Abstract
In this paper, we investigate the numerical solution of a model equation u xx =\(\frac{1}{{\varepsilon ^2 }}\)exp(−\(\frac{x}{\varepsilon }\)) (and several slightly more general problems) when ∈≪1 using the standard central difference scheme on nonuniform grids. In particular, we are interested in the error behaviour in two limiting cases: (i) the total mesh point number N is fixed when the regularization parameter ∈→0, and (ii) ∈ is fixed when N→∞. Using a formal analysis, we show that a generalized version of a special piecewise uniform mesh 12 and an adaptive grid based on the equidistribution principle share some common features. And the “optimal” meshes give rates of convergence bounded by |log(∈)| as ∈→0 and N is given, which are shown to be sharp by numerical tests.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Beckett, G., 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(2), 109–131.
Carey, C. F., and Dinh, H. T. (1985). Grading functions and mesh redistribution. SIAM J.Numer.Anal. 22, 1028–1040.
Chorin, A. J. (1973). Numerical study of slightly viscous flow. J.Fluid Mech. 57, 785–796.
Connett, W. C., Golik, W. L., and Schwartz, A. L. (1993). A. superconvergent scheme on irregular grids for systems of two-point boundary value problems. Comp.Appl.Math. 12, 227–246.
Crank, J. (1984). Free and Moving Boundary Problems, Clarendon Press, New York.
deBoor, C. (1973). Good approximation by splines with variable knots. II. In Springer Lecture Note Series, Vol. 363, Springer-Verlag, Berlin.
Hou, T. Y., and Wetton, B. T. R. (1992). Convergence of a finite difference scheme for the Navier-Stokes equations using vorticity boundary conditions. SIAM J.Numer.Anal. 29, 615–639.
Huang, H., and Wetton, B. R. (1996). Discrete compatibility in finite difference methods for viscous incompressible fluid flow. J.Comput.Phys. 126, 468–478.
Huang, H., and Li, Z. (1999). Convergence analysis of immersed interface method. IMA J.Numer.Anal. 19, 583–608.
Huang, H., and Seymour, B. R. (2000). Finite difference solutions of incompressible flow problems with corner singularities. J.Sci.Comput. 15, 265–292.
Huang, W., Ren, Y., and Russell, R. D. (1994). Moving mesh partial differential equations (MMPDES) based on the equidistribution principle. SIAM.J.Numer.Anal. 31, 709.
Miller, J. J. H., O'Riordan, E., and Shishkin, G. I. (1995). On piecewise-uniform meshes for upwind-and central-difference operators for solving singular perturbed problems. IMA J.Numer.Anal. 15, 89–99.
Pereyra, V., and Sewell, E. G. (1975). Mesh selection for discrete solution of boundary value problems in ordinary differential equations. Numer.Math. 23, 261–268.
Peskin, C. S., and McQueen, D. M. (1992). Cardiac fluid dynamics. Critical Reviews in Biomedical Engineering 20, 451.
Qiu, Y., Sloan, D. M., and Tang, T. (2000). Convergence analysis of an adaptive finite difference method for a singular perturbation problem. J.Comput.Appl.Math. 116, 121–143.
Rights and permissions
About this article
Cite this article
Budd, C.J., Huang, H. & Russell, R.D. Mesh Selection for a Nearly Singular Boundary Value Problem. Journal of Scientific Computing 16, 525–552 (2001). https://doi.org/10.1023/A:1013250525615
Issue Date:
DOI: https://doi.org/10.1023/A:1013250525615