Computable Error Bounds for the Finite Element Method for Elliptic Boundary Value Problems

• R. E. Barnhill
• J. R. Whiteman
Part of the International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique book series (ISNM, volume 19)

Abstract

The purpose of this paper is to determine more nearly computable error bounds for finite element solutions to two-dimensional elliptic boundary value problems defined on simply connected polygonal regions. In the appropriate norm, the interpolation remainder is an upper bound on the finite element remainder. This follows from a best approximation property of finite element solution (Section 2). The SARD kernel theorems  provide representations of admissible linear functionals defined on function spaces of prescribed smoothness. These theorems are defined for rectangles and in Section 4 are extended to triangles. The method can be extended to more general regions . In this paper we calculate the constants in interpolation error bounds for triangles. This is done in Section 5 for the particular case of piecewise linear interpolation. Finally in Section 6 the results of Section 5 are applied to a specific boundary value problem in order to obtain numerical results.

References

1. 1.
Barnhill, R. E., and J. A. Gregory: Sard kernel theorems on triangular and rectangular domains with extensions and applications to finite element error bounds. TR/11, Department of Mathematics, Brunei University, 1972.Google Scholar
2. 2.
Barnhill, R. E., J. A. Gregory and J. R. Whiteman: The extension and application of Sard kernel theorems to compute finite element error bounds. Proc. of O. N. R. Regional Symposium “Mathematical foundations of the finite element method with applications to partial differential equations”, Univ. of Maryland, Baltimore County, June 1972.Google Scholar
3. 3.
Barnhill, R. E. and J. R. Whiteman: Error analysis of finite element methods with triangles for elliptic boundary value problems. In Whiteman (ed.), The Mathematics of Finite Elements and Applications. Acad. Press, London 1973.Google Scholar
4. 4.
Barnhill, R. E. and J. R. Whiteman: Singularities due to re-entrant boundaries in elliptic problems. Proc. of Symposium “Numerische Methoden bei Differentialgleichungen”. Math. Forschungsinstitut, Oberwolfach, June 1972.Google Scholar
5. 5.
Birkhoff, G., M. H. Schultz and R.S. Varga: Piecewise Hermite interpolation in one and two variables with applications to partial differential equations. Numer. Math. 11 (1968), 232–256.
6. 6.
Bramble, J. H. and M. Zlamal: Triangular elements in the finite element method. Math. Comp. 24 (1970), 809–820.
7. 7.
Sard, A.: Linear Approximation. Mathem. Survey 9, Americ. Math. Society, Providence, R. I., 1963.
8. 8.
Smirnov, V. I.: A Course of Higher Mathematics. Vol. V. Pergamon Press, Oxford, 1964.Google Scholar
9. 9.
Varga, R. S.: The role of interpolation and approximation theory in variational and projectional methods for solving partial differential equations. IFIP Congr. 71 (1971), 14–19, North Holland, Amsterdam.Google Scholar
10. 10.
Zenisek, A.: Interpolation polynomials on the triangle. Numer. Math. 15 (1970), 283–296.
11. 11.
Zlamal, M.: On the finite element method. Numer. Math. 12 (1968), 394–409.