Wavelet Numerical Solutions for a Class of Elliptic Equations with Homogeneous Boundary Conditions

: This paper focuses on a method to construct wavelet Riesz bases with homogeneous boundary condition and use them to a kind of second-order elliptic equation. First, we construct the splines on the interval [ 0,1 ] and consider their approximation properties. Then we deﬁne the wavelet bases and illustrate the condition numbers of stiffness matrices are small and bounded. Finally, several numerical examples show that our approach performs efﬁciently.

To solve the variational problem (3), we use finite-dimensional subspaces V = span{v 1 , v 2 , · · · , v m } to approximate H 1 0 (Ω). Suppose x k ∈ C, k = 1, 2, · · · , m, such that u(x) := m ∑ k=1 x k v k (x) satisfies the following equation a(u, v j ) = f , v j L 2 (Ω) , ∀v j ∈ H 1 0 (Ω), j = 1, 2, · · · , m, or equivalently, m ∑ k=1 a jk x k = b j , j = 1, 2, · · · , m, where a jk := a(v j , v k ), b j =: f , v j , j, k ∈ {1, · · · , m}. In 1992, Chui and Wang [1] studied semi-orthogonal wavelets generated from cardinal spline. Dahmen, Kunoth and Urban [2] gave biorthogonal spline wavelets in 1999. In 2006, Jia and Liu [3] constructed wavelet bases on the interval [0, 1] and applied them to the Sturm-Liouville Equation with the Dirichlet boundary condition. In 2011, Jia and Zhao [4] applied the wavelets bases on the unit square to the biharmonic equation and extended the method to general elliptic equation of fourth-order. However, to our knowledge, there is no numerical schemes based on wavelet bases to be derived. This paper is organized as follows. In Section 2, we construct splines on the interval [0, 1] with homogeneous boundary condition and then investigate their approximation properties. The sufficient condition for norm equivalence is provided in Section 3. In Section 4, we describe the wavelet method and show that the condition number of the wavelet stiffness matrix is not only relatively small but also uniformly bounded. Finally, some numerical examples are given in Section 5 so as to demonstrate that our wavelet bases are very useful and efficient.

Splines and Approximation Property
In this section, we construct splines which satisfy the homogeneous boundary conditions on the interval [0, 1] and then investigate their some properties.
Let j, d ∈ N and I a countable index set. Suppose that t j := (t j k ) is the sequence such that t j k < t j k+1 for all k ∈ I. The B-spline of order d is given by , . . . , t j k+d ; f (t)] t denotes the dth order divided difference at the points t j k , t j k+1 , . . . , t j k+d , and x + := max{x, 0}, x m + := (x + ) m . From now on, suppose that d = 3, j 2, t j : k=1 is given by Many useful properties of B-spline can be established. For example, there exist complex numbers c k , (k ∈ Z) such that p(x) = ∑ k∈Z c k B j k,3 (x), j ≥ 3, k = 2 j + 3. i.e., a polynomial p whose degree is at most 2 can be represented as a B-spline series.
Moreover, the above properties, we also have the following Lemma.

Proof. The conclusions (i)-(v) can be obtainen easily.
(vi) It is necessary to show that there exist C 1 > 0, C 2 > 0 which are independent of j such that Thanks to the properties of the Rayleigh quotient [6], one has where λ min (G j ) and λ max (G j ) denote the minimal and maximal eigenvalue (in absolute value) of the matrix G j respectively.
To estimate the eigenvalues of G j , one uses Gerschgorin's theorem [7] to obtain that Hence, one obtains i.e., Φ j is a Riesz sequence in L 2 (Ω), and the Riesz bounds are independent of the level j.
(iv) P 2 (Ω) ⊂ V j , where P 2 (Ω) denotes the space of all polynomials p of degree at most 2 on Ω.
Obviously, there exists a positive constant M which is independent of j, such that B j m,3 L 2 (Ω) M. Use the notation Define a family of projector P j : then we have the following theorem.
where 0 < µ < 3 2 . Here and throughout, the notation A B indicates that A ≤ cB with a positive constant c which is independent of A and B. If A B and B A, we call A and B are equivalent, denoted by A ∼ B.
Then we have the following properties.
Theorem 3. For any j 2, one has For all j 2, let Q j be the linear projection from V j+1 onto V j given as follows: for f j+1 ∈ V j+1 , f j := Q j f j+1 is the unique element in V j determined by the interpolation condition [4] . . , 2 j−1 , then ψ j,k ∈ V j+1 , and Q j ψ j,k = 0. Define W j := span Ψ j , then dim W j = 2 j . Moreover, Ψ j is a Riesz basis of W j and its Riesz bounds are independent of j. Since dim V j + dim W j = dim V j+1 , one obtains Suppose that M j := (M j,0 , M j,1 ), then from Lemma 3 (v) and Theorem 3 (iv), one obtains Recursively, one has . . where I j is a 2 j × 2 j unity matrix. Since the determinant of M j is equal to 0.5, one obtains Ψ j+1 := Φ 2 ∪ Ψ 2 ∪ Ψ 3 ∪ · · · ∪ Ψ j is a basis of the space V j+1 .

