Sard kernel theorems on triangular domains with application to finite element error bounds
- 67 Downloads
Error bounds for interpolation remainders on triangles are derived by means of extensions of the Sard Kernel Theorems. These bounds are applied to the Galerkin method for elliptic boundary value problems. Certain kernels are shown to be identically zero under hypotheses which are, for example, fulfilled by tensor product interpolants on rectangles. This removes certain restrictions on how the sides of the triangles and/or rectangles tend to zero. Explicit error bounds are computed for piecewise linear interpolation over a triangulation and applied to a model problem.
KeywordsMathematical Method Tensor Product Linear Interpolation Galerkin Method Model Problem
Unable to display preview. Download preview PDF.
- 1.Babuska, I.: Private communication, Salt Lake City, August, 1974Google Scholar
- 2.Barnhill, R. E., Gregory, J. A.: Sard kernel theorems on triangular and rectangular domains with extensions and applications to finite element error bounds. Technical Report 11, Department of Mathematics, Brunel University, Uxbridge, Middlesex, England, July, 1972Google Scholar
- 3.Barnhill, R. E., Gregory, J. A., Whiteman, J. R.: The extension and application of Sard kernel theorems to compute finite element error bounds. Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A. K. Aziz (ed.), p. 749–755, New York: Academic Press 1972Google Scholar
- 4.Barnhill, R. E., Mansfield, L.: Error bounds for smooth interpolation in triangles. J. Approx. Theory11, 306–318 (1974)Google Scholar
- 5.Barnhill, R. E., Whiteman, J. R.: Error analysis of finite element methods with triangles for elliptic boundary value problems. The Mathematics of Finite Elements and Applications, J. R. Whiteman (ed.), p. 83–112, London: Academic Press 1972Google Scholar
- 6.Birkhoff, G., Schultz, M. H., Varga, R. S.: Piecewise Hermite interpolation in one and two variables with applications to partial differential equations. Numer. Math.11, 232–256 (1968)Google Scholar
- 7.Gregory, J. A.: Piecewise interpolation theory for functions of two variables. Ph. D. Thesis, Brunel Univ. 1975Google Scholar
- 8.Sard, A.: Linear Approximation. Mathematical Survey 9, American Mathematical Society, Providence, Rhode Island, 1963Google Scholar
- 9.Varga, R. S.: The role of interpolation and approximation theory in variational and projectional methods for solving partial differential equations. IFIP Congress 71, p. 14–19, Amsterdam: North Holland 1971Google Scholar