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

Authors: Jiankang Liu; Zhoushun Zheng; Qinwu Xu
Journal: Journal of Applied Mathematics
 (2012)

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## 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 ofN.


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## Body

## 1.  Introduction

In this paper, we consider the following two-point boundary value problem:(1.1)(βux)x+u=f+vδ(x-α)+12(β-+β+)wδ'(x-α),x∈Ω=(a,b),
with boundary conditions
(1.2)u(a)=ua,u(b)=ub,
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]:
(1.3)[u]=w,[βux]=v,
where the jump notation [·] is defined as
(1.4)[u]=limx→α+u(x)-limx→α-u(x)=u+-u-.
This problem is referred to as the interface problem and is used in various applications of physics, engineering, and biological sciences, see [2–4] 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 Section2, 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 ofnth degree on interval [a,b] is defined as [12]
(2.1)Bi,n(x)=(ni)(x-a)i(b-x)n-i(b-a)n,i=0,1,…,n,
where the binomial coefficients are given by (ni)=n!/i!(n-i)!, with n!=1×2×⋯×n for n≥1 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:
(2.2)Bi,n(x)=b-xb-aBi,n-1(x)+xb-aBi-1,n-1(x).
The derivatives of the nth degree B-polynomials are combinations of B-polynomials of degree n-1, which can be formulated as
(2.3)Bi,n′=nb-a(Bi-1,n-1-Bi,n-1),Bi,n′′=n(n-1)(b-a)2(Bi-2,n-2-2Bi-1,n-2+Bi,n-2),Bi,n′′′=n(n-1)(n-2)(b-a)3(Bi-3,n-3-3Bi-2,n-3+3Bi-1,n-3-Bi,n-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 inΩ1 and Ω2 are u1 and u2, respectively, then the problem (1.1)–(1.3) can be regarded as the following two smooth problems:
(3.1)(β1u1′)′+u1=f,x∈Ω1,withu1(a)=ua,(β2u2′)′+u2=f,x∈Ω2,withu2(b)=ub,
together with jump conditions (1.3), which make (3.1) closed.In each subdomainΩk,k=1,2, the solution uk(x) can be approximated by
(3.2)Uk(x)=∑i=0Nk‍Ck,iBi,Nk(x),x∈Ωk,k=1,2,
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
(3.3)-∫aα‍β1u1′v1′dx+∫aα‍u1v1dx=∫aα‍fv1dx-β1u1′v1|aα,-∫αb‍β2u2′v2′dx+∫αb‍u2v2dx=∫αb‍fv2dx-β2u2′v1|αb,
where vk∈H1(Ωk) is arbitrary and satisfies certain boundary conditions.Plugging (3.2) into (3.3) and replacing vk with B-polynomials basis produce
(3.4)-∑i=0N1‍(∫aα‍β1Bi,N1′Bj,N1′dx+∫aα‍Bi,N1Bj,N1dx)C1,i=∫aα‍f(x)Bj,N1dx-β1u1′Bj,N1|aα,-∑i=0N2‍(∫αb‍β2Bi,N2′Bm,N2′dx+∫αb‍Bi,N2Bm,N2dx)C2,i=∫αb‍f(x)Bm,N2dx-β2u2′Bm,N2|αb,
with j=0,1,…,N1,m=0,1,…,N2.Observing thatC1,0=ua, C2,N2=ub and Bj,Nk(x), j=1,2,…,Nk-1, have zeros at the endpoints, (3.4) can be reformulated as
(3.5)∑i=1N1‍C1,i∫aα‍(-β1Bi,N1′Bj,N1′+Bi,N1Bj,N1)dx=∫aα‍(f(x)Bj,N1+β1uaB0,N1′Bj,N1′-uaB0,N1Bj,N1)dx,j=1,…,N1-1,(3.6)∑i=0N2-1‍C2,i∫αb‍(-β2Bi,N2′Bm,N2′+Bi,N2Bm,N2)dx=∫αb‍(f(x)Bm,N2+β2ubBN2,N2′Bm,N2′-ubBN2,N2Bm,N2)dx,m=1,…,N2-1.
Let j=N1 in (3.5) and m=0 in (3.6), we get
(3.7)∑i=1N1‍C1,i∫aα‍(-β1Bi,N1′BN1,N1′+Bi,N1BN1,N1)dx=∫aα‍(f(x)BN1,N1+β1uaB0,N1′BN1,N1′-uaB0,N1BN1,N1)dx-β1ux-,(3.8)∑i=0N2-1‍C2,i∫αb‍(-β2Bi,N2′B0,N2′+Bi,N2B0,N2)dx=∫αb‍(f(x)B0,N2+β2ubBN2,N2′B0,N2′-ubBN2,N2B0,N2)dx+β2ux+.
Adding (3.7) and (3.8) together, combined with jump condition [βux]=β2ux+-β1ux-=v, yields
(3.9)∑i=1N1‍C1,i∫aα‍(-β1Bi,N1′BN1,N1′+Bi,N1BN1,N1)dx+∑i=0N2-1‍C2,i∫αb‍(-β2Bi,N2′B0,N2′+Bi,N2B0,N2)dx=∫aα‍(f(x)BN1,N1+β1uaB0,N1′BN1,N1′-uaB0,N1BN1,N1)dx+∫αb‍(f(x)B0,N2+β2ubBN2,N2′B0,N2′-ubBN2,N2B0,N2)dx+v.
Another jump condition [u]=w implies
(3.10)-C1,N1+C2,0=w.
Equations (3.5), (3.6), (3.9), and (3.10) form a linear system with N1+N2 equations and N1+N2 unknowns:
(3.11)C1,1,C1,2,…,C1,N1,C2,0,C2,1,…,C2,N2-1.


