# zbMATH — the first resource for mathematics

A nonconforming finite element method for a singularly perturbed boundary value problem. (English) Zbl 0813.65105
This paper analyzes a new nonconforming Petrov-Galerkin finite element method for solving linear singularly perturbed two-point boundary value problems without turning points. The method is shown to be convergent, uniformly in the perturbation parameter, of order $$h^{1/2}$$ in a norm slightly stronger than the energy norm. The proof uses a new abstract convergence theorem for Petrov-Galerkin finite element methods.

##### MSC:
 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 34B05 Linear boundary value problems for ordinary differential equations 34E15 Singular perturbations for ordinary differential equations
Full Text:
##### References:
 [1] Brooks, A. N., Hughes, T. J. R.: Streamline upwind Petrov-Galerkin methods for advection dominated flows. Proc. third-international conference on finite element methods in fluid flow. Banff, Canada 1980. · Zbl 0449.76077 [2] Doolan, E. P., Miller, J. J. H., Schilders, W. H. A.: Uniform numerical methods for problems with initial and boundary layers. Dublin: Boole Press 1980. · Zbl 0459.65058 [3] Felgenhauer, A., Adam, D., Roos, H.-G.: A novel nonconforming uniformly convergent finite element method in two dimensions (in preparation). · Zbl 0869.65068 [4] Felgenhauer, A.: Dissertation B. TH Magdeburg 1988. [5] de Groen, P. P. N., Hemker, P. W.: Error bounds for exponentially fitted Galerkin methods applied to stiff two point boundary value problems. In: Numerical analysis of singular perturbation problems (Hemker, P. W., Miller, J. J. H., eds.), pp. 217–249. Nijmegen: Academic Press 1979. · Zbl 0429.65082 [6] Hemker, P. W.: A numerical study of stiff two-point boundary problems. Amsterdam: Mathematical Centrum 1977. · Zbl 0426.65043 [7] Hughes, T. J. R., Brooks, A. N.: A multidimensional upwind scheme with no crosswind diffusion. In: Finite element methods for convection dominated flows (Hughes, T. J. R., ed.), pp. 19–35. AMP34, ASME 1980. [8] Johnson, C.: Numerical solution of partial differential equations by the finite element method. Cambridge: Cambridge University Press 1987. · Zbl 0628.65098 [9] Johnson, C.: Streamline diffusion methods for problems in fluid mechanics. In: Finite elements in fluids (Gallagher, R. H., Carey, G. F., Oden, T. J., Zienkiewicz, O. C., eds.) vol. 6, pp. 251–261. Chicester: Wiley 1986. [10] Morton, K. W.: Galerkin finite element methods and their generalizations. In: The state of the art in numerical analysis (Jserles, A., Powell, M. J. D., eds.), pp. 645–680. Oxford: Clarendon Press 1987. [11] O’Riordan, E., Stynes, M.: An analysis of a superconvergence result for a singularly perturbed boundary value problem. Math. Comp.46, 81–92 (1986). · Zbl 0612.65043 [12] O’Riordan, E., Stynes, M.: A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions. Math. Comp.57, 47–62 (1991). · Zbl 0733.65063 [13] O’Riordan, E., Stynes, M.: A uniformly convergent difference scheme for an elliptic singular perturbation problem. In: Discretization methods of singular perturbation and flow problems (Tobiska, L., ed.), pp. 48–55. Magdeburg: Technical University ”Otto von Guericke” Magdeburg 1989. [14] Roos, H.-G., Adam, D., Felgenhauer, A.: A nonconforming uniformly convergent finite element method in two dimensions (to appear in: Proceedings of the International Workshop on Numerical Methods for the Navier-Stokes Equations, Heidelberg). · Zbl 0869.65068 [15] Roos, H.-G.: Global uniformly convergent schemes for a singularly perturbed boundary value problem using patched spline-functions. J. Comp. Appl. Math.29, 69–77 (1990). · Zbl 0686.65048 [16] Roos, H.-G.: An analytically oriented discretization technique for boundary value problems. Math. Seminar der Univ. Hamburg61, 139–152 (1991). · Zbl 0776.65057 [17] Shishkin, G. I.: Grid approximation of singularly perturbed elliptic and parabolic equations (in Russian). Second Doctoral thesis, Moscow: Keldysh Institute 1990. [18] Shishkin, G. I.: Methods of constructing grid approximations for singularly perturbed boundary-value problems. Condensing-grid methods. Russ. J. Numer. Anal. Math. Modell.7, 537–562 (1992). · Zbl 0816.65072 [19] Stynes, M., O’Riordan, E.: An analysis of a singularly perturbed two-point boundary value problem using only finite element techniques. Math. Comp.56, 663–675 (1991). · Zbl 0718.65062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.