Sharp estimates for finite element approximations to elliptic problems with Neumann boundary data of low regularity
HTML articles powered by AMS MathViewer
- by Aaron Solo PDF
- Math. Comp. 76 (2007), 1787-1800 Request permission
Abstract:
Consider a second order homogeneous elliptic problem with smooth coefficients, $Au = 0$, on a smooth domain, $\Omega$, but with Neumann boundary data of low regularity. Interior maximum norm error estimates are given for $C^0$ finite element approximations to this problem. When the Neumann data is not in $L^1(\partial \Omega )$, these local estimates are not of optimal order but are nevertheless shown to be sharp. A method for ameliorating this sub-optimality by preliminary smoothing of the boundary data is given. Numerical examples illustrate the findings.References
- Ivo Babuska, Joseph E. Flaherty, William D. Henshaw, John E. Hopcroft, Joseph E. Oliger, and Tayfun Tezduyar (eds.), Modeling, mesh generation, and adaptive numerical methods for partial differential equations, The IMA Volumes in Mathematics and its Applications, vol. 75, Springer-Verlag, New York, 1995. MR 1370242, DOI 10.1007/978-1-4612-4248-2
- Ivo Babuška and Victor Nistor, Boundary value problems in spaces of distributions on smooth and polygonal domains, preprint.
- Ivo Babuška and Victor Nistor, Interior numerical approximation of boundary value problems with a distributional data, Numer. Methods Partial Differential Equations 22 (2006), no. 1, 79–113. MR 2185526, DOI 10.1002/num.20086
- Christine Bernardi, Optimal finite-element interpolation on curved domains, SIAM J. Numer. Anal. 26 (1989), no. 5, 1212–1240 (English, with French summary). MR 1014883, DOI 10.1137/0726068
- J. H. Bramble and A. H. Schatz, Higher order local accuracy by averaging in the finite element method, Math. Comp. 31 (1977), no. 137, 94–111. MR 431744, DOI 10.1090/S0025-5718-1977-0431744-9
- James H. Bramble and J. Thomas King, A robust finite element method for nonhomogeneous Dirichlet problems in domains with curved boundaries, Math. Comp. 63 (1994), no. 207, 1–17. MR 1242055, DOI 10.1090/S0025-5718-1994-1242055-6
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
- George J. Fix, Max D. Gunzburger, and Janet S. Peterson, On finite element approximations of problems having inhomogeneous essential boundary conditions, Comput. Math. Appl. 9 (1983), no. 5, 687–700. MR 726817, DOI 10.1016/0898-1221(83)90126-8
- Donald A. French and J. Thomas King, Approximation of an elliptic control problem by the finite element method, Numer. Funct. Anal. Optim. 12 (1991), no. 3-4, 299–314. MR 1143001, DOI 10.1080/01630569108816430
- Ju. P. Krasovskiĭ, Properties of Green’s functions, and generalized solutions of elliptic boundary value problems, Dokl. Akad. Nauk SSSR 184 (1969), 270–273 (Russian). MR 0237956
- J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR 0350177
- Joachim A. Nitsche and Alfred H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974), 937–958. MR 373325, DOI 10.1090/S0025-5718-1974-0373325-9
- A. H. Schatz and L. B. Wahlbin, Interior maximum norm estimates for finite element methods, Math. Comp. 31 (1977), no. 138, 414–442. MR 431753, DOI 10.1090/S0025-5718-1977-0431753-X
- Alfred H. Schatz, Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. I. Global estimates, Math. Comp. 67 (1998), no. 223, 877–899. MR 1464148, DOI 10.1090/S0025-5718-98-00959-4
- Lars B. Wahlbin, Local behavior in finite element methods, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 353–522. MR 1115238
Additional Information
- Aaron Solo
- Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
- Address at time of publication: Susquehanna International Group, 401 City Line Avenue, Bala Cynwyd, Pennsylvania 19004
- Email: als54@cornell.edu
- Received by editor(s): April 11, 2006
- Received by editor(s) in revised form: August 2, 2006
- Published electronically: May 3, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 76 (2007), 1787-1800
- MSC (2000): Primary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-07-01993-X
- MathSciNet review: 2336268