### 3.2. Bernstein Collocation Methods

Substitution of (3.2) into (3.1) produces residuals
(3.12)Rk=∑i=0Nk‍Ck,iβkBi,Nk′′+∑i=0Nk‍Ck,iBi,Nk-f,k=1,2.
In each interval [a,α] and [α,b], define the collocation points:
(3.13)X1={x1,i∣a=x1,0<x1,1<⋯<x1,N1-1<x1,N1=α},X2={x2,i∣α=x2,0<x2,1<⋯<x2,N2-1<x2,N2=b}.
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 thatC1,0 and C2,N2 in (3.2) are known. Collocation of (3.12) at points Xk yields
(3.14)∑i=1N1‍C1,i(β1Bi,N1′′(x1,j)+Bi,N1(x1,j)β1Bi,N1′′(x1,j)+Bi,N1(x1,j))=f(x1,j)-ua(β1B0,N1′′(x1,j)+B0,N1(x1,j)),j=1,…,N1-1,∑i=0N2-1‍C2,i(β2Bi,N2′′(x2,m)+Bi,N2(x2,m))=f(x2,m)-ub(β2BN2,N2′′(x2,m)+BN2,N2(x2,m)),m=1,…,N2-1.
From (3.2) we can easily get
(3.15)Uk′(x)=∑i=0Nk‍Ck,iBi,Nk′(x),x∈Ωk,k=1,2.
Jump condition [u]=w and [βux]=v imply
(3.16)-C1,N1+C2,0=w,-∑i=1N1‍C1,iβ1Bi,N1′(α)+∑i=0N2-1‍C2,iβ2Bi,N2′(α)=v+β1uaB0,N1′(α)-β2ubBN2,N2′(α).
Equations (3.14) and (3.16) produce the linear systems of N1+N2 equations with N1+N2 unknowns
(3.17)C1,1,C1,2,…,C1,N1,C2,0,C2,1,…,C2,N2-1.


