# Preconditioning analysis of nonuniform incremental unknowns method for two dimensional elliptic problems

@article{Yang2015PreconditioningAO, title={Preconditioning analysis of nonuniform incremental unknowns method for two dimensional elliptic problems}, author={Ai-Li Yang and Yujiang Wu and Zhengda Huang and Jinyun Yuan}, journal={Applied Mathematical Modelling}, year={2015}, volume={39}, pages={5436-5451} }

Abstract For the linear system obtained by discretizing two dimensional elliptic boundary value problems on nonuniform meshes, the condition number of the coefficient matrix preconditioned by nonuniform incremental unknowns (NUIUs) method, abbreviated as NUIUs matrix, is carefully analyzed. Comparing to the original coefficient matrix, the condition number of the NUIUs matrix is reduced from O ( a d ) to O ( d 2 ) with a ≥ 4 and d being the level of discretization. Numerical experiments are… Expand

#### One Citation

A non-alternating preconditioned HSS iteration method for non-Hermitian positive definite linear systems

- Mathematics
- 2017

By utilizing the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration technique, we establish a non-alternating PHSS (NPHSS) iteration method for solving large sparse non-Hermitian… Expand

#### References

SHOWING 1-10 OF 38 REFERENCES

Algebraic preconditioning analysis of the multilevel block incremental unknowns method for anisotropic elliptic operators

- Computer Science, Mathematics
- Math. Comput. Model.
- 2013

A diagonal preconditioner is introduced such that the condition number of the BIU matrix is further reduced to O ( 1 + 4 e π − 2 h − 2 ) , which means that thecondition number is optimal if e = O ( h 2 ) . Expand

Algebraic conditioning analysis of the incremental unknowns preconditioner

- Mathematics
- 1998

Abstract Incremental unknowns are efficient in the numerical solution of elliptic linear differential equations but no rigorous theoretical justification was available. Hereafter, we establish that… Expand

A class of modified block SSOR preconditioners for symmetric positive definite systems of linear equations

- Mathematics, Computer Science
- Adv. Comput. Math.
- 1999

A class of modified block SSOR preconditioners is presented for solving symmetric positive definite systems of linear equations, which arise in the hierarchical basis finite element discretizations… Expand

Incremental unknowns for solving partial differential equations

- Mathematics
- 1991

SummaryIncremental unknowns have been proposed in [T] as a method to approximate fractal attractors by using finite difference approximations of evolution equations. In the case of linear elliptic… Expand

Staggered incremental unknowns for solving Stokes and generalized Stokes problems

- Mathematics
- 2000

This article is devoted to the presentation of a multilevel method using finite differences that is well adapted for solving Stokes and Navier-Stokes problems in primitive variables. We use Uzawa… Expand

A class of hybrid algebraic multilevel preconditioning methods

- Mathematics
- 1996

Abstract A class of hybrid algebraic multilevel preconditioning methods is presented for solving systems of linear equations with symmetric positive-definite matrices resulting from the… Expand

Bi-parameter incremental unknowns ADI iterative methods for elliptic problems

- Mathematics, Computer Science
- Numerical Algorithms
- 2011

Theoretical analysis shows that the condition numbers are reduced significantly by IU method, and the iterative sequences produced by the bi-parameter incremental unknowns ADI methods converge to the unique solution of the linear system if the two parameters belong to a given parameter region. Expand

Analysis of a Galerkin finite element method on a Bakhvalov–Shishkin mesh for a linear convection–diffusion problem

- Mathematics
- 2000

We consider a Galerkin finite element method that uses piecewise bilinears on a modified Shishkin mesh for a model singularly perturbed convection-diffusion problem on the unit square. The method is… Expand

Incremental unknowns in finite differences: condition number of the matrix

- Mathematics
- 1993

The utilization of incremental unknowns (IU) with multilevel finite differences was proposed in [R. Temam, SIAM J. Math. Anal., 21 (1991), pp. 154–178] for the integration of elliptic partial… Expand

Algebraic Analysis of the Hierarchical Basis Preconditioner

- Mathematics, Computer Science
- SIAM J. Matrix Anal. Appl.
- 1995

It is shown that for uniform grids the result can be obtained using a purely linear algebraic argument. Expand