Characterization Theorem
In this section, we use wavelet bases to characterize Sobolev space H The modulus of continuity of f is given by The mth modulus of smoothness of f is given by Let µ > 0, 1 ≤ p, q ≤ ∞, the Besov space B µ p,q (Ω) is the collection of the functions f ∈ L p (Ω) satisfying where m is the least integer greater than µ. The norm for B µ p,q (Ω) is defined by If p = q = 2, the space B µ 2,2 (R) is the same as the Sobolev space H µ (R), and the semi-norm | · | B µ 2,2 (R) and | · | H µ (R) are equivalent [9]. In the following theorem, we give a characterization of the space H µ 0 (Ω) via the B-spline wavelets constructed before.
(i) First, we show that the left part in (9) is established. Since g j ∈ V j , one obtains Please note that {2 −jµ φ j,k | j 2, k ∈ I j } is a Bessel sequence in space H µ 0 (Ω) [11]. Therefore, Since Φ j is a Riesz bases of V j and the Riesz bounds are independent of j, one obtains ∑ k∈I j |b j,k | 2 C 2 2 g j L 2 (Ω) . Therefore, (ii) Next we show that the right part of inequality (9) is established. For any f ∈ H µ 0 (Ω), according to Poincare inequality, one has For j 3, one obtains From the definition of Besov space, one obtains

Wavelet Preconditioning
In this section, we show that our wavelet bases constructed in the previous section are very useful and efficient.
Let u j ∈ V j such that Recall that Φ j is the base of V j , then the above Equation (10) is equivalent to the systems where A j := (a(φ j,k , φ j,m )) k,m∈I j , f T j := ( f , φ j,m L 2 (Ω) ) m∈I j , u T j := (u j,k ) k∈I j .
However, the following example shows that the condition number of the stiffness matrix A j is almost O(2 2j ) (j → ∞). Hence, the system (11) is very difficult to solve without preconditioning. (1), let the coefficient function q(x) ≡ c, x ∈ Ω. For c ∈ {0, 0.1, 1, 10, 100}, the condition numbers of the matrix A j are shown in Table 1. It is clearly seen that the condition numbers increase exponentially with respect to the level j. In Table 2, the factors between two successive levels are indicated. The numbers show that the growth is by a factor of 4. Moreover, the increase rate is independent on the particular choice of c. To overcome the above difficulty, we use the wavelets preconditioning method. In fact, as we know, the collection of functions Ψ j := Φ 2 ∪ Ψ 2 ∪ Ψ 3 ∪ · · · ∪ Ψ j−1 is also a basis of V j , so we suppose that

Example 1. In Equation
where the vector b T j = {a 2,k , b i,m | k ∈ I 2 , i = 2, . . . , j − 1, m ∈ I i } is the solution of the following equation where the matrix B j := (a(σ, η)) σ,η∈Ψ j , F T j := ( f , η L 2 (Ω) ) η∈Ψ j . In fact, we use the PCG (Preconditioned Conjugate Gradient) algorithm (please see, e.g., [12] pp. 94-95) to solve the system (13), that is where the preconditioner is a diagonal matrix , . . . , 2 j−1 , . . . , 2 j−1 Therefore, the system (13) is equivalent to the following equation where To show the condition number of the matrix B j is uniformly bounded, we give the following example. (1), let the coefficient function q(x) ≡ c, x ∈ Ω. For c ∈ {0, 0.1, 1, 10, 100}, the condition numbers of the matrix B j are shown in Table 3. Comparing with the values in Table 1, it is clearly seen that the condition numbers of the matrix B j are uniformly bounded with respect to the level j. Therefore, using Ψ j as a basis for V j yields an asymptotically preconditioned system (15). However, Figure 1 shows that the matrix B j is not sparse. The so-called finger structure is visible. Now we are facing the following situation: the stiffness matrix A j with respect to the single-scale basis Φ j is sparse but ill conditioned; the wavelet stiffness matrix B j is asymptotically optimal preconditioned, but not sparse.

Example 2. In Equation
Till now, both methods cannot be used immediately. However, where the matrix T j and D j given by (8) (14) respectively are both sparse. Therefore, we know that the matrix Hence B j := D −1 j T j A j T T j D −1 j , F j := D −1 j T j f j , can be expressed as the product of some sparse matrices. that is Equation (15) can be written as In fact, the above Equation (16) combines both positive effects.

Example 4. In Equation
The exact solutions for the above two examples and their numerical solutions with j = 6 are showed respectively in Figure 2. The error estimates in L 2 norm and convergence factor between successive level are given in Table 4 which demonstrates that the growth factor is almost by 4 which is the same as in (17).  Institutional Review Board Statement: Not applicable.

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