## 3.1. Galerkin Formulations

The variational formulation of (3.1) reads
(3.3)-∫aα‍β1u1′v1′dx+∫aα‍u1v1dx=∫aα‍fv1dx-β1u1′v1|aα,-∫αb‍β2u2′v2′dx+∫αb‍u2v2dx=∫αb‍fv2dx-β2u2′v1|αb,
where vk∈H1(Ωk) is arbitrary and satisfies certain boundary conditions.Plugging (3.2) into (3.3) and replacing vk with B-polynomials basis produce
(3.4)-∑i=0N1‍(∫aα‍β1Bi,N1′Bj,N1′dx+∫aα‍Bi,N1Bj,N1dx)C1,i=∫aα‍f(x)Bj,N1dx-β1u1′Bj,N1|aα,-∑i=0N2‍(∫αb‍β2Bi,N2′Bm,N2′dx+∫αb‍Bi,N2Bm,N2dx)C2,i=∫αb‍f(x)Bm,N2dx-β2u2′Bm,N2|αb,
with j=0,1,…,N1,m=0,1,…,N2.Observing thatC1,0=ua, C2,N2=ub and Bj,Nk(x), j=1,2,…,Nk-1, have zeros at the endpoints, (3.4) can be reformulated as
(3.5)∑i=1N1‍C1,i∫aα‍(-β1Bi,N1′Bj,N1′+Bi,N1Bj,N1)dx=∫aα‍(f(x)Bj,N1+β1uaB0,N1′Bj,N1′-uaB0,N1Bj,N1)dx,j=1,…,N1-1,(3.6)∑i=0N2-1‍C2,i∫αb‍(-β2Bi,N2′Bm,N2′+Bi,N2Bm,N2)dx=∫αb‍(f(x)Bm,N2+β2ubBN2,N2′Bm,N2′-ubBN2,N2Bm,N2)dx,m=1,…,N2-1.
Let j=N1 in (3.5) and m=0 in (3.6), we get
(3.7)∑i=1N1‍C1,i∫aα‍(-β1Bi,N1′BN1,N1′+Bi,N1BN1,N1)dx=∫aα‍(f(x)BN1,N1+β1uaB0,N1′BN1,N1′-uaB0,N1BN1,N1)dx-β1ux-,(3.8)∑i=0N2-1‍C2,i∫αb‍(-β2Bi,N2′B0,N2′+Bi,N2B0,N2)dx=∫αb‍(f(x)B0,N2+β2ubBN2,N2′B0,N2′-ubBN2,N2B0,N2)dx+β2ux+.
Adding (3.7) and (3.8) together, combined with jump condition [βux]=β2ux+-β1ux-=v, yields
(3.9)∑i=1N1‍C1,i∫aα‍(-β1Bi,N1′BN1,N1′+Bi,N1BN1,N1)dx+∑i=0N2-1‍C2,i∫αb‍(-β2Bi,N2′B0,N2′+Bi,N2B0,N2)dx=∫aα‍(f(x)BN1,N1+β1uaB0,N1′BN1,N1′-uaB0,N1BN1,N1)dx+∫αb‍(f(x)B0,N2+β2ubBN2,N2′B0,N2′-ubBN2,N2B0,N2)dx+v.
Another jump condition [u]=w implies
(3.10)-C1,N1+C2,0=w.
Equations (3.5), (3.6), (3.9), and (3.10) form a linear system with N1+N2 equations and N1+N2 unknowns:
(3.11)C1,1,C1,2,…,C1,N1,C2,0,C2,1,…,C2,N2-1.


## 3.2. Bernstein Collocation Methods

Substitution of (3.2) into (3.1) produces residuals
(3.12)Rk=∑i=0Nk‍Ck,iβkBi,Nk′′+∑i=0Nk‍Ck,iBi,Nk-f,k=1,2.
In each interval [a,α] and [α,b], define the collocation points:
(3.13)X1={x1,i∣a=x1,0<x1,1<⋯<x1,N1-1<x1,N1=α},X2={x2,i∣α=x2,0<x2,1<⋯<x2,N2-1<x2,N2=b}.
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 thatC1,0 and C2,N2 in (3.2) are known. Collocation of (3.12) at points Xk yields
(3.14)∑i=1N1‍C1,i(β1Bi,N1′′(x1,j)+Bi,N1(x1,j)β1Bi,N1′′(x1,j)+Bi,N1(x1,j))=f(x1,j)-ua(β1B0,N1′′(x1,j)+B0,N1(x1,j)),j=1,…,N1-1,∑i=0N2-1‍C2,i(β2Bi,N2′′(x2,m)+Bi,N2(x2,m))=f(x2,m)-ub(β2BN2,N2′′(x2,m)+BN2,N2(x2,m)),m=1,…,N2-1.
From (3.2) we can easily get
(3.15)Uk′(x)=∑i=0Nk‍Ck,iBi,Nk′(x),x∈Ωk,k=1,2.
Jump condition [u]=w and [βux]=v imply
(3.16)-C1,N1+C2,0=w,-∑i=1N1‍C1,iβ1Bi,N1′(α)+∑i=0N2-1‍C2,iβ2Bi,N2′(α)=v+β1uaB0,N1′(α)-β2ubBN2,N2′(α).
Equations (3.14) and (3.16) produce the linear systems of N1+N2 equations with N1+N2 unknowns
(3.17)C1,1,C1,2,…,C1,N1,C2,0,C2,1,…,C2,N2-1.


