomials in computer-aided geometric design because of their excellent properties [10]. Recently there are some works that used Bernestein polynomials as basis for numerically solving differential equations [11, 12], integral equations [13, 14], and so on, but none of them is about interface problems. Our method is different from Shin and Jung’s in three aspects. (i)The trial functions are chosen as Bernstein polynomials due to its nice properties. These polynomials defined on an interval form a complete basis over the interval. Each of these polynomials is positive and their sum is unity. (ii)The Galerkin formulation is constructed for this problem. Since the Bernstein polynomials are algebraic polynomials, the mass matrix and stiff matrices can be evaluated efficiently. (iii)The B-polynomial-based collocation formulation is given collocated with both equidistant points and spectral points, respectively. Unlike Lagrange basis functions, the B-polynomial differential matrix can be computed easily and the condition number of resulting linear system is smaller than that of Lagrange basis, which is shown in the numerical examples.

The rest of this paper is organized as follows. In Section 2, the Bernstein polynomials on interval [𝑎,𝑏] and its derivatives are given. The Galerkin formulation and B-polynomial-based collocation formulation are shown in Section 3. Two numerical examples are given and analyzed in Section 4 before the conclusions are made in Section 5.

2. Bernstein Polynomials Basis

The general form of Bernstein polynomials of 𝑛th degree on interval [𝑎,𝑏] is defined as [12] 𝐵𝑖,𝑛𝑛𝑖((𝑥)=𝑥𝑎)𝑖(𝑏𝑥)𝑛𝑖(𝑏𝑎)𝑛,𝑖=0,1,,𝑛,(2.1) where the binomial coefficients are given by (𝑛𝑖)=𝑛!/𝑖!(𝑛𝑖)!, with 𝑛!=1×2××𝑛 for 𝑛1 and 0!=1. These (𝑛+1) B-polynomials of degree 𝑛 form a complete basis over the interval [𝑎,𝑏]. It is easy to show that any given polynomial of degree 𝑛 can be expressed in terms of linear combination of the B-polynomials basis functions. The B-polynomials can be generated by a recursive definition: 𝐵𝑖,𝑛(𝑥)=𝑏𝑥𝐵𝑏𝑎𝑖,𝑛1𝑥(𝑥)+𝐵𝑏𝑎𝑖1,𝑛1(𝑥).(2.2) The derivatives of the 𝑛th degree B-polynomials are combinations of B-polynomials of degree 𝑛1, which can be formulated as 𝐵𝑖,𝑛=𝑛𝐵𝑏𝑎𝑖1,𝑛1𝐵𝑖,𝑛1,𝐵𝑖,𝑛=𝑛(𝑛1)(𝑏𝑎)2𝐵𝑖2,𝑛22𝐵𝑖1,𝑛2+𝐵𝑖,𝑛2,𝐵𝑖,𝑛=𝑛(𝑛1)(𝑛2)(𝑏𝑎)3𝐵𝑖3,𝑛33𝐵𝑖2,𝑛3+3𝐵𝑖1,𝑛3𝐵𝑖,𝑛3,(2.3) where we set 𝐵𝑖,𝑛=0 if 𝑖<0 or 𝑖>𝑛. In favor of these recursive relations, the differentiation matrix of Bernstein basis can be evaluated conveniently, while the computation of differentiation matrix of high-order Lagrange basis function may suffer from certain difficulties [15].

3. Numerical Methods

In this section, we give the numerical method for solving interface problem (1.1)–(1.3). It includes Galerkin formulation and collocation formulation. Without loss of generality, we assume that only one interface 𝑥=𝛼 exists in Ω and 𝛽 is piecewise constant in each subdomain. The multiple interface can be handled analogously. Thus the entire domain is divided by 𝛼 into two parts Ω1=(𝑎,𝛼) and Ω2=(𝛼,𝑏).

