LEGENDRE MULTIWAVELET GALERKIN METHODS FOR DIFFERENTIAL EQUATIONS

Abstract

The multiresolution analysis for Legendre multiwavelets are given, anti-derivatives of Legendre multiwavelets are used for the numerical solution of differential equations, a special form of multilevel augmentation method algorithm is proposed to solve the disrete linear system efficiently, convergence rate of the Galerkin methods is given and numerical examples are presented.

1. Introduction

The idea of applying wavelet bases to discretize differential equations in wavelets-Galerkin methods has been explored by many authors [6,10,11,14]. Among them, Xu and Shann [14] presented a thorough study of one dimensional problems. Instead of using wavelets directly, they took the anti-derivatives of wavelets as trial functions. In this way, singularity of wavelets is smoothed, the boundary condition can be treated easily. Since the introduction of multiwavelets into the numerical solution of the integral equation [2], multiwavelets bases have also been applied to discretize the differential equations because some difficulties of using wavelets for the representation of differential operators may be eliminated by using multiwavelets that may possess all the following properties: orthogonality on a finite interval, symmetry, compact support without overlap and high order of vanishing moments. In recent years, Legendre wavelets and multiwavelets, which are not differentiable on [0,1], has drawn a lot of attention in this direction [3,9,12,13]. Recent developments include a special multilevel augmentation method (MAM), which was proposed by Chen, Wu and Xu [5] to solve some linear system of equations arising from discretizing differential equations that requires the use of special multiscale bases. Under certain conditions it leads to an efficient, stable and accurate solver for the discrete linear system. The most important point is that this MAM is not an iterative method.

In this paper we shall study Legendre multiwavelets-Galerkin methods based on variational principles. We shall take anti-derivatives of multiwavelets as trial functions as in [14], and then we show that the problem of evaluating the integrals of multiwavelets (cf. [7,11]) can be resolved by using the properties of Legendre polynomials. The convergence rate of the method is given, and MAM algorithm [5] is applied for the two-point boundary value problem. The paper is organized as follows. Section 2 reviews the constructions of Legendre multiscaling functions and multiwavelets. Section 3 defines the anti-derivatives of Legendre multiscaling functions and multiwavelets, and a general operational matrix of integration is derived by using a derivative formula of Legendre polynomials. Section 4 discusses how to calculate the product matrix of basis functions and resolve the problem of evaluating the integrals of multiwavelets. In section 5 we use the antiderivatives of Legendre multiwavelets as trial functions in the Galerkin methods for the two-point boundary value problems of ODEs, and propose a special form of MAM algorithm to solve the discrete linear system of equations. In section 6 we use the tensor product of Legendre multiwavelet basis to solve a Dirichlet problem for the elliptic equation on a rectangle. Finally, section 7 presents some numerical examples and conclusion.

 

2. Legendre multiwavelets

Let L2[0, 1] be the Hilbert space equipped with the inner product

and the norm

We will define a multiresolution approximation of L2[0,1] of multiplicity r generated by Legendre multiwavelets (cf. [2,3,9]).

2.1. Legendre multiscaling functions

The Legendre polynomial is given by the following Rodrigues formula:

Clearly Pm(x) is a polynomial of degree m. It has the following properties on the interval [-1,1] (cf. [1]):

Legendre multiscaling functions are defined as follows:

Then they have the following properties on the interval [0,1]:

Now let r > 1 be an integer, and take the first r multiscaling functions φ1, φ2, · · ·, φr, which form an orthonormal bases of the function space

Let be the vector of the multiscaling functions:

and denote for integers j > 0 and k = 0, 1, · · · , 2j − 1

the dilates and translates of the multiscaling functions, where

is supported on the interval [2−jk, 2−j(k + 1)] ⊂ [0, 1]. For a fixed j > 0, all elements of the vector

form an orthonormal bases of the 2jr-dimensional space

then one has and (iv) r functions φ1, φ2, · · · , φr form an orthonormal bases of the space V0. Therefore, {Vj}j≥0 is an orthonormal multiresolution approximation of L2[0, 1] of multiplicity r.