## 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 whichv≠0 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]
(4.1)(βux)x+u=1+vδ(x-α),x∈(a,b),u(a)=u(b)=0,
with the following exact solution:
(4.2)u(x)={C1cos(xβ1)+C2sin(xβ1)+1,x∈(a,α),C3cos(xβ2)+C4sin(xβ2)+1,x∈(α,b),
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. Table1 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 Tables2–5, 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.Table 1
Convergence analysis of Example4.1 by Galerkin formulation.
N
β
1
=
100,  β2=10
β
1
=
10, β2=100
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
4
2.8902
e
+
002
8.0336
e
-
006
3.1656
e
-
005
1.2005
e
+
002
6.0268
e
-
007
4.4909
e
-
006
6
1.9948
e
+
003
1.2627
e
-
008
7.1402
e
-
008
5.0076
e
+
002
2.3531
e
-
010
2.6075
e
-
009
8
2.6058
e
+
004
1.1896
e
-
011
8.7994
e
-
011
6.5562
e
+
003
5.5614
e
-
014
8.1463
e
-
013
10
3.6110
e
+
005
8.9538
e
-
014
7.2848
e
-
013
9.0944
e
+
004
1.2798
e
-
014
2.0911
e
-
013
12
5.1689
e
+
006
5.7748
e
-
014
5.2493
e
-
013
1.3025
e
+
006
5.5197
e
-
015
8.2017
e
-
014Table 2
Convergence analysis of Example4.1 by Bernstein collocation method (β1=100,  β2=10).
N
Equidistant collocation points
L-G-L collocation points
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
4
8.0697
e
+
002
3.2739
e
-
005
7.2959
e
-
005
9.7943
e
+
002
9.3669
e
-
006
3.5591
e
-
005
6
1.7192
e
+
003
1.0254
e
-
007
2.8205
e
-
007
2.6631
e
+
003
1.8687
e
-
008
7.6800
e
-
008
8
3.5459
e
+
003
2.7004
e
-
010
6.8954
e
-
010
5.9340
e
+
003
2.7678
e
-
011
9.5494
e
-
011
10
2.0224
e
+
004
6.0013
e
-
013
1.1492
e
-
012
1.1258
e
+
004
2.5097
e
-
014
7.5211
e
-
014
12
1.2993
e
+
005
8.5704
e
-
013
3.2062
e
-
012
2.0852
e
+
004
1.4182
e
-
015
3.3308
e
-
015Table 3
Convergence analysis of Example4.1 by Lagrange collocation method (β1=100,  β2=10).
N
Equidistant collocation points
L-G-L collocation points
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
4
3.3057
e
+
003
5.2000
e
-
005
7.2298
e
-
005
3.9362
e
+
003
1.4511
e
-
005
2.8433
e
-
005
6
1.5322
e
+
004
1.9556
e
-
007
2.7515
e
-
007
1.5295
e
+
004
3.1861
e
-
008
5.9502
e
-
008
8
8.3734
e
+
004
4.5539
e
-
010
6.4269
e
-
010
4.6202
e
+
004
3.9798
e
-
011
7.1989
e
-
011
10
4.6854
e
+
005
6.4179
e
-
013
9.4096
e
-
013
1.1468
e
+
005
1.8912
e
-
014
4.8567
e
-
014
12
4.0396
e
+
006
8.1907
e
-
014
1.8001
e
-
013
2.4580
e
+
005
2.2058
e
-
014
3.3532
e
-
014Table 4
Convergence analysis of Example4.1 by Bernstein collocation method (β1=10,  β2=100).
N
Equidistant collocation points
C-G-L collocation points
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
