Optimal 𝐿^{∞} estimates for the finite element method on irregular meshes

Scott, Ridgway

doi:10.1090/S0025-5718-1976-0436617-2

Optimal $L^{\infty }$ estimates for the finite element method on irregular meshes
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Math. Comp. 30 (1976), 681-697 Request permission

Abstract:

Uniform estimates for the error in the finite element method are derived for a model problem on a general triangular mesh in two dimensions. These are optimal if the degree of the piecewise polynomials is greater than one. Similar estimates of the error are also derived in ${L^p}$. As an intermediate step, an ${L^1}$ estimate of the gradient of the error in the finite element approximation of the Green’s function is proved that is optimal for all degrees.

References

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
J. H. Bramble and S. R. Hilbert, Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. Math. 16 (1970/71), 362–369. MR 290524, DOI 10.1007/BF02165007
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 10.1090/S0025-5718-1975-0398120-7
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 A. H. Schatz, Higher order local accuracy by averaging in the finite element method, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974) Publication No. 33, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 1–14. MR 0657964
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
A.-P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Sympos. Pure Math., Vol. IV, American Mathematical Society, Providence, R.I., 1961, pp. 33–49. MR 0143037
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 10.1016/0045-7825(73)90019-4
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 10.1090/S0025-5718-1975-0371077-0
Jim Douglas Jr., Todd Dupont, and Mary Fanett Wheeler, An $L^{\infty }$ estimate and a superconvergence result for a Galerkin method for elliptic equations based on tensor products of piecewise polynomials, Rev. Française Automat. Informat. Recherche Opérationnelle Sér Rouge 8 (1974), no. R-2, 61–66 (English, with Loose French summary). MR 0359358
Todd Dupont and Ridgway Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comp. 34 (1980), no. 150, 441–463. MR 559195, DOI 10.1090/S0025-5718-1980-0559195-7
Isaac Fried, Finite-element method: accuracy at a point, Quart. Appl. Math. 32 (1974/75), 149–161. MR 436623, DOI 10.1090/S0033-569X-1974-0436623-1
F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. MR 131498, DOI 10.1002/cpa.3160140317
Frank Natterer, Über die punktweise Konvergenz finiter Elemente, Numer. Math. 25 (1975/76), no. 1, 67–77 (German, with English summary). MR 474884, DOI 10.1007/BF01419529
J. Nitsche, Lineare Spline-Funktionen und die Methoden von Ritz für elliptische Randwertprobleme, Arch. Rational Mech. Anal. 36 (1970), 348–355 (German). MR 255043, DOI 10.1007/BF00282271
J. A. Nitsche, $L_{\infty }$-convergence of finite element approximation, Journées “Éléments Finis” (Rennes, 1975) Univ. Rennes, Rennes, 1975, pp. 18. MR 568857
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
G. O. Okikiolu, Aspects of the theory of bounded integral operators in $L^{p}$-spaces, Academic Press, London-New York, 1971. MR 0445237
Jaak Peetre, Espaces d’interpolation et théorème de Soboleff, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 279–317 (French). MR 221282, DOI 10.5802/aif.232
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
Ridgway Scott, Finite element convergence for singular data, Numer. Math. 21 (1973/74), 317–327. MR 337032, DOI 10.1007/BF01436386

A Fourier Analysis of the Finite Element Variational Method

Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
Lars Wahlbin, On maximum norm error estimates for Galerkin approximations to one-dimensional second order parabolic boundary value problems, SIAM J. Numer. Anal. 12 (1975), 177–182. MR 383785, DOI 10.1137/0712016
Mary Fanett Wheeler, $L_{\infty }$ estimates of optimal orders for Galerkin methods for one-dimensional second order parabolic and hyperbolic equations, SIAM J. Numer. Anal. 10 (1973), 908–913. MR 343658, DOI 10.1137/0710076