Reduktionsverfahren f�r Differenzengleichungen bei Randwertaufgaben I

Abstract

Summary. This paper describes a fast and numerically stable method for solving the discrete Dirichlet problem for Poisson's equation in case of a rectangle (and mainly, a square). By using a special calculus for difference operators, the system of linear equations is reduced to a block-triangular system such that the diagonal blocks are heavily diagonally dominant. For a standard version of the algorithm, the number of operations and the computing time are proportional to h -~ (h =mesh width). The method is one of total reduction compared with the method of block-cyclic reduction (odd-even reduction) [2], which we describe as a method of partial reduction.-Due to the developed calculus, many generalizations are possible.-In a following part II of the paper, the algorithm and numerical results will be described in detail.

Direct Multi-Grid Methods for Linear Systems With Harmonic Aliasing Patterns

“…When solving Poisson's equation, this solution is equivalent to the total reduction method by Schröder and Trottenberg [7], [8]. The derivation of total reduction methods was based on the structure of a stationary impulse response for LSI systems and thus imposes stronger assumptions on the system.…”

“…8a changes from a W-cycle into a V-cycle [16], reducing the computational complexity from O(n log n) to O(n). This is actually what happens when solving Poisson's equation, where this method is equivalent to total reduction methods [7], [8]. In general, this depends on the structure of the system and it will not be study here in depth.…”

“…11. The same problem appears in other direct multi-grid solvers like total and partial (or cyclic) reduction methods [7]- [9]. , is solved in a square domain with periodic boundary conditions for two source vectors.…”

“…In the same category there are other direct multi-level solvers like: total reduction methods [7], [8], partial (cyclic) reduction methods [9] and LU factorization of non-standard forms [10]. Besides their structural differences, each method works under certain limitations.…”

Approximate Cyclic Reduction Preconditioning

“…This method can be used, for example, for solving a linear system with a tridiagonal matrix or with a special block tridiagonal matrix (d. [14], [17], [25], [26]). We explain the cyclic reduction principle by considering an n x n linear system with a tridiagonal matrix:…”