4
2.4186
e
+
002
1.3416
e
-
005
1.8145
e
-
005
2.7609
e
+
002
3.7065
e
-
006
6.8234
e
-
006
6
4.6720
e
+
002
1.4350
e
-
008
1.9575
e
-
008
6.8378
e
+
002
1.5692
e
-
009
3.5133
e
-
009
8
8.0785
e
+
002
9.1093
e
-
012
1.2390
e
-
011
1.4516
e
+
003
4.1601
e
-
013
1.0452
e
-
012
10
1.8803
e
+
003
1.1076
e
-
014
1.3953
e
-
014
2.6872
e
+
003
1.3912
e
-
015
1.9211
e
-
015
12
9.7253
e
+
003
2.9255
e
-
014
3.5622
e
-
014
4.4882
e
+
003
1.5746
e
-
015
2.4994
e
-
015Table 5
Convergence analysis of Example4.1 by Lagrange collocation method (β1=10,  β2=100).
N
Equidistant collocation points
C-G-L collocation points
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
4
9.8076
e
+
002
1.3416
e
-
005
1.8145
e
-
005
1.1152
e
+
003
3.7065
e
-
006
6.8234
e
-
006
6
4.1388
e
+
003
1.4350
e
-
008
1.9575
e
-
008
4.6618
e
+
003
1.5692
e
-
009
3.5133
e
-
009
8
2.0589
e
+
004
9.1456
e
-
012
1.2443
e
-
011
1.4939
e
+
004
3.9966
e
-
013
1.0327
e
-
012
10
1.2877
e
+
005
1.4536
e
-
014
2.5636
e
-
014
3.8643
e
+
004
1.8444
e
-
014
2.2488
e
-
014
12
1.4925
e
+
006
2.5445
e
-
013
3.2439
e
-
013
8.5365
e
+
004
1.5287
e
-
014
2.1577
e
-
014Example 4.2.
In this example, the solutionu has nonzero jump across α. The following interface problem is considered
(4.3)(βux)x+u=1+vδ(x-α)+12(β1+β2)wδ′(x-α),x∈(a,b),u(a)=u(b)=0.
The jump conditions are [u]=w, [βux]=v. The exact solution is
(4.4)u(x)={C1cos(xβ1)+C2sin(xβ1)+1,x∈(a,α),C3cos(xβ2)+C4sin(xβ2)+1,x∈(α,b),
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 Example4.1. 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 1–5, while the error in Tables 6–10 is much larger than that in Tables 1–5. The regularity of the solution can affect the accuracy of the numerical algorithm enormously.Table 6
Convergence analysis of Example4.2 by Galerkin formulation.
N
β
1
=
100, β2=10
β
1
=
10, β2=100
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
4
2.8902
e
+
002
3.5634
e
-
004
1.3781
e
-
003
1.2005
e
+
002
1.4132
e
-
005
1.0699
e
-
004
6
1.9948
e
+
003
5.6809
e
-
007
3.1820
e
-
006
5.0076
e
+
002
5.6321
e
-
009
6.2488
e
-
008
8
2.6058
e
+
004
5.3895
e
-
010
3.9608
e
-
009
6.5562
e
+
003
1.3353
e
-
012
1.9540
e
-
011
10
3.6110
e
+
005
3.4189
e
-
013
3.0718
e
-
012
9.0944
e
+
004
4.3556
e
-
014
3.5483
e
-
013
12
5.1689
e
+
006
7.9344
e
-
014
7.1592
e
-
013
1.3025
e
+
006
2.6693
e
-
014
4.7276
e
-
013Table 7
Convergence analysis of Example4.2 by Bernstein collocation method (β1=100, β2=10).
N
Equidistant collocation points
L-G-L collocation points
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
4
8.0697
e
+
002
1.4582
e
-
003
