Avoiding slave points in an adaptive refinement procedure for convection-diffusion problems in 2D

A finite difference method is presented for singularly perturbed convection-diffusion problems with discretization error estimate of nearly second order. In a standard patched adaptive refinement method certain slave nodes appear where the approximation is done by interpolating the values of the approximate solution at adjacent nodes. This deteriorates the accuracy of truncation error. In order to avoid the slave points we change the stencil at the interface points from a cross to a skew one. The efficiency of this technique is illustrated by numerical experiments in 2D.

[1]  J. W. Schmidt,et al.  L. Collatz, Funktionalanalysis und numerische Mathematik. (Die Grundlehren der mathematischen Wissenschaften, Band 120) VI + 371 S. m. 96 Abb. u. 2 Porträts. Berlin/Göttingen/Heidelberg 1964. Springer-Verlag. Preis geb. DM 58,— , 1965 .

[2]  Martin Stynes,et al.  A Uniformly Convergent Galerkin Method on a Shishkin Mesh for a Convection-Diffusion Problem☆ , 1997 .

[3]  P. Hemker,et al.  Mixed Defect Correction Iteration for the Solution of a Singular Perturbation Problem , 1983 .

[4]  Hans-Görg Roos.  Layer‐Adapted Grids for Singular Perturbation Problems , 1998 .

[5]  O. Axelsson,et al.  Defect correction methods for convection dominated convection-diffusion problems , 1990 .

[6]  Claes Johnson,et al.  Computational Differential Equations , 1996 .

[7]  Klaus Böhmer,et al.  Defect Correction Methods , 1984, Computing Supplementum.

[8]  A. A. Samarskii.  Monotonic difference schemes for elliptic and parabolic equations in the case of a non-selfadjoint elliptic operator☆ , 1965 .

[9]  Shagi-Di Shih,et al.  Asymptotic anaylsis of a singular perturbation problem , 1987 .

[10]  R. Kellogg,et al.  Analysis of some difference approximations for a singular perturbation problem without turning points , 1978 .

[11]  Eugene O'Riordan,et al.  Numerical Methods for Singular Perturbation Problems , 1996 .

[12]  Owe Axelsson,et al.  Adaptive refinement for convection-diffusion problems based on a defect-correction technique and finite difference method , 1997, Computing.

[13]  K. Bohmer.  Defect Correction Methods: Theory and Applications , 1984 .

[14]  U MartinStynes.  A Uniformly Convergent Galerkin Method on a Shishkin Mesh for a Convection-Diffusion Problem , 1997 .

[15]  R. B. Kellogg,et al.  Numerical analysis of singular perturbation problems , 1983 .

[16]  R. B. Kellogg,et al.  Differentiability properties of solutions of the equation -ε 2 δ u + ru = f ( x,y ) in a square , 1990 .

[17]  L. Collatz.  Funktionalanalysis und numerische Mathematik , 1964 .

[18]  M. Stynes,et al.  Numerical methods for singularly perturbed differential equations : convection-diffusion and flow problems , 1996 .

[19]  Martin Stynes,et al.  A hybrid difference scheme on a Shishkin mesh for linear convection-diffusion problems , 1999 .