A Survey of Error Estimates for Iterative Solutions of Systems of Linear Equations with an Application to the Solution of Poisson's Equation

AbstractWe consider the iterative solution of the discrete Poisson’s equation with Dirichlet boundary conditions. The discrete domain is embedded into an extended domain and the resulting system of linear equations is solved using a fixed point iteration combined with a multilevel circulant preconditioner. Our numerical results show that the rate of convergence is independent of the grid’s step sizes and of the number of spatial dimensions, despite the fact that the iteration operator is not bounded as the grid is refined. The embedding technique and the preconditioner is derived with inspiration from theory of boundary integral equations. The same theory is used to explain the behaviour of the preconditioned iterative method.

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New error estimates of Adini's elements for Poisson's equation

Applied Numerical Mathematics ◽

10.1016/j.apnum.2003.10.009 ◽

2004 ◽

Vol 50 (1) ◽

pp. 49-74 ◽

Cited By ~ 7

Author(s):

Hung-Tsai Huang ◽

Zi-Cai Li ◽

Ningning Yan

Keyword(s):

Error Estimates ◽

Poisson’S Equation ◽

Poisson's Equation

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Algorithms for Solving Systems of Linear Equations in the Field Fpk

Journal of Automation and Information Sciences ◽

10.1615/jautomatinfscien.v51.i10.10 ◽

2019 ◽

Vol 51 (10) ◽

pp. 1-22

Author(s):

Sergey L. Kryvyi ◽

H. I. Hoherchak

Keyword(s):

Linear Equations ◽

Systems Of Linear Equations

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On Some Properties of Semi-rings

Mechanical Engineering and Computer Science ◽

10.24108/0318.0001379 ◽

2018 ◽

pp. 35-50

Author(s):

A. I. Belousov

Keyword(s):

Linear Equations ◽

Oriented Graph ◽

Theoretical Computer Science ◽

Alternative Methods ◽

Main Role ◽

Cost Matrix ◽

Sequential Elimination ◽

The Matrix ◽

Systems Of Linear Equations ◽

The Cost

The main objective of this paper is to prove a theorem according to which a method of successive elimination of unknowns in the solution of systems of linear equations in the semi-rings with iteration gives the really smallest solution of the system. The proof is based on the graph interpretation of the system and establishes a relationship between the method of sequential elimination of unknowns and the method for calculating a cost matrix of a labeled oriented graph using the method of sequential calculation of cost matrices following the paths of increasing ranks. Along with that, and in terms of preparing for the proof of the main theorem, we consider the following important properties of the closed semi-rings and semi-rings with iteration.We prove the properties of an infinite sum (a supremum of the sequence in natural ordering of an idempotent semi-ring). In particular, the proof of the continuity of the addition operation is much simpler than in the known issues, which is the basis for the well-known algorithm for solving a linear equation in a semi-ring with iteration.Next, we prove a theorem on the closeness of semi-rings with iteration with respect to solutions of the systems of linear equations. We also give a detailed proof of the theorem of the cost matrix of an oriented graph labeled above a semi-ring as an iteration of the matrix of arc labels.The concept of an automaton over a semi-ring is introduced, which, unlike the usual labeled oriented graph, has a distinguished "final" vertex with a zero out-degree.All of the foregoing provides a basis for the proof of the main theorem, in which the concept of an automaton over a semi-ring plays the main role.The article's results are scientifically and methodologically valuable. The proposed proof of the main theorem allows us to relate two alternative methods for calculating the cost matrix of a labeled oriented graph, and the proposed proofs of already known statements can be useful in presenting the elements of the theory of semi-rings that plays an important role in mathematical studies of students majoring in software technologies and theoretical computer science.

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