*Uniform convergence analysis of finite difference approximations for general singular perturbed problem on adaptive grids*

**DOI:**10.54647/physics14429 83 Downloads 2096 Views

**Author(s)**

**Abstract**

In this paper we consider a more general singular perturbation problem, that is, -epsilon u ''(x) - a(x)u'(x) + b(x)u(x) = f(x) (0 < epsilon << 1) on an adaptive grid. The mesh is constructed adaptively by equidistributing a monitor function based on the arc-length of the approximated solutions. Our analysis provide insight into the convergence behaviour on such mesh, and the posterior error estimates of piecewise linear interpolation about the approximate solution is investigated and an epsilon-uniform error estimate for the first-order upwind discretization of general singular perturbed problem is derived at last. We extend the relevant results of the document to a more general case.

**Keywords**

adaptive grids; general singular perturbed problem; convergence analysis; posterior error estimates

**Cite this paper**

Linan Sun, Antao Wang,
Uniform convergence analysis of finite difference approximations for general singular perturbed problem on adaptive grids
, *SCIREA Journal of Physics*.
Volume 7, Issue 2, April 2022 | PP. 57-67.
10.54647/physics14429

**References**

[ 1 ] | Miller J J H, O’Riordan E and Shishkin G I 1996 Fitted Numerical Methods for Singular Perturbation Problems World Scientific Singapore |

[ 2 ] | Beckett G M and Mackenzie J A 2000 Convergence analysis of finite difference approximations on equidistributed grids to a singularly perturbed boundary value problem Appl. Numer. Math. 35 87-109 |

[ 3 ] | Mackenzie J A 1999 Uniform convergence analysis of an upwind finite-difference approximation of a convection-diffusion boundary value problem on an adaptive grid IMA J. Numer. Anal. 19 233-249 |

[ 4 ] | Kopteva N and Stynes M 2001 A robust adaptive method for quasi-linear one-dimensional convection-diffusion problem SIAM J. Numer. Anal. 39 1446-1467. |

[ 5 ] | Kopteva N 2001 Maximum norm a posteriori error estimates for a one-dimensional convection-diffusion problem SIAM J. Numer. Anal. 39 423-441 |

[ 6 ] | Linss T 2001 Uniform pointwise convergence of Knite diIerence schemes using grid equidistribution Computing 66 27-39 |

[ 7 ] | Chen Y 2006 Uniform convergence analysis of finite difference approximations for singular perturbation problems on an adaptive grid Adv. Comput. Math. 24 197-212 |

[ 8 ] | Chen Y 2003 Uniform pointwise convergence for a singularly perturbed problem using arc-length equidistribution Comput. Appl. Math. 159 25-34 |

[ 9 ] | Qiu Y, Sloan D M and Tang T 2000 Numerical solution of a singularly perturbed two-point boundary value problem using equidistribution: Analysis of convergence Comput. Appl. Math. 116 121-143 |

[ 10 ] | Qiu Y and Sloan D M 1999 Analysis of difference approximations to a singularly perturbed two-point boundary value problem on an adaptively generated grid Comput. Appl. Math. 101 1-25 |

[ 11 ] | Tang T 2005 Moving Mesh Methods for Computational Fluid Dynamics In Recent Advances in Adaptive Computations AMS 383 141-173 |

[ 12 ] | Tan Z, Lim K M and Khoo B C 2007 An Adaptive Mesh Redistribution Method For The Incompressible Mixture Flows Using Phase-field Model Comput. Phys. 225 1137-1158 |

[ 13 ] | Mackenzie J A and Mekwi W R 2007 On the Use of Moving Mesh Methods to Solve PDEs. in T.Tang and J. Xu eds., Adaptive Computations: Theory and Algorithms, Science Press 243-278 Beijing |

[ 14 ] | Di Y N, Li R and Tang T 2008 A General Moving Mesh Framework In 3D And Its Application For Simulating TheMixture Of Multi-Phase Flows Com. Comput. Phys. 3(3) 582-602 |

[ 15 ] | Budd C J, Huang W Z and Russell R D 2009 Adaptivity with Moving Grids Acta Numerica 1-131 |

[ 16 ] | Delzanno G L and Finn J M 2011 The fluid dynamic approach to equidistribution methods for grid adaptation Comput. Phy. Comm. 182 330-346 |

[ 17 ] | Sun L N, Wu Y J and Yang A L 2011 Uniform convergence analysis of finite difference approximations for advection-reaction-diffusion problem on adaptive grids Int. J Comput. Math. 88 3292-3307 |