Abstract
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.
1. 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. [1–4]) 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.
2. Problem and Its Equivalent Form
We consider the following 3D exterior Helmholtz problem [12]:where is a bounded domain in with regular boundary . 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 . In order to assure the existence and uniqueness of the solution of (1), the solution satisfies the Sommerfeld radiation condition:where denotes the spherical coordinates, , . Condition (2) asserts that the scattered wave is outgoing at infinity.
The corresponding variational form of problems (1)-(2) iswhere
Particularly, if 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 is called the Poisson integral operator of Helmholtz equation in , is the first kind of Hankel function, denotes the associated Legendre function of the first kind, denotes the complex conjugate of , 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 is called the natural integral operator of Helmholtz equation in .
3. Schwartz Alternating Algorithm and Its Convergence
Taking two sphere surfaces and , such that and is surrounded by , denote
Construct the following Schwartz alternating algorithm.
Step 1. Select the initial and let , .
Step 2. Solve the problem in :
Step 3. Solve the problem in :
Step 4. Let .
Step 5. Let , and go to Step 2.
Where and are the th and th approximate solutions in and , respectively, is an arbitrary function in . Note that, on interface , only is needed. So it is unnecessary to solve (10). Actually we can obtain by making use of the following Poisson integral equation: where is the trace operator on . Because of and , and can be regarded as the subspaces of , if elements of are extended by zero. Let and ; obviously , , and . (9) and (10) are, respectively, equivalent to the following variational forms: denotes projected to ; change (12) intoBy the definition of projection, we obtainThis is equivalent towhere is the orthogonal complementary space of ; that is, Let denote the errors. Then (15) can be rewritten asThis yieldsIt is easy to know that if are convergent, then their limits are in .
Lemma 1. , .
Lemma 2. and for any there exists a positive constant such thatwhere denotes energy norm.
Proof. If , there exists a positive constant such that Simultaneously,
Theorem 3. Suppose that ; then
Proof. Following (15), we have which means thatLet and be weakly convergent subsequence, so Following (24), we can obtain Following (17), we have namely
Theorem 4. There exists a constant , such thathold true.
Proof. Substituting into (19), we have Equivalently where namely Similarly Following (17) and (18), it comes that By recursion we obtain Similarly, we obtain
4. Analysis of Convergence Rate
Absolutely, the convergence rate of the above Schwartz alternating algorithm is closely related to the overlapping degree of and . 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 . However, under certain assumptions, we can find out the relationship between contraction factor and overlapping degree of and .
Consider the following boundary value problem in domain :
Lemma 5. If and , then is the solution of (38).where
Theorem 6. If we apply the Schwartz alternating algorithm given in Section 3 to problems (1)-(2) and , then hold true, where constants and are independent of , while
Proof. On one hand, for , we have where Following [9], we obtainOn the other hand, for , we have , while where According to Lemma 5, it comes thatwhere Suppose that ; we haveFollowing trace theorem, we haveBy Lemma 5 again, we obtainBy induction, it comes thatDenoting(53) is changed intoClearlynoting thatFollowing (56) and (57), there exists independent of , such that Similarly, we obtain It can be seen from Theorem 6 that the larger the overlapping part of and is, the smaller the contraction factor is and, consequently, the faster the Schwartz alternating algorithm converges.
5. Error Estimates of Algorithms
Subdivide into hexahedrons. Let denote the linear finite element space over . Putting can be regarded as the subspace of if its elements are extended by zero. We first establish the following discrete Schwartz alternating algorithm:withBy the Poisson integral formula, the solution of (62) can be given aswhere denotes the Dirichlet trace operator and denotes the Poisson integral operator. It is easy to verify that and the term corresponding to vanishes while . Define Then, extending the elements of 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.
Theorem 7. For the discrete Schwartz alternating algorithms (61)-(62) and the constant in Theorem 4, there hold the following error estimatesSince , (70) also hold true if the norm in their left hand sides is replaced by the norm of .
6. Numerical Examples
To test the effectiveness of the method in this paper, we give some numerical examples, using the discrete Schwartz alternating algorithm in Section 5. In all examples, the exact solutions are known. The purpose of these examples is to check the convergence in terms of iteration and mesh size .
Suppose that is the exterior domain of sphere . The artificial boundary is , . Split the bounded domain into finite element mesh as follows. We subdivide the bounded domain using the following family of planes , , and , where is a positive integer. Now we obtain a uniform hexahedrons partition of with mesh size of . Let be the piecewise trilinear finite element space on . Substitute for (where denotes the interpolation basis function corresponding to node ).
Denote the maximum node–error on denotes the maximum node-error of the adjacent two-steps on nodes is the convergence rate . We substitute for in the computing of the entries of stiffness matrix.
Example 1. We take that the exact solution of problem (1) is where
Let , , , , and ; the numerical results are shown in Tables 1 and 2.
Example 2. We take that the exact solution of problem (1) is where
Let , , , , and ; the numerical results are shown in Tables 3 and 4.
As can be seen from Tables 3 and 4, the discrete Schwartz alternating algorithm is geometrically convergent and is nearly not affected by mesh parameter . With the increase of , 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.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The research is partly supported by the National Natural Sciences Foundation of China (Contact/Grant nos. 11401296 and 11371198), Jiangsu Provincial Natural Science Foundation of China (Contact/Grant no. BK20141008), and the Natural Science Fund for Colleges and Universities in Jiangsu Province (Contact/Grant no. 14KJB110007). And it is also partly sponsored by Qing Lan Project.