Domain decomposition methods for the wave Heimholtz equation

A number of different versions of the domain decomposition method are discussed for solving the Helmholtz equation in the case of an alternating spectrum of the operator. Using a new class of spectrally equivalent operators, it is shown that the convergence rates of the proposed methods have the same order of magnitude as those of similar methods used to solve mesh Poisson's equation. The proposed methods are realized as numerical processes in subspaces by using the partial solutions approach, so that it is possible to obtain estimates of the computational cost; these coincide (again by order of magnitude) with the estimates known for similar methods in the case of Poisson's equation. This paper treats the domain decomposition methods applied to solving systems of mesh equations obtained by approximating the Helmholtz equation in domains partitioned into rectangles. The main distinguishing feature of the problem is the assumption that the spectrum of the Helmholtz operator of the original equation contains both positive and negative eigenvalues. Such systems can obviously be solved by some two-stage iterative method using as an inner iterative process a technique, including the domain decomposition method, for mesh Poisson's equation. The convergence rate of this two-stage iterative method is equal to that of the inner iterative method [4,5]. This approach is traditionally regarded as sufficiently efficient. A large group of competing decomposition methods were suggested in 1978 [9] and later for solving mesh Poisson's equation in domains partitioned into rectangular subdomains [5,6,20]. These methods can be regarded as numerical processes in subspaces [10,11,17,18]; their realization has two distinctive features. First, the realization of these methods requires the storage of only those components of the vectors which correspond to the mesh points that either lie on the sides of the rectangles or are at a distance of, at most, one mesh size from these sides. In other words, it is possible to store only One-dimensional' mesh functions when solving a twodimensional mesh problem. The second distinctive feature of these methods is the use of the partial solutions approach [2,9,13]; as a result, the computational cost per step is proportional to the number of mesh points in the domain, as in the majority of the frequently used iterative methods. The above-described spectacular properties, stemming from the characteristics of the specific domain decomposition chosen for the inner iterative process, cannot be extended to the traditionally constructed two-stage iterative method as a whole. Originally the versions of domain decomposition methods for mesh Helmholtz equations, preserving the two features described in the case of an alternating spectrum, Originally published in Russian as Preprint No. 125 of the Department of Numerical Mathematics of the USSR Academy of Sciences, 1986. 114 Yu. A. Kuznetsov and 0. D. Trufanov were pioneered in our publications [14, 15]. Extension to a more general case met with considerable difficulties. In this paper we suggest a new approach to construct domain decomposition methods for the problem outlined, using the method, proposed in [12], for constructing new classes of spectrally equivalent matrices. The main results are presented in Section 3. Sections 4 and 5 are based on the results of Section 3 and on familiar information from the theory of iterative methods in subspaces and domain decomposition methods used to solve mesh systems with symmetric positive definite matrices. The main result of the paper can be briefly formulated as follows. The partition of the domain into subdomains must be such that the spectrum of the operator of the original Helmholtz equation with specially chosen boundary conditions (which are generally different in different subdomains) have only positive eigenvalues in each subdomain. The subsequent sequence of steps of constructing the method is nearly identical to that required for operators with a non-alternating spectrum, although some specificities remain. 1. DIFFERENTIAL PROBLEM Let a polygonal domain Ω, such that each linear segment of its boundary <9Ω = Ω\Ω is parallel to one of the coordinate axes, be given in a plane E2 with Cartesian coordinates xi and x2. We partition Ω into ρ pairwise disjoint rectangular elements Ω, (t = 1, . . . , p) with sides parallel to the coordinate axes. Therefore, Ω = (Jf= ̂ and Ω, ηΩ5 = 0 for s φ t. Such partitions are possible for all sufficiently large values of p. Now fix on δΩ two disjoint sets Γ1 and Γ2 such that Γι u Γ2 = δΩ, Γ\ being closed and consisting of a finite number of segments, and Γ2 being open and also consisting of a finite number of linear open segments. We assume that the set 3Ω\Γ, where Γ = Γ! u Γ2, belongs to the set of vertices of the rectangles Ω, (t = 1, . . . , p). Consider a wave Helmholtz equation Au + cou=-/ in Ω (1.1) with homogeneous boundary conditions du w = 0 on Γ! , ^ = 0 on Γ2. (1.2) ov Here, ν is the vector of the outward normal to <5Ω at points in Γ2, and feL2(Q) is a given function. We demand that in each rectangle in Ω, the function ω = ω(χ), χ = (x1? x2), equal the sum of two positive piecewise-continuous functions in one variable, ω(χ) = ω,(χ) = ω^(χι) + co2tt(x2)9 χεΩ, (1.3) and ίηίΩί cot(x) > Ο, £ = 1,. , p. Obviously, the formulation of problem (1.1), (1.2) includes the form of ω but not the partition of the domain Ω into rectangles. Consequently, the assumption imposed on ω implies an additional condition on the construction of a required partition of Ω into rectangular elements, provided such a partition is at all possible. Define the space a),v = Q on Γ,} (1.4) Domain decomposition methods 115 as a subspace of the Sobolev space Η^2(Ω) and also a bilinear form f jSf(ti,t? )= [VwVi? an«;] d£l, u.veW^r^). (1.5) Jn Instead of problem (1.1), (1.2), we consider the following variational problem: find Ί)> h that \). (1.6) Here ( , ) is the usual scalar product in ί>2(Ω). Throughout the paper, we assume that the following conditions are satisfied. Condition 1. At least one eigenvalue λ of the eigenvalue problem Δι; -tων = — λων in Ω υ = 0 on Γ! (1.7) -^ = Ο on Γ2 δν is negative, and zero is not an eigenvalue of problem (1.7). Note 1. Under the condition as stated, problem (1.6) always has a unique solution. Note 2. For the sake of convenience, we enumerate the eigenvalues of problem (1.7) in the following order: where q is a positive integer. Condition 2. Problems Δι; + cov = — / in Ω, υ-O on rltl (1.8) dv ^ — = 0 on Γ2 §l where δΩ, = Ω,\Ω,, ), r2,f = (^,nr2)\ru (1.9) and ν, is the vector of the outward normal to Γ2 „ t — l,. ,p, can be solved by the conventional separation-of-variables method. Note 3. Figure 1 shows three partitions of an L-shaped domain Ω with δΩ = Γ2 into rectangles. The partitions (a) and (b) do not satisfy Condition 2 because problem (1.8), (1.9) cannot be solved for the rectangles Ωχ by the method of separation of variables (both the Dirichlet and the Neumann boundary condition are imposed at the right vertical side of Ω^. Partition (c) satisfies Condition 2. Note that Condition 2 is obligatory for the methods discussed in Section 4. We will change this condition for the method presented in Section 5. 116 Υ u. A. Kuznetsov and 0. D. Trufanov