Two-stage fictitious components method for solving the Dirichlet boundary value problem

The Dirichlet boundary value problem is considered for an elliptic equation with piecewisesmooth coefficients in twoand three-dimensional domains with complex internal and external curvilinear boundaries. To approximate the boundary value problem, the standard finite element method is used on rectangular meshes locally adapted to the boundaries. Systems of mesh equations arised are solved by a two-stage iterative method. This method involves the use of spectrally equivalent operators with constant coefficients as an outer iterative procedure and of the nonsymmetric version of the fictitious components method as an inner iterative procedure. The paper contains convergence rate estimates for the method discussed, proposes algorithms of its realization as a computational process in a subspace, gives estimates for the arithmetic and communication complexity of the algorithms suggested. The paper ends with the results of a numerical experiment to solve a specific three-dimensional problem of electrostatics. The fictitious components method has already a 15-year long history. Symmetric versions of this method were independently proposed in [2] and [17]. These papers also contained some convergence rate estimates for the symmetric fictitious components method. Especially note that the technique proposed in [2] and then developed in [3] for estimating the convergence rate of the symmetric fictitious components method for the Neumann boundary value problems makes up the foundation of present-day investigations in convergence of fictitious components methods and of many versions of domain decomposition methods. This technique is based on the theorems on continuation and traces of mesh functions, which are mesh analogues of the corresponding theorems in the theory of functions and functional analysis. The proof of such theorems has been given recently, for example, in [7,24,30]. The convergence rate estimates for the symmetric version of the fictitious components method for elliptic problems with the Dirichlet boundary conditions were obtained in [20,22]. The results of these investigations are used in this paper. The nonsymmetric version of the fictitious components method was independently proposed and investigated for solving elliptic problems with the Dirichlet boundary conditions in [11] and [23]. In [23], the author also established its close relation (duality, in a certain sense) to the symmetric fictitious components method for the Neumann boundary conditions. It was shown that the convergence rate estimates for the method constructed were implied by [2,3] and were also independent of the mesh size. Note that the symmetric and nonsymmetric versions of the fictitious components method are very close to capacitance matrix methods [4,25-27] by their idea, technique of constructing convergence rate estimates and realization algorithms. The two approaches have considerably enriched each other recently both in theory and in realization algorithms and applications. This paper is devoted to further development of theory and to applications of the fictitious components method (the capacitance matrix method) for elliptic problems 302 S. A. Finogenov and Yu. A. Kuznetsov with the Dirichlet boundary conditions for domains with curvilinear boundaries. It is a continuation of the previous investigations [13,14,19] made by the authors. The distinctive feature of these investigations lies in considering the fictitious components method for equations with piecewise-constant coefficients as a computational process in a subspace of a considerably smaller dimension as compared with the dimension of the original algebraic problem. The practical foundation for the realization of such computational processes is made up by the partial solution algorithms [5,13,15]. In Section 1 of this paper, the nonsymmetric fictitious components method is considered as an inner iterative procedure of another iterative method. The matrix properties which will be later used as preconditioners for an outer iterative method are investigated, and the estimates are given for the arithmetic and communication complexity of the algorithms. The results obtained in Section 1 are used for the construction of a two-stage iterative method for two-dimensional elliptic problems in Section 2 and for three-dimensional problems in Section 3. Algorithms for realizing two-stage processes are also discussed and the estimates of their complexity are given therein. Section 4 contains the results of a numerical experiment to solve a threedimensional problem with complex internal and external boundaries. 1. FICTITIOUS COMPONENTS METHOD Let Ω be a two-dimensional polygonal domain with the boundary ΟΩ. According to [8], let us consider the following Dirichlet boundary value problem: for a given geL2(il) find νεΗ^Ω) such that (1.1) Vi?°Vwdu= Jn Jo Embed Ω into a rectangle Π with the sides parallel to the coordinate axes, denote by ΟΠ the boundary of Π, and define a domain G = Π\Ω with the boundary dG. Construct in Π a rectangular, in general, non-uniform mesh ΠΛ assuming that for each mesh cell the ratio of the radius of the circle circumscribed to that of the circle inscribed is bounded from above by a mesh-independent constant. Then, partitioning the rectangular cells of the mesh ΠΛ into triangles, as shown, for example, in Fig. 1, Figure 1. A case of the mesh ΠΛ. Two-stage fictitious components method 303 construct a triangular mesh ΠΛ, and also triangular meshes ΩΛ = ΠήπΩ and Let us define 7Λ(Ω) as a space of functions continuous in Ω, linear in each triangle ΩΛ and vanishing on 5Ω. Consider the following approximate problem [8]: find ι/ΈΚΛ(Ω) such that Vv°Vwd&= gwdu VweVh(U). (1.2) Jn Jn This problem leads to a system of linear algebraic equations