Interior maximum norm estimates for finite element methods

Schatz, A. H.; Wahlbin, L. B.

doi:10.1090/S0025-5718-1977-0431753-X

Interior maximum norm estimates for finite element methods

Authors: A. H. Schatz and L. B. Wahlbin
Journal: Math. Comp. 31 (1977), 414-442
MSC: Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-1977-0431753-X
MathSciNet review: 0431753
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Interior a priori error estimates in the maximum norm are derived from interior Ritz-Galerkin equations which are common to a class of methods used in approximating solutions of second order elliptic boundary value problems. The estimates are valid for a large class of piecewise polynomial subspaces used in practice, which are defined on quasi-uniform meshes. It is shown that the error in an interior domain ${\Omega _1}$ can be estimated with the best order of accuracy that is possible locally for the subspaces used plus the error in a weaker norm over a slightly larger domain which measures the effects from outside of the domain ${\Omega _1}$.

Ivo Babuška and A. K. Aziz, Survey lectures on the mathematical foundations of the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR 0421106
James H. Bramble, A survey of some finite element methods proposed for treating the Dirichlet problem, Advances in Math. 16 (1975), 187–196. MR 381348, DOI https://doi.org/10.1016/0001-8708%2875%2990150-4
James H. Bramble, Joachim A. Nitsche, and Alfred H. Schatz, Maximum-norm interior estimates for Ritz-Galerkin methods, Math. Comput. 29 (1975), 677–688. MR 0398120, DOI https://doi.org/10.1090/S0025-5718-1975-0398120-7
J. H. Bramble and J. E. Osborn, Rate of convergence estimates for nonselfadjoint eigenvalue approximations, Math. Comp. 27 (1973), 525–549. MR 366029, DOI https://doi.org/10.1090/S0025-5718-1973-0366029-9
J. H. Bramble and A. H. Schatz, Estimates for spline projections, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 10 (1976), no. R-2, 5–37. MR 0436620
J. H. Bramble and V. Thomée, Interior maximum norm estimates for some simple finite element methods, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 5–18 (English, with French summary). MR 359354
James H. Bramble and Miloš Zlámal, Triangular elements in the finite element method, Math. Comp. 24 (1970), 809–820. MR 282540, DOI https://doi.org/10.1090/S0025-5718-1970-0282540-0
P. G. Ciarlet and P.-A. Raviart, General Lagrange and Hermite interpolation in ${\bf R}^{n}$ with applications to finite element methods, Arch. Rational Mech. Anal. 46 (1972), 177–199. MR 336957, DOI https://doi.org/10.1007/BF00252458
P. G. Ciarlet and P.-A. Raviart, Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Engrg. 2 (1973), 17–31. MR 375802, DOI https://doi.org/10.1016/0045-7825%2873%2990019-4
Jim Douglas Jr., Todd Dupont, and Lars Wahlbin, Optimal $L_{\infty }$ error estimates for Galerkin approximations to solutions of two-point boundary value problems, Math. Comp. 29 (1975), 475–483. MR 371077, DOI https://doi.org/10.1090/S0025-5718-1975-0371077-0
Stephen Hilbert, A mollifier useful for approximations in Sobolev spaces and some applications to approximating solutions of differential equations, Math. Comp. 27 (1973), 81–89. MR 331715, DOI https://doi.org/10.1090/S0025-5718-1973-0331715-3
Fritz John, General properties of solutions of linear elliptic partial differential equations, Proceedings of the Symposium on Spectral Theory and Differential Problems, Oklahoma Agricultural and Mechanical College, Stillwater, Okla., 1951, pp. 113–175. MR 0043990
Carlo Miranda, Partial differential equations of elliptic type, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Springer-Verlag, New York-Berlin, 1970. Second revised edition. Translated from the Italian by Zane C. Motteler. MR 0284700
Frank Natterer, Über die punktweise Konvergenz finiter Elemente, Numer. Math. 25 (1975/76), no. 1, 67–77 (German, with English summary). MR 474884, DOI https://doi.org/10.1007/BF01419529

J. A. NITSCHE, "${L_\infty }$-convergence for finite element approximation," 2nd Conf. on Finite Elements (Rennes, France, May 12-14, 1975).

Joachim A. Nitsche and Alfred H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974), 937–958. MR 373325, DOI https://doi.org/10.1090/S0025-5718-1974-0373325-9
Martin Schechter, On $L^{p}$ estimates and regularity. I, Amer. J. Math. 85 (1963), 1–13. MR 188615, DOI https://doi.org/10.2307/2373179
Ridgway Scott, Optimal $L^{\infty }$ estimates for the finite element method on irregular meshes, Math. Comp. 30 (1976), no. 136, 681–697. MR 436617, DOI https://doi.org/10.1090/S0025-5718-1976-0436617-2
Gilbert Strang, Approximation in the finite element method, Numer. Math. 19 (1972), 81–98. MR 305547, DOI https://doi.org/10.1007/BF01395933

G. STRANG & G. FIX, "A Fourier analysis of the finite element variational method." (Unpublished manuscript.)

Mary Fanett Wheeler, An optimal $L_{\infty }$ error estimate for Galerkin approximations to solutions of two-point boundary value problems, SIAM J. Numer. Anal. 10 (1973), 914–917. MR 343659, DOI https://doi.org/10.1137/0710077