Suppose that the solutions in Ω1 and Ω2 are 𝑢1 and 𝑢2, respectively, then the problem (1.1)–(1.3) can be regarded as the following two smooth problems: 𝛽1𝑢1+𝑢1=𝑓,𝑥Ω1,with𝑢1(𝑎)=𝑢𝑎,𝛽2𝑢2+𝑢2=𝑓,𝑥Ω2,with𝑢2(𝑏)=𝑢𝑏,(3.1) together with jump conditions (1.3), which make (3.1) closed.

In each subdomain Ω𝑘,𝑘=1,2, the solution 𝑢𝑘(𝑥) can be approximated by 𝑈𝑘(𝑥)=𝑁𝑘𝑖=0𝐶𝑘,𝑖𝐵𝑖,𝑁𝑘(𝑥),𝑥Ω𝑘,𝑘=1,2,(3.2) where 𝑁𝑘 is the degree of B-polynomials on each subdomain. According to the interpolation property of B-polynomials at two endpoints, we can easily get 𝐶1,0=𝑢𝑎 and 𝐶2,𝑁2=𝑢𝑏.

3.1. Galerkin Formulations

The variational formulation of (3.1) reads 𝛼𝑎𝛽1𝑢1𝑣1d𝑥+𝛼𝑎𝑢1𝑣1d𝑥=𝛼𝑎𝑓𝑣1d𝑥𝛽1𝑢1𝑣1||𝛼𝑎,𝑏𝛼𝛽2𝑢2𝑣2d𝑥+𝑏𝛼𝑢2𝑣2d𝑥=𝑏𝛼𝑓𝑣2d𝑥𝛽2𝑢2𝑣1||𝑏𝛼,(3.3) where 𝑣𝑘𝐻1(Ω𝑘) is arbitrary and satisfies certain boundary conditions.

Plugging (3.2) into (3.3) and replacing 𝑣𝑘 with B-polynomials basis produce 𝑁1𝑖=0𝛼𝑎𝛽1𝐵𝑖,𝑁1𝐵𝑗,𝑁1d𝑥+𝛼𝑎𝐵𝑖,𝑁1𝐵𝑗,𝑁1𝐶d𝑥1,𝑖=𝛼𝑎𝑓(𝑥)𝐵𝑗,𝑁1d𝑥𝛽1𝑢1𝐵𝑗,𝑁1||𝛼𝑎,𝑁2𝑖=0𝑏𝛼𝛽2𝐵𝑖,𝑁2𝐵𝑚,𝑁2d𝑥+𝑏𝛼𝐵𝑖,𝑁2𝐵𝑚,𝑁2𝐶d𝑥2,𝑖=𝑏𝛼𝑓(𝑥)𝐵𝑚,𝑁2𝑑𝑥𝛽2𝑢2𝐵𝑚,𝑁2||𝑏𝛼,(3.4) with 𝑗=0,1,,𝑁1,𝑚=0,1,,𝑁2.

Observing that 𝐶1,0=𝑢𝑎, 𝐶2,𝑁2=𝑢𝑏 and 𝐵𝑗,𝑁𝑘(𝑥), 𝑗=1,2,,𝑁𝑘1, have zeros at the endpoints, (3.4) can be reformulated as 𝑁1𝑖=1𝐶1,𝑖𝛼𝑎𝛽1𝐵𝑖,𝑁1𝐵𝑗,𝑁1+𝐵𝑖,𝑁1𝐵𝑗,𝑁1=d𝑥𝛼𝑎𝑓(𝑥)𝐵𝑗,𝑁1+𝛽1𝑢𝑎𝐵0,𝑁1𝐵𝑗,𝑁1𝑢𝑎𝐵0,𝑁1𝐵𝑗,𝑁1d𝑥,𝑗=1,,𝑁11,(3.5)𝑁21𝑖=0𝐶2,𝑖𝑏𝛼𝛽2𝐵𝑖,𝑁2𝐵𝑚,𝑁2+𝐵𝑖,𝑁2𝐵𝑚,𝑁2=d𝑥𝑏𝛼𝑓(𝑥)𝐵𝑚,𝑁2+𝛽2𝑢𝑏𝐵𝑁2,𝑁2𝐵𝑚,𝑁2𝑢𝑏𝐵𝑁2,𝑁2𝐵𝑚,𝑁2d𝑥,𝑚=1,,𝑁21.(3.6) Let 𝑗=𝑁1 in (3.5) and 𝑚=0 in (3.6), we get 𝑁1𝑖=1𝐶1,𝑖𝛼𝑎𝛽1𝐵𝑖,𝑁1𝐵𝑁1,𝑁1+𝐵𝑖,𝑁1𝐵𝑁1,𝑁1=d𝑥𝛼𝑎𝑓(𝑥)𝐵𝑁1,𝑁1+𝛽1𝑢𝑎𝐵0,𝑁1𝐵𝑁1,𝑁1𝑢𝑎𝐵0,𝑁1𝐵𝑁1,𝑁1d𝑥𝛽1𝑢𝑥,(3.7)𝑁21𝑖=0𝐶2,𝑖𝑏𝛼𝛽2𝐵𝑖,𝑁2𝐵0,𝑁2+𝐵𝑖,𝑁2𝐵0,𝑁2=d𝑥𝑏𝛼𝑓(𝑥)𝐵0,𝑁2+𝛽2𝑢𝑏𝐵𝑁2,𝑁2𝐵0,𝑁2𝑢𝑏𝐵𝑁2,𝑁2𝐵0,𝑁2d𝑥+𝛽2𝑢+𝑥.(3.8) Adding (3.7) and (3.8) together, combined with jump condition [𝛽𝑢𝑥]=𝛽2𝑢+𝑥𝛽1𝑢𝑥=𝑣, yields 𝑁1𝑖=1𝐶1,𝑖𝛼𝑎𝛽1𝐵𝑖,𝑁1𝐵𝑁1,𝑁1+𝐵𝑖,𝑁1𝐵𝑁1,𝑁1+d𝑥𝑁21𝑖=0𝐶2,𝑖𝑏𝛼𝛽2𝐵𝑖,𝑁2𝐵0,𝑁2+𝐵𝑖,𝑁2𝐵0,𝑁2=d𝑥𝛼𝑎𝑓(𝑥)𝐵𝑁1,𝑁1+𝛽1𝑢𝑎𝐵0,𝑁1𝐵𝑁1,𝑁1𝑢𝑎𝐵0,𝑁1𝐵𝑁1,𝑁1+d𝑥𝑏𝛼𝑓(𝑥)𝐵0,𝑁2+𝛽2𝑢𝑏𝐵𝑁2,𝑁2𝐵0,𝑁2𝑢𝑏𝐵𝑁2,𝑁2𝐵0,𝑁2d𝑥+𝑣.(3.9) Another jump condition [𝑢]=𝑤 implies 𝐶1,𝑁1+𝐶2,0=𝑤.(3.10) Equations (3.5), (3.6), (3.9), and (3.10) form a linear system with 𝑁1+𝑁2 equations and 𝑁1+𝑁2 unknowns: 𝐶1,1,𝐶1,2,,𝐶1,𝑁1,𝐶2,0,𝐶2,1,,𝐶2,𝑁21.(3.11)

3.2. Bernstein Collocation Methods

Substitution of (3.2) into (3.1) produces residuals 𝑅𝑘=𝑁𝑘𝑖=0𝐶𝑘,𝑖𝛽𝑘𝐵𝑖,𝑁𝑘+𝑁𝑘𝑖=0𝐶𝑘,𝑖𝐵𝑖,𝑁𝑘𝑓,𝑘=1,2.(3.12) In each interval [𝑎,𝛼] and [𝛼,𝑏], define the collocation points: 𝑋1=𝑥1,𝑖𝑎=𝑥1,0<𝑥1,1<<𝑥1,𝑁11<𝑥1,𝑁1,𝑋=𝛼2=𝑥2,𝑖𝛼=𝑥2,0<𝑥2,1<<𝑥2,𝑁21<𝑥2,𝑁2.=𝑏(3.13) Here the points in 𝑋𝑘 can be equidistant points, Legendre-Gauss-Lobatto (L-G-L) points, or Chebyshev-Gauss-Lobatto (C-G-L) points.

Note that 𝐶1,0 and 𝐶2,𝑁2 in (3.2) are known. Collocation of (3.12) at points 𝑋𝑘 yields 𝑁1𝑖=1𝐶1,𝑖𝛽1𝐵𝑖,𝑁1𝑥1,𝑗+𝐵𝑖,𝑁1𝑥1,𝑗𝑥=𝑓1,𝑗𝑢𝑎𝛽1𝐵0,𝑁1𝑥1,𝑗+𝐵0,𝑁1𝑥1,𝑗,𝑗=1,,𝑁11,𝑁21𝑖=0𝐶2,𝑖𝛽2𝐵𝑖,𝑁2𝑥2,𝑚+𝐵𝑖,𝑁2𝑥2,𝑚𝑥=𝑓2,𝑚𝑢𝑏𝛽2𝐵𝑁2,𝑁2𝑥2,𝑚+𝐵𝑁2,𝑁2𝑥2,𝑚,𝑚=1,,𝑁21.(3.14) From (3.2) we can easily get 𝑈𝑘(𝑥)=𝑁𝑘𝑖=0𝐶𝑘,𝑖𝐵𝑖,𝑁𝑘(𝑥),𝑥Ω𝑘,𝑘=1,2.(3.15) Jump condition [𝑢]=𝑤 and [𝛽𝑢𝑥]=𝑣 imply 𝐶1,𝑁1+𝐶2,0=𝑤,𝑁1𝑖=1𝐶1,𝑖𝛽1𝐵𝑖,𝑁1(𝛼)+𝑁21𝑖=0𝐶2,𝑖𝛽2𝐵𝑖,𝑁2(𝛼)=𝑣+𝛽1𝑢𝑎𝐵0,𝑁1(𝛼)𝛽2𝑢𝑏𝐵𝑁2,𝑁2(𝛼).(3.16) Equations (3.14) and (3.16) produce the linear systems of 𝑁1+𝑁2 equations with 𝑁1+𝑁2 unknowns 𝐶1,1,𝐶1,2,,𝐶1,𝑁1,𝐶2,0,𝐶2,1,,𝐶2,𝑁21.(3.17)

4. Numerical Experiments

In this section, we give two examples to verify the accuracy of proposed numerical method. The first example is the one in which 𝑣0 and 𝑤=0, while in the second example both 𝑤 and 𝑣 are not zeros. We compare our results (B-polynomials-based) with that of Shin and Jung’s (Lagrange-polynomials-based) [5]. In all cases, we take 𝑁1=𝑁2=𝑁 and the resulting linear systems are almost block diagonal and solved by BiCGStab algorithm.

Example 4.1. Consider the interface problem given in [5] 𝛽𝑢𝑥𝑥+𝑢=1+𝑣𝛿(𝑥𝛼),𝑥(𝑎,𝑏),𝑢(𝑎)=𝑢(𝑏)=0,(4.1) with the following exact solution: 𝐶𝑢(𝑥)=1𝑥cos𝛽1+𝐶2𝑥sin𝛽1𝐶+1,𝑥(𝑎,𝛼),3𝑥cos𝛽2+𝐶4𝑥sin𝛽2+1,𝑥(𝛼,𝑏),(4.2) where 𝑎=0, 𝑏=5, 𝛼=5/3, and 𝑣=10. The jump conditions read [𝑢]=0, [𝛽𝑢𝑥]=𝑣. 𝐶𝑖s can be determined by the boundary and jump conditions.
Both Galerkin formulation and collocation method are used to solve this problem numerically. The condition number (Cond) of the resulting linear system is given in each computation. Table 1 gives the convergence analysis of Galerkin formulation with different coefficients 𝛽1 and 𝛽2, in which the 𝐿2 and 𝐻1 norms are shown. It can be seen that the error reduces rapidly as the order of the B-polynomials increases.
The convergence analysis of collocation formulation is shown in Tables 25, in which the B-polynomials basis and Lagrange basis are compared. In each table, we use two types of collocation points to compare the results. It shows that the results of spectral collocation points (L-G-L or C-G-L points) are more accurate than the equidistant collocation points. Comparing Table 2 with Table 3, we can conclude that the condition number of linear systems derived from B-polynomials is much smaller than that derived from Lagrange polynomials. And the error from the former is smaller than the latter. Similar analysis can be got by comparing Table 4 with Table 5.

tab1
Table 1: Convergence analysis of Example 4.1 by Galerkin formulation.
tab2
Table 2: Convergence analysis of Example 4.1 by Bernstein collocation method (𝛽1=100, 𝛽2=10).
tab3
Table 3: Convergence analysis of Example 4.1 by Lagrange collocation method (𝛽1=100, 𝛽2=10).
tab4
Table 4: Convergence analysis of Example 4.1 by Bernstein collocation method (𝛽1=10, 𝛽2=100).
tab5
Table 5: Convergence analysis of Example 4.1 by Lagrange collocation method (𝛽1=10, 𝛽2=100).

Example 4.2. In this example, the solution 𝑢 has nonzero jump across 𝛼. The following interface problem is considered 𝛽𝑢𝑥𝑥1+𝑢=1+𝑣𝛿(𝑥𝛼)+2𝛽1+𝛽2𝑤𝛿(𝑥𝛼),𝑥(𝑎,𝑏),𝑢(𝑎)=𝑢(𝑏)=0.(4.3) The jump conditions are [𝑢]=𝑤, [𝛽𝑢𝑥]=𝑣. The exact solution is 𝐶𝑢(𝑥)=1𝑥cos𝛽1+𝐶2𝑥sin𝛽1𝐶+1,𝑥(𝑎,𝛼),3𝑥cos𝛽2+𝐶4𝑥sin𝛽2+1,𝑥(𝛼,𝑏),(4.4) where 𝑎=0, 𝑏=5, 𝛼=5/3, 𝑤=10, 𝑣=10, and 𝐶𝑖s can be determined by boundary and jump conditions.
The similar convergence analysis results can be obtained compared with Example 4.1. Since the nonzero jump 𝑤 just affects the right-hand side of the resulting linear systems, the condition numbers in Tables 6, 7, 8, 9, and 10 are unchanged compared with that in Tables 15, while the error in Tables 610 is much larger than that in Tables 15. The regularity of the solution can affect the accuracy of the numerical algorithm enormously.

tab6
Table 6: Convergence analysis of Example 4.2 by Galerkin formulation.
tab7
Table 7: Convergence analysis of Example 4.2 by Bernstein collocation method (𝛽1=100, 𝛽2=10).
tab8
Table 8: Convergence analysis of Example 4.2 by Lagrange collocation method (𝛽1=100, 𝛽2=10).
tab9
Table 9: Convergence analysis of Example 4.2 by Bernstein collocation method (𝛽1=10, 𝛽2=100).
tab10
Table 10: Convergence analysis of Example 4.2 by Lagrange collocation method (𝛽1=10, 𝛽2=100).

5. Conclusions

In this paper, a new numerical method based on B-polynomials expansion is proposed for solving one-dimensional interface problems. We give two methods to evaluate the expansion coefficients, the Galerkin formulation, and the collocation formulation. Both methods can yield highly accurate results with small number of B-polynomials. In collocation method, the Lagrange polynomials are used to compare with B-polynomials. It is shown by numerical examples that B-polynomials are superior to Lagrange polynomials in both condition number and accuracy, especially when collocated with equidistant points. In theoretical aspect, since the B-polynomials basis is equivalent to power basis or Lagrange basis under certain invertible transformations, theoretical analysis of the proposed method may be done similarly, which is a part of our future research plan. The method can be extended to problems with multiple interfaces easily.

Acknowledgments

The authors thank the supports from the Institute of Applied Software of Central South University. The authors also thank the anonymous reviewers and the editor for their valuable comments. This work was supported partially by National Natural Science Foundation of China (no. 51174236), National Basic Research Program of China (no. 2011CB606306), and Hunan Provincial Innovation Foundation For Postgraduate (no. CX2011B080).

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