Stability of a streamline diffusion finite element method for turning point problems

A one-dimensional singularly perturbed problem with a boundary turning point is considered in this paper. Let V^h be the linear finite element space on a suitable grid T"h. A variant of streamline diffusion finite element method is proved to be almost uniform stable in the sense that the numerical approximation u"h satisfies @?u-u"[email protected]?"~=

[1]  L. Wahlbin,et al.  On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions , 1983 .

[2]  George G. Lorentz,et al.  Constructive Approximation , 1993, Grundlehren der mathematischen Wissenschaften.

[3]  Silvia Bertoluzza,et al.  Stable Discretizations of Convection-Diffusion Problems via Computable Negative-Order Inner Products , 2000, SIAM J. Numer. Anal..

[4]  Zhimin Zhang,et al.  Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems , 2003, Math. Comput..

[5]  Jinchao Xu,et al.  An Optimal Streamline Diffusion Finite Element Method for a Singularly Perturbed Problem , 2005 .

[6]  Arnold Reusken Maximum norm convergence of multigrid methods for elliptic boundary value problems , 1992 .

[7]  Natalia Kopteva Maximum Norm A Posteriori Error Estimates for a One-Dimensional Convection-Diffusion Problem , 2001, SIAM J. Numer. Anal..

[8]  Ping Lin,et al.  Numerical solution of quasilinear attractive turning point problems , 1992 .

[9]  On maximum norm convergence of multigrid methods for two-point boundary value problems , 1992 .

[10]  Giancarlo Sangalli,et al.  Analysis of the advection-diffusion operator using fractional order norms , 2004, Numerische Mathematik.

[11]  C. D. Boor,et al.  Good approximation by splines with variable knots. II , 1974 .

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

[13]  T. Linß,et al.  Uniform methods for semilinear problems with attractive boundary turning points , 2002 .

[14]  Ping Lin,et al.  A numerical method for quasilinear singular perturbation problems with turning points , 1991, Computing.

[15]  Giancarlo Sangalli Quasi Optimality of the SUPG Method for the One-Dimensional Advection-Diffusion Problem , 2003, SIAM J. Numer. Anal..

[16]  L. D. Marini,et al.  A Priori Error Analysis of Residual-Free Bubbles for Advection-Diffusion Problems , 1999 .

[17]  Endre Süli,et al.  Residual-free bubbles for advection-diffusion problems: the general error analysis , 2000, Numerische Mathematik.

[18]  Hans-Görg Roos,et al.  The streamline-diffusion method for a convection-diffusion problem with a point source , 2003 .

[19]  Lutz Tobiska,et al.  Numerical Methods for Singularly Perturbed Differential Equations , 1996 .

[20]  Long Chen,et al.  Stability and accuracy of adapted finite element methods for singularly perturbed problems , 2008, Numerische Mathematik.

[21]  Zhimin Zhang,et al.  Finite element superconvergence approximation for one‐dimensional singularly perturbed problems , 2002 .

[22]  Silvia Bertoluzza,et al.  Negative norm stabilization of convection-diffusion problems , 2000, Appl. Math. Lett..