On the 𝐿^{∞}-convergence of Galerkin approximations for second-order hyperbolic equations

Baker, Garth A.; Dougalis, Vassilios A.

doi:10.1090/S0025-5718-1980-0559193-3

On the $L^{\infty }$-convergence of Galerkin approximations for second-order hyperbolic equations

Authors: Garth A. Baker and Vassilios A. Dougalis
Journal: Math. Comp. 34 (1980), 401-424
MSC: Primary 65M15; Secondary 65N30
DOI: https://doi.org/10.1090/S0025-5718-1980-0559193-3
MathSciNet review: 559193
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that certain classes of high order accurate Galerkin approximations for homogeneous second-order hyperbolic equations, known to possess optimal order rate of convergence in ${L^2}$, also possess optimal order rate of convergence in ${L^\infty }$. This is attainable with particular smoothness assumptions on the initial data. We establish sufficient conditions for optimal ${L^\infty }$-convergence of the approximations to the solution and also the approximation to its time derivative. This is done for both semidiscrete approximations and for single-step fully discrete approximations generated by rational functions.

Garth A. Baker, Error estimates for finite element methods for second order hyperbