A DN Alternating Algorithm for Solving 3 D Exterior Helmholtz Problems

The nonoverlapping domain decomposition method, which is based on the natural boundary reduction, is applied to solve the exterior Helmholtz problem over a three-dimensional domain. The basic idea is to introduce a spherical artificial boundary; the original unbounded domain is changed into a bounded subdomain and a typical unbounded region; then, a Dirichlet-Nuemann (D-N) alternating method is presented; the finite element method and natural boundary element methods are alternately applied to solve the problems in the bounded subdomain and the typical unbounded subdomain. The convergence of the D-N alternating algorithm and its discretization are studied. Some numerical experiments are presented to show the performance of this method.


Introduction
Many scientific and engineering problems can be reduced to exterior boundary value problems of partial differential equations.Although the numerical methods for solving boundary value problems, such as the finite element method and the finite difference method, are very effective on a bounded domain, yet we often find it 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] and references therein for more details.
There have been studied for Dirichlet-Nuemann alternating algorithm, or D-N alternating algorithm on 2D unbounded domain problems extensively, such as Possion equation [4] and Helmholtz equation [5].Wu and Yu [6] studied the natural integral equations of 3D Helmholtz problems.Jia et al. [7] investigated the coupled natural boundary element-finite element method for solving 3D exterior Helmholtz problem.
In this paper, based on the natural boundary reduction, a D-N alternating algorithm is devised for the numerical solution of three dimensional Helmholtz problem in an infinite region with an inner spheroid boundary.Firstly, we turn the D-N alternating algorithm into the Richardson iterative algorithm which is equivalent to the original method.Secondly, we prove the convergence of algorithm in the general exterior domain.Then, we give the weak form and discretization of the original equations and prove the convergence of the discretization form.Finally, some numerical examples are presented to illustrate the feasibility and efficiency of this method.

D-N Alternating Algorithm
Let (, , ) denote the spherical coordinates; we consider the following 3D exterior Helmholtz problem: where Ω is a bounded domain in R 3 , Ω  = R 3 \ Ω, and Γ  = Ω. denotes the wave number related to the wavelength  of the incident wave through  = 2/.And assume that  is a known function.In order to assure the existence and uniqueness of the solution of (1), problem (1) satisfies the 2

Mathematical Problems in Engineering
Sommerfeld radiation condition at infinity, which imposes that the scattered wave is outgoing: where i = √ −1,  = |x| = √ 2 +  2 +  2 .Take a sphere ( (iv) Step 4. Input   , and let where   1 and   2 are the th approximate solutions in Ω int and Ω ext , respectively. 1 and  2 denote the unit outward normals of Γ 1 with respect to two neighboring subdomains;   denotes the th relaxation factor and  0 is an arbitrary function in  1/2 (Γ 1 ).Note that, on interface Γ 1 , only the value of the normal derivative of the solution of (3) is needed in solving (4).So it is unnecessary to solve (3).Actually, we can obtain   2 / 2 directly from   by making use of the following natural integral equation [6]: where K is the natural integral operator of Helmholtz equation in Ω ext : Λ      (, ) , (7) ℎ  (⋅) is the first-kind Hankel function of order  and and    (⋅) denotes the associated Legendre function of the first kind,    (⋅) denotes the complex conjugate of    (⋅).

Equivalent Iterative Method
To discuss the convergence of the D-N alternating algorithm in Section 2, we first establish a law of deciding.Let  be the exact solution of (1).Denote  =  0  = | Γ 1 ,  1 = | Ω int ,  2 = | Ω ext ; it is easy to know  ∈  1/2 (Γ 1 ), and  1 and  2 are solutions of the following equations, respectively: By Green formula, we have According to superimposition principle, the solutions of ( 9) and ( 10) can be denoted where T   ( = 1, 2) satisfy, respectively, It is easy to verify that Extend elements of Ω 1 and Ω 2 by zeros to R 1  and R 2 , respectively; therefore R 1  satisfies while R 2  satisfies where R 2 is the Possion integral operator of Helmholtz equation in the exterior spherical domain Ω ext .Define S 2 is K, which is the natural integral operator of Helmholtz equation in the exterior spherical domain Ω ext .Substituting (12) in (11), we have Namely, Define the right-hand term in (19)  = −T 1 / 1 ; obviously, , a function independent of , can be solved independently by the problem in subdomains in advance.Then, ( 19) is equivalent to equation where Then, Proof.We note that  1/2 (Γ 1 ) ⊂  2 (Γ 1 ) ⊂  −1/2 (Γ 1 ); therefore, for any ,  ∈  1/2 (Γ 1 ) ⊂  2 (Γ 1 ).By Green formula, we know where This proves S  is symmetric positive operator.
Corollary 2. There exists S −1  , which is the inverse operator of S  .
Corollary 3. The bilinear form   (⋅, ⋅) on a normed linear space   is said to be coercive on  1/2 (Γ 1 ); that is, there exists a positive constant  such that where By Poincaré-Steklov operator, solving the original problem ( 1) is rewritten as solving the operator equation (20).Generally speaking, it is easier to solve S −1 1 than to solve the inverse of S. In fact, is equivalent to finding  1 ∈  1 , such that Let It is easy to verify that Now we introduce S 1 as preprocessor and preconditioned Richardson iterative method.Given  0 ∈  1/2 (Γ 1 ), the procedure of iterative where Following [9], to analyze the convergence of algorithm ( 30), what we need to do is just to estimate the eigenvalues of operator S −1 1 S, namely, to estimate the upper bound and the lower bound of ⟨S, ⟩/⟨S 1 , ⟩.Lemma 4.There exist two positive constants  and , such that Particularly, optimal relaxation factor is   = 2/( + ).
, and let V  =  −   .Following ( 3)-( 10), we have Meanwhile, Substituting the above equation to the third equation of (34), we can obtain Therefore, It comes that Substituting the above equation in (39), we have
For (, ) ∈  1/2 (Γ 1 ), in the  2 -sense, the following expansion holds where    () denotes the associated Legendre function of the first kind;    (, ) denotes the complex conjugate of    .On the one hand, satisfies on the other hand, set Following [11] and (48), we have lim According to Green formula and orthogonality of    (, ), it comes that Theorem 10.If relaxation factor   satisfies then the D-N alternating algorithm (3)-( 5) converges.Particularly, the optimal relaxation factor is while the optimal compression ratio of the iteration is Remark 11.We have analyzed the convergence for exterior spherical domain.Following [10], the convergence analysis for the general exterior domain can be extended from above content.Obviously, the optimal relaxation factor and optimal compression ratio of the iteration depend on  1 and the geometry of Ω  in some way, which is confirmed by our numerical results.
In the following, we consider the discretization of this weak form.Discretize Ω int into a finite number of element domains.Let  ℎ (Ω int ) denote the linear subspace of  1 0 (Ω int ) corresponding to this partition.Define  0 ,  1 , and   to be the sets of all nodes belonging to Γ  , Γ 1 , and Ω int , respectively.Denote    the approximate value and   (x) a basis function at node .Obviously, The finite element approximation of   1 can be expressed as By (61) and ( 5), we can obtain the following discrete D-N alternating algorithm: where ,  = 0, 1, ,

Convergence Analysis for the Discretization Form
In the following, we consider the convergence of the discretization form.

Numerical Examples
To test the effectiveness of the method in this paper, we give two numerical examples, using the discrete D-N alternating algorithm in Section 2. In the two examples, the exact solutions are known.The purpose of these examples is to check the convergence in terms of iteration and mesh size ℎ.Suppose that Split the bounded domain Ω int into finite element mesh as follows.Firstly, make every inner boundary edge into  1 equivalent parts and connect nodes by using segments which are parallel to the corresponding coordinate axis; then, the grid located in the inner boundary Γ 1 is generated.Secondly, along the radial direction, rays starting from each node of located in the inner boundary intersect artificial boundaries at some nodes, and all the other nodes are produced by dividing segments along the radial direction into  2 equivalent parts.At last, we get an eightnode trilinear isoparametric finite elements of Ω int .
Denote  the maximum node-error on Ω  : We substitute ∑  =0 for ∑ ∞ =0 in the computing of the entries of  ℎ .Denote by  the total number of nodes on Ω int .By computing, the results are as follows.

Table 3 :
The relation between convergence rate and relaxation factor.