The paper treats a version of the block bordering method for solving elliptic equations with piecewise-constant coefficients in domains composed of rectangles. An iterative process whose convergence is independent of the mesh step size is constructed, and the number of arithmetic operations required for realization of the method with a given accuracy is estimated.For the sake of simplicity, let us consider the following problem: ueW 2 (fl) Ln(u,9) where /eL 2 (Q), and Ω is a rectangle composed of three squares Ω ί? Ω = Uf el £ :Ω={(χ,3θ| 0
This paper suggests a technique for constructing a preconditioner for grid elliptic problems with the mixed boundary condition (the main boundary condition is imposed on a part of the boundary and the natural boundary condition is imposed on the remaining part). Here, we have managed to construct for domains with the piecewise-smooth boundary such a preconditioner that the convergence rate of the corresponding iterative processes is independent of the parameter of discretization of the original differential problem. The main technique for constructing preconditioners is the method of splitting the original functional space into a vector sum of subspaces with the subsequent construction of preconditioners in these subspaces. This paper is aimed at constructing the preconditioner B for solving systems of grid equations approximating elliptic boundary value problems in domains with complex geometry. The preconditioner B can be used, for example, in iterative processes of the following form:where A is the coefficient matrix of the original system of grid equations. The convergence rate of iterative process (0.1) is dependent on the constants c and c from the inequalitiesvalid for any vector (A is considered to be a symmetric and positive definite matrix). In [5], a technique has been suggested for constructing the preconditioner B in the case of uniform grids in the rectangle. In addition, the constants c and c from (0.2) are independent of /ι, and to perform multiplication B~l by the vector, it is necessary to solve the system of grid equations corresponding to the five-point approximation of the Laplace operator in the rectangle on the uniform grid. Of interest is the construction of the preconditioner Β with similar characteristics in the case of boundary value problems in domains with complex geometry. The most efficient preconditioners for solving boundary value problems in domains with complex geometry can be constructed, as a rule, by 'simplifying' geometry of the original domain. Here, we can point out two approaches. The first approach consists in partitioning the original domain into simpler subdomains (domain decomposition methods), and the second one consists in embedding the original domain into a domain of the canonical form, for example, into the rectangle in the two-dimensional case and into the parallelepiped in the three-dimensional case and in introducing additional equations (the fictitious domain method and its matrix counterparts) [3,4,6,9,10,12,[23][24][25]. In the second group of methods, major gains have been obtained for problems with the natural boundary condition [1].Using the matrix counterpart of the fictitious domain method we have managed to construct the preconditioner Β such that the constants c and c from (0.2) are independent of /z, and the main operation in performing multiplication of B~l by the Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 7/7/15 11:03 PM 152 5. V. Nepomnyaschikh vector consists in solving the five-point f...
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