Approximation Theory and its Applications

, Volume 12, Issue 2, pp 86–98

# Optimally accurate Petrov-Galerkin method of finite elements

• M. Stojanović
Article

## Abstract

We perform analysis for a finite elements method applied to the singular self-adjoint problem. This method uses continuous piecewise polynomial spaces for the trial and the test spaces. We fit the trial polynomial space by piecewise exponentials and we apply so exponentially fitted Galerkin method to singular self-adjoint problem by approximating driving terms by Lagrange piecewise polynomials, linear, quadratic and cubic. We measure the erroe in max norm. We show that method is optimal of the first order in the error estimate. We also give numerical results for the Galerkin approximation.

## Keywords

Difference Scheme Truncation Error Galerkin Approximation Polynomial Space Singular Perturbation Problem

## References

1. [1]
de Groen P.P.N., Hemker P.W., Error Bounds for Exponentially Fitted Galerkin Methods Applied to Stiff Two-Point Boundary Value Problems, Proc. Conf. on Numerical Analysis of Singular Perturbation (in: P. W. Hemker and J. J. Miller, eds.), Academic Press, (1979), 217–249.Google Scholar
2. [2]
Doolan E. P., Miller J. J. H., Schilders W. H. A., Uniform Numerical Methods for Problems with Initial and Boundary Layers, (1980), Boole Press, Dublin, 223–238.
3. [3]
Farrell A. P., Sufficient Conditions for Uniform Convergence of a Class of Difference Schemes for a Singularly Perturbed Problem, IMA J. Num. Anal. 7(1987), 459–472.
4. [4]
Griffiths D. S., An Analysis of the Petrov-Galerkin Finite Element Method, Comp. Meths. Appl. Math. Eng. 14(1978), 39–64.
5. [5]
Griffiths D.F., Towards Time-stepping Algorithms for Convective-diffusion, Proc. Conf. on Numerical Analysis of Singular Perturbation Problems, (in: P. W. Hemker and J. J. Miller, eds). Academic Press, (1979) 216–249.Google Scholar
6. [6]
Mitchell, A. R. and Christie I., Finite Difference/Finite Element Methods at the Parabolic-Hyperbolic Interface, Proc. Conf. on Numerical Analysis of Singular Perturbation Problems, (in: P. W. Hemker and J. J. Miller, eds.), Academic Press, (1979), 339–361.Google Scholar
7. [7]
Stojanovic, M., A Uniformly Accurate Spline Collocation Method for Singular Perturbation Problem, Calcolo, 1–2, 27(1990), 81–88.

## Authors and Affiliations

• M. Stojanović
• 1