2.8374
e
-
003
9.7943
e
+
002
5.7278
e
-
004
1.4832
e
-
003
6
1.7192
e
+
003
7.2538
e
-
006
1.3324
e
-
005
2.6631
e
+
003
1.4192
e
-
006
3.5539
e
-
006
8
3.5459
e
+
003
1.9942
e
-
008
3.4812
e
-
008
5.9340
e
+
003
1.9544
e
-
009
4.5723
e
-
009
10
2.0224
e
+
004
3.4252
e
-
011
5.7468
e
-
011
1.1258
e
+
004
1.6657
e
-
012
3.6562
e
-
012
12
1.2993
e
+
005
7.4279
e
-
012
1.0651
e
-
011
2.0852
e
+
004
1.1997
e
-
014
2.0057
e
-
014Table 8
Convergence analysis of Example4.2 by Lagrange collocation method (β1=100, β2=10).
N
Equidistant collocation points
L-G-L collocation points
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
4
3.3057
e
+
003
1.4582
e
-
003
2.8374
e
-
003
3.9362
e
+
003
5.7278
e
-
004
1.4832
e
-
003
6
1.5322
e
+
004
7.2538
e
-
006
1.3324
e
-
005
1.5295
e
+
004
1.4192
e
-
006
3.5539
e
-
006
8
8.3734
e
+
004
1.9942
e
-
008
3.4812
e
-
008
4.6202
e
+
004
1.9544
e
-
009
4.5723
e
-
009
10
4.6854
e
+
005
3.4418
e
-
011
5.7661
e
-
011
1.1468
e
+
005
2.1240
e
-
012
3.9355
e
-
012
12
4.0396
e
+
006
1.5353
e
-
013
5.0360
e
-
013
2.4580
e
+
005
2.7272
e
-
013
4.6544
e
-
013Table 9
Convergence analysis of Example4.2 by Bernstein collocation method (β1=10, β2=100).
N
Equidistant collocation points
C-G-L collocation points
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
4
2.4186
e
+
002
1.7618
e
-
004
2.8143
e
-
004
2.7609
e
+
002
4.9906
e
-
005
1.3489
e
-
004
6
4.6720
e
+
002
2.6676
e
-
007
3.8509
e
-
007
6.8378
e
+
002
2.9181
e
-
008
7.8955
e
-
008
8
8.0785
e
+
002
2.0042
e
-
010
2.7687
e
-
010
1.4516
e
+
003
9.1270
e
-
012
2.4596
e
-
011
10
1.8803
e
+
003
1.4215
e
-
013
1.8609
e
-
013
2.6872
e
+
003
8.3967
e
-
015
1.7718
e
-
014
12
9.7253
e
+
003
6.1320
e
-
014
8.4108
e
-
014
4.4882
e
+
003
6.8128
e
-
015
1.5746
e
-
014Table 10
Convergence analysis of Example4.2 by Lagrange collocation method (β1=10, β2=100).
N
Equidistant collocation points
C-G-L collocation points
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
Cond
∥
u
-
U
∥
L
2
∥
u
-
U
∥
H
1
4
9.8076
e
+
002
1.7618
e
-
004
2.8143
e
-
004
1.1152
e
+
003
4.9906
e
-
005
1.3489
e
-
004
6
4.1388
e
+
003
2.6676
e
-
007
3.8509
e
-
007
4.6618
e
+
003
2.9181
e
-
008
7.8955
e
-
008
8
2.0589
e
+
004
2.0054
e
-
010
2.7699
e
-
010
1.4939
e
+
004
9.1229
e
-
012
2.4579
e
-
011
10
1.2877
e
+
005
2.4447
e
-
013
4.5634
e
-
013
3.8643
e
+
004
7.1025
e
-
014
9.6688
e
-
014
12
1.4925
e
+
006
3.0489
e
-
012
4.3164
e
-
012
8.5365
e
+
004
8.5589
e
-
013
1.2211
e
-
012


## 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.



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*Source: 859315-2012-09-08.xml*