The Schwartz Alternating Algorithm for Solving 3 D Exterior Helmholtz Problems

Based on the natural boundary reduction, an overlapping domain decomposition method is discussed for solving exterior Helmholtz problem over a three-dimensional (3D) domain. By introducing two different artificial boundaries, the original unbounded domain is divided into a bounded subdomain and a typical unbounded region, and a Schwartz alternating method is presented. The finite element method and natural boundary element method are alternately applied to solve the problems in the bounded subdomain and the typical unbounded subdomain. Moreover, the convergence of the Schwartz alternating algorithm is studied. Finally, some numerical experiments are presented to show the performance of this method.


Introduction
When solving a problem modelled by a linear partial differential equation in the bounded or unbounded domain, many existing methods can be adopted.For the problem in the bounded domain, the finite element method and the finite difference method and so on are very effective.Yet we often find them difficult to be applied to unbounded problem directly.To solve such problems in infinite region numerically, there are a variety of numerical methods (cf.[1][2][3][4]) and references therein for more details.
Schwartz alternating algorithm is one of the most efficient techniques for solving problems in the unbounded domain, such as harmonic equation [5,6], Stokes equation [7], and Helmholtz equation [8].Over exterior 3D domain, Wu and Yu studied the natural integral equations of Helmholtz problems [9] and overlapping domain decomposition method for harmonic equation [10].We gave a D-N alternating algorithm for solving 3D exterior Helmholtz problems [11].
In this paper, a Schwartz alternating algorithm based on the natural boundary reduction is devised for the numerical solution of exterior three-dimensional Helmholtz problem.Firstly, an exterior Helmholtz problem is introduced and the corresponding variational form is obtained.Secondly, the Schwartz alternating algorithm is posed.Then, the convergence rate is discussed and the relationship between contraction factor  and overlapping extent is given.Besides, the error estimate of the algorithm is offered.Finally, some numerical examples are presented to illustrate the feasibility and efficiency of this method.

Problem and Its Equivalent Form
We consider the following 3D exterior Helmholtz problem [12]: where Ω is a bounded domain in R 3 with regular boundary Ω = Γ 0 .(x) is the unknown function and  is a known function, which satisfies appropriate conditions. denotes the wave number, related to the wavelength  of the incident wave through  = 2/.In order to assure the existence and uniqueness of the solution of (1), the solution (x) satisfies the Sommerfeld radiation condition: where (, , ) denotes the spherical coordinates,  = |x| = √ 2 +  2 +  2 , i = √ −1.Condition (2) asserts that the scattered wave is outgoing at infinity.The corresponding variational form of problems ( 1)-( 2) is where Particularly, if Ω is a spherical domain with radius  whose centre is the origin of coordinates, the solution of ( 1)-( 2) is given by the following Poisson integral formula: where P is called the Poisson integral operator of Helmholtz equation in Ω  , ℎ ℓ () = √/2 (1)  +1/2 () is the first kind of Hankel function,   ℓ (⋅) denotes the associated Legendre function of the first kind,   ℓ (⋅) denotes the complex conjugate of   ℓ (⋅), and By using (5), we will develop an overlapping domain decomposition method (Schwartz alternating algorithm based on the natural boundary reduction) for problems (1)- (2).Taking the normal derivative of (5), we have the following natural integral equation: where K is called the natural integral operator of Helmholtz equation in Ω  .
Actually we can obtain  (+1) by making use of the following Poisson integral equation: where  2 is the trace operator on Γ 2 .Because of  (2)  9) and (10) are, respectively, equivalent to the following variational forms: denotes  projected to   ; change (12) into By the definition of projection, we obtain This is equivalent to where  ⊥  is the orthogonal complementary space of   ; that is, Let ( = 1, 2) denote the errors.Then (15) can be rewritten as This yields It is easy to know that if Lemma 2.  =  1 + 2 and for any V ∈  there exists a positive constant  0 such that where | ⋅ | 1 = √(⋅, ⋅) denotes energy norm.

Analysis of Convergence Rate
Absolutely, the convergence rate of the above Schwartz alternating algorithm is closely related to the overlapping degree of Ω 1 and Ω 2 .Although it can be deduced intuitively that the larger the overlapping part is, the faster the convergence rate will be; yet we find it difficult to analyse the convergence rate for general unbounded domain Ω  .However, under certain assumptions, we can find out the relationship between contraction factor  and overlapping degree of Ω 1 and Ω 2 .
Consider the following boundary value problem in domain Ω 1 : where  2) and hold true, where constants  1 and  2 are independent of , while Proof.On one hand, for  0 ∈  1/2 (Γ 0 ), we have where Following [9], we obtain On the other hand, for ũ ∈  1 0 (Ω  ), we have ũ| where According to Lemma 5, it comes that where Suppose that  ∈  − ; we have Following trace theorem, we have By Lemma 5 again, we obtain By induction, it comes that Denoting Clearly noting that Following ( 56) and (57), there exists  1 independent of , such that Similarly, we obtain It can be seen from Theorem 6 that the larger the overlapping part of Ω 1 and Ω 2 is, the smaller the contraction factor  is and, consequently, the faster the Schwartz alternating algorithm converges.

Error Estimates of Algorithms
Subdivide Ω 1 into hexahedrons.Let  ℎ (Ω 1 ) denote the linear finite element space over Ω 1 .Putting Mathematical Problems in Engineering ∘  ℎ (Ω 1 ) can be regarded as the subspace of  if its elements are extended by zero.We first establish the following discrete Schwartz alternating algorithm: Find with By the Poisson integral formula, the solution of (62) can be given as where  2 :  1 (Ω 1 ) →  1/2 (Γ 2 ) denotes the Dirichlet trace operator and P :  1/2 (Γ 2 ) →  1 0 (Ω 2 ) denotes the Poisson integral operator.It is easy to verify that and the term corresponding to ∑ −1 =0 vanishes while  = 0. Define Then, extending the elements of  ℎ (Ω 2 ) by zero, we have Now we introduce the following variational problem: Obviously, (68) exists as a unique solution.Assume that Similarly in [10], we have the following error estimates.
Since  −  () ℎ ∈ , (70) also hold true if the norm in their left hand sides is replaced by the norm of  1 0 (Ω).
As can be seen from Tables 3 and 4, the discrete Schwartz alternating algorithm is geometrically convergent and  ℎ () is nearly not affected by mesh parameter ℎ.With the increase of  2 ,  ℎ () is smaller.So the larger the overlapping part is, the faster the convergence rate will be.All these are in accord with the theoretical analyses.