2.2. Legendre multiwavelet functions

Since V0 ⊂ V1, we denote W0 the orthonormal complement of V0 in V1, V1 = V0⊕W0, then an orthonormal basis {ψ1, · · · , ψr} of the space W0 can be obtained by the Gram-Schmidt process: for m = 1, 2, · · · , r we define inductively

For instance, when r = 3, we take Legendre polynomials

By formula (2.2), we have the multiscaling functions on their support [0, 1]

and through formula (2.10) the multiwavalet functions can be constructed as following

2.3. Legendre multiwavelet basis

Let be the vector of the multiwavelet functions:

and denote for integers j ≥ 0 and k = 0, 1, · · · , 2j − 1

the dilates and translates of the multiwavelet functions, where

is supported on the interval [2−jk, 2−j(k + 1)] ⊂ [0, 1]. Let ℓ = 2j − 1, and

then Wj is the orthonormal complement of Vj in Vj+1: Vj+1 = Vj ⊕ Wj, so we inductively obtain the decomposition

and

Hence is an orthonormal multiwavelet basis of L2[0, 1], and all elements of the vector

form another orthonormal bases of the space Vj (see (2.8)-(2.9)).

For any f(x) ∈ L2[0, 1], we have

the projection of f(x) in Vn is

which can also be expanded by the orthonormal basis Փn(x) defined on (2.8):

where (ℓ = 2n−1 − 1, ȷ = 2n − 1)

Since ψm(1 ≤ m ≤ r) is orthogonal to {φ1, φ2, · · · , φr},which is equivalent to the basis {1, x, x2, · · · , xr−1}, the first r moments of ψm vanish: Thus, if f(x) ∈ Cr[0,1], then [2]

2.4. The transformation matrix between two bases

Now we give out the transformation matrix Tj between the two orthonormal basis Փj(x) and Ψj(x) of the space Vj defined on (2.8) and (2.15) respectively, such that

Rewrite formula (2.10) as

From the formula (2.6) and V0 ⊥ W0 one has ψm(1 − x) = (−1)mψm(x), m = 1, 2, · · · , then for m = 1, 2, · · · , r

Define matrices

and lower triangular matrices

then

The second equation in (2.21) is usually called dilation equation that leads to

or in its matrix form

where

Finally, from (2.15),(2.21), and (2.22) we obtain (by using induction)

 

3. Anti-derivatives of Legendre multiwavelets

Let From (2.11) we have

From the derivative formula (2.5) and the formula (2.7) it is readily seen that

Define the coefficients

then

where L is a r × r matrix and vector,

Dilating and translating we get

Let then

where P is a (ℓ + 1) × (ℓ + 1) block matrix and a (ℓ + 1) × 1 block matrix,

Now we define as the anti-derivative of Ψn(x):

then by (2.20) and (3.8)

Remark 3.1. The matrix P in (3.9) is called operational matrix of integration for an approximate formula [13]. Therefore, (3.8) gives the exact definition of the operational matrix of integration.

 

4. Evaluation of the connection coefficients

The integrals of the products of several basis functions are called the connection coefficients [7,11]. By orthonormality of the multiscaling functions and multiwavelet functions one has and by (2.20) one has so the integrals of the products of two basis functions are obtained. Next we want to evaluate the integrals of the products of three basis functions

For this purpose, we show that the product φm(x)φi(x) of two multiscaling functions can be expressed as a linear combination of finite number of functions among {φm(x),m ≥ 1}. Note that φ1(x) = 1, one has

By defining the coefficients

the recurrence relation (2.4) can be rewritten as (note that

Multiplying this equation with φi(x) we obtain

Exchanging the index m and i we obtain

Then equalling right hand sides of above two equations, we get

This means if all products {φm(x)φi−1(x),m ≥ i − 1} and {φm(x) φi(x),m ≥ i} are linear combinations of finite number of functions among {φm(x),m ≥ 1}, then by induction all products {φm(x)φi+1(x),m ≥ i + 1} are also linear combinations of finite number of functions among {φm(x),m ≥ 1}. Starting from (4.2) and (4.4), we inductively deduce the following equation

where the coefficients for i = 1, 2 are from (4.2) and (4.4)

and the coefficients for i ≥ 2 are

Inserting (4.5) into (4.1), the integrals of the products of three basis functions are obtained:

Other integrals of the products of three basis functions can be obtained by combining above formula with one of the formula (2.20),(3.8) and (3.12).

 

5. Applications to two-point boundary value problems

In this section, we apply Legendre multiwavelets to numerical solutions of a two-point boundary value problem of ODE

with either one of the boundary conditions:

We assume that f ∈ L2[0, 1], the coefficients p(x) and q(x) are continuously differentiable in I = [0, 1] with

Let Hs(I) denote the standard Sobolev space with the norm ∥ · ∥s and seminorm | · |s given by

We define

It is well-known that the seminorm | · |1 is a norm in these two spaces and is equivalent to the norm ∥ · ∥1.

5.1. Error estimates

The variational form of (5.1) is

where a(·, ·) is a bilinear form defined by

Clearly a(·, ·) is continuous and coercive on It is well-known that, by Lax-Milgram lemma, (5.3) admits a unique weak solution u ∈ H1(I).

Now we apply Legendre multiwavelets to Galerkin methods for solving (5.3). According to the boundary condition, we construct a finite dimensional space Sn, then we solve numerically the Galerkin projection of the solution u on Sn defined by

It is also clear that (5.5) admits a unique solution un ∈ Sn such that [14]

By (3.11) the elements of the vector and a part of the functions of the set

where We have

Lemma 5.1. The three sets {q2(x), · · · , qr(x)} ∪Ξ, {q1(x), · · ·, qr(x)}∪Ξ, and {q1(x), · · ·, qr(x)} ∪Ξ∪ {1} form a basis for the spaces respectively.

Proof. We only prove for space the proofs for other two spaces are similar. First, it is easy to see that the functions of {q2(x), · · · , qr(x)} ∪ Ξ are linearly independent. Then, for any Since w ∈ L2(I), by (2.16)-(2.17), there are numbers such that

Since and

we have c1 = 0. Hence, if we take

then and

As | · |1 is a norm on the proof is completed.

Let the set

then Sn can be defined by its basis as

By (3.11)-(3.12) the basis of Sn consists of the elements of the vectors

respectively according to boundary conditions, where is Tn with its first line deleted. So we also use these vectors to represent the basis of Sn.

From the proof of lemma 5.1 we know that for any by (2.19)

This means for any

Combining (5.6) with (5.8), we have proved

Theorem 5.2. Let u and un be the solutions of (5.3) and (5.5) respectively. If then we have

5.2. The solution of the linear system of equations

In this subsection we suppose that Dirichlet condition is imposed on the boundary. Let and v go through all elements of in (5.5), then we have

If we denote the matrices

then a linear system of equations is obtained:

By assumption (5.2), the coefficient matrix An is a symmetric and positive definite matrix. Since the basis is an orthonormal basis of in the sense of the inner product < u, v >1:=< u′, v′ >, it is easy to prove that the condition number of An is bounded:

Proposition 5.3. For any n ≥ 0,

Proof. see [4], page 166.

Now we discuss how to solve the linear system (5.10). In practise,we expand the functions f(x), p(x), q(x) in Vn through the basis Փn(x):

By virtue of (2.19), the above approximation of p(x) and q(x) still keep their property (5.2) for sufficiently large n, so An is still a symmetric and positive definite matrix. We have (P is defined in (3.9))

Let

Then

Dn is a block diagonal matrix by virtue of (4.1) and (4.6), thus the inverses of Dn and Bn can be computed very efficiently.

In recent years a special multilevel augmentation method (MAM) has been developed [5] to solve some linear system of equations arising from discretizing differential equations that requires the use of special multiscale bases and leads to an efficient, stable and accurate solver for the discrete linear system. Our basis is just a basis of this type that meets all requirements for this MAM algorithm, but it is different from those bases in [5]. Besides, The MAM algorithm in an example in [5] can be applied to (5.10) only when p(x) = q(x) = 1 is valid. Here we propose a special form of the MAM algorithm for (5.10) as follows:

MAM algorithm for (5.10) Let m0 > 0 be a fixed integer.

Step 1 Solve um0 ∈ RN1 (N1 = 2m0r) from equation Am0um0 = fm0.

Step 2 Set um0,0 := um0 and split the matrix where

Step 3 For integer m ≥ 1, suppose that um0, m-1 ∈ RN2 (N2 = 2m0+m-1r) has been obtained and do the following:

where and

respectively;

(ii): Augment um0,m-1 by setting

(iii): Solve from the algebraic equations

Noting that Dm0,m is a block diagonal matrix, and Tn is an orthogonal matrix, the computation of the inverse of can be reduced to the computation of the inverse of one 2m0r × 2m0r matrix by using inverse operation for block matrices in basic linear algebra. Therefore this algorithm is very efficient. As for its accuracy, the following proposition ensures that the approximate solution um0,m generated by the MAM has the same order of approximation as that of the subspaces Sn.

Proposition 5.4. Let u and be the solutions of (5.3) and (5.5) respectively, where um0,m is the solution of (5.10). If hen there exists m0 ≥ 1, such that

Proof. From the assumption (5.2) and theorem 5.2, it is obvious that the operator T of (5.1) satisfies the conditions (H1) − (H5) in [5]. Then the conclusion comes from theorem 2.3 of [5].

 

6. Application to Dirichlet problem for the elliptic equation on a rectangle

In this section we discuss the application of Legendre multiwavelets to numerical solution of the Dirichlet problem for the elliptic equation on the rectangle Ω = [0, 1]2

Let L2(Ω) be the Hilbert space equipped with the inner product

and the norm

We assume that f ∈ L2(Ω). Let H1(Ω) denotes the standard Sobolev space with the norm ∥ · ∥1 and semi-norm | · |1 given by

We define

It is well-known that | · |1 is equivalent to ∥ · ∥1 in

The variational form of (6.1) is

where a(·, ·) is a bilinear form defined by

Clearly a(·, ·) is continuous and coercive on and (6.2) admits a unique weak solution by Lax-Milgram lemma.

Now we apply Legendre multiwavelets to Galerkin methods to solve (6.2). Some results in section 2 can be extended to L2(Ω). Firstly, the tensor products {Vn[0, 1]⊗Vn[0, 1]} form a multiresolution analysis (MRA) of the space L2(Ω) = L2[0, 1] ⊗ L2[0, 1], the elements of Փn(x) ⊗ Փn(y) and the elements of Ψn(x) ⊗ Ψn(y) are two equivalent bases of {Vn[0, 1] ⊗ Vn[0, 1]}. (Here and afterward we use tensor product, or Kronecker product, of two matrices. For its definition and properties we refer readers to [8]). Secondly, each function f ∈ L2(Ω) can be approximated by

and the error of the approximation for f ∈ Cr+1,r+1(Ω) is

i.e., the rate of convergence is of order r/2 (see [2]).

Following the line of section 5, we want to seek the approximate solution on a finite dimensional space i.e., we will solve numerically the Galerkin projection un of the solution u on Sn defined by

It is easy to see that is a basis of Sn. We suppose that and let in (6.5), then we have a linear system of equations

where (P is defined in (3.9))

This linear system can be solved by using the method of separation of variables. Since Kn is a symmetric matrix with positive eigenvalues, there exists an orthonormal matrix Gn and an invertible diagonal matrix Λ such that then the solution of (6.6) is given by

where H = I ⊗ Λ + Λ ⊗ I + Λ ⊗ Λ is an invertible diagonal matrix. Thus the problem of solving (6.6) is reduced to an one-dimensional eigenvalue problem for a symmetric matrix Kn, which size is (2nr − 1) × (2nr − 1).

If then we have the error estimate

Remark 6.1. The operator T1 of (6.1) satisfies the conditions (T1) − (T5) in [5] so the MAM algorithm can be applied to solve (6.6). The elements of must be reordered according to the standard construction of the bases of L2[0,1] ⊗ L2[0,1] (see [2] for an outline of the construction). However, further investigations are needed to see how to split An of (6.6) such that the inverse of the matrix in Step 3 of MAM algorithm in section 5 can be computed rapidly.

 

7. Numerical Examples and Conclusions

We present three numerical examples.

Example 7.1. Two-point boundary value problem

The exact solution is u(x) = |2x−1|3−1. We take r = 2, this means the Legendre multiwavelet basis is a linear basis. And we take m0 = 2 to apply the special form of MAM algorithm of section 5. The numerical results are summarized in Table 1.

Example 7.2. Two-point boundary value problem

The exact solution is u(x) = sin(πx). We take r = 3, this means the Legendre multiwavelet basis is a quadratic basis. And we take m0 = 1 to apply the special form of MAM algorithm of section 5. The numerical results are also summarized in Table 1.

TABLE 1.Numerical results for (7.1) and (7.2)

Example 7.3. Dirichlet problem for the elliptic equation

The exact solution is u(x, y) = sin(πx)y(y − 1). We make use of linear basis (r = 2) which means each basis function is a product of two linear functions g1(x) and g2(y), and cubic basis (r = 4) respectively. The numerical results are summarized in Table 2.

TABLE 2.Numerical results for (7.3)

Conclusions. Legendre multiwavelets together with its anti-derivatives are suitable for constructing orthonormal bases to solve some boundary value problems of ODEs and PDEs using Galerkin methods. In one dimensional case, applying multilevel augmentation method leads to an efficient, stable, and accurate algorithm, the rate of convergence in Galerkin methods is of order r. In two dimensional case, it is convenient to use tensor product of Legendre multiwavelet bases to form a basis for boundary value problems on a rectangle, the rate of convergence in Galerkin methods is of order r/2, and further investigations are needed to see how to apply the MAM algorithm to solve the discrete linear system more efficiently.