An analysis of a singularly perturbed two-point boundary value problem using only finite element techniques

Authors:
Martin Stynes and Eugene O’Riordan

Journal:
Math. Comp. **56** (1991), 663-675

MSC:
Primary 65L60; Secondary 34E15

DOI:
https://doi.org/10.1090/S0025-5718-1991-1068809-4

MathSciNet review:
1068809

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a new analysis of Petrov-Galerkin finite element methods for solving linear singularly perturbed two-point boundary value problems without turning points. No use is made of finite difference methodology such as discrete maximum principles, nor of asymptotic expansions. On meshes which are either arbitrary or slightly restricted, we derive energy norm and norm error bounds. These bounds are uniform in the perturbation parameter. Our proof uses a variation on the classical Aubin-Nitsche argument, which is novel insofar as the bound is obtained independently of the energy norm bound.

**[1]**O. Axelsson,*Stability and error estimates of Galerkin finite element approximations for convection-diffusion equations*, IMA J. Numer. Anal.**1**(1981), 329-345. MR**641313 (83a:65105)****[2]**A. E. Berger, J. M. Solomon, and M. Ciment,*An analysis of a uniformly accurate difference method for a singular perturbation problem*, Math. Comp.**37**(1981), 79-94. MR**616361 (83f:65121)****[3]**K. V. Emelyanov,*Difference schemes for singularly perturbed boundary value problems*, BAIL IV (S. K. Godunov, J. J. H. Miller, and V. A. Novikov, eds.), Boole Press, Dublin, 1986, pp. 51-60. MR**926697 (89c:65112)****[4]**E. C. Gartland, Jr.,*An analysis of a uniformly convergent finite difference/finite element scheme for a model singular-perturbation problem*, Math. Comp.**51**(1988), 93-106. MR**942145 (89b:65187)****[5]**-,*Graded-mesh difference schemes for singularly perturbed two-point boundary value problems*, Math. Comp.**51**(1988), 631-657. MR**935072 (89d:65073)****[6]**P. P. N. de Groen,*A finite element method with a large mesh-width for a stiff two-point boundary value problem*, J. Comput. Appl. Math.**7**(1981), 3-15. MR**611944 (82d:65059)****[7]**A. F. Hegarty, E. O'Riordan, and M. Stynes,*A comparison of uniformly convergent difference schemes for two-dimensional convection-diffusion problems*(in preparation).**[8]**A. M. Il'in,*Differencing scheme for a differential equation with a small parameter affecting the highest derivative*, Mat. Zametki**6**(1969), 237-248; English transl. in Math. Notes**6**596-602. MR**0260195 (41:4823)****[9]**R. B. Kellogg and A. Tsan,*Analysis of some difference approximations for a singular perturbation problem without turning points*, Math. Comp.**32**(1978), 1025-1039. MR**0483484 (58:3485)****[10]**K. W. Morton,*Galerkin finite element methods and their generalizations*, The State of the Art in Numerical Analysis (A. Iserles and M. J. D. Powell, eds.), Clarendon Press, Oxford, 1987, pp. 645-680. MR**921681 (89e:65100)****[11]**K. W. Morton and B. W. Scotney,*Petrov-Galerkin methods and diffusion-convection problems in 2D*, The Mathematics of Finite Elements and Applications V (MAFELAP 1984) (J. R. Whiteman, ed.), Academic Press, London, 1985, pp. 343-366. MR**811047 (87i:76049)****[12]**K. Niederdrenk and H. Yserentant,*Die gleichmässige Stabilität singulär gestörter diskreter und kontinuierlicher Randwertprobleme*, Numer. Math.**41**(1983), 223-253. MR**703123 (84j:65049)****[13]**E. O'Riordan and M. Stynes,*A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions*, Math. Comp.**57**(1991) (to appear). MR**1079029 (92j:65174)****[14]**M. Stynes,*An adaptive uniformly convergent numerical method for a semilinear singular perturbation problem*, SIAM J. Numer. Anal.**26**(1989), 442-455. MR**987400 (90f:65137)****[15]**M. Stynes and E. O'Riordan,*A finite element method for a singularly perturbed boundary value problem*, Numer. Math.**50**(1986), 1-15. MR**864301 (88e:65101)****[16]**-,*Finite element methods for elliptic convection-diffusion problems*, BAIL V, Proc 5th Internat. Conf. on Boundary and Interior Layers (B. Guo, J. J. H. Miller, and Z. Shi, eds.), Boole Press, Dublin, 1988, pp. 65-76. MR**990253 (90d:65200)****[17]**-,*An analysis of a two-point boundary value problem with a boundary layer, using only finite element techniques*, Technical Report, Department of Mathematics, University College, Cork, October 1989.**[18]**W. G. Szymczak and I. Babuška,*Adaptivity and error estimation for the finite element method applied to convection diffusion problems*, SIAM J. Numer. Anal.**21**(1984), 910-954. MR**760625 (86c:65143)****[19]**M. Van Veldhuizen,*Higher order schemes of positive type for singular perturbation problems*, Numerical Analysis of Singular Perturbation Problems (P. W. Hemker and J. J. H. Miller, eds.), Academic Press, New York, 1979, pp. 361-383. MR**556526 (81g:65114)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65L60,
34E15

Retrieve articles in all journals with MSC: 65L60, 34E15

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1991-1068809-4

Article copyright:
© Copyright 1991
American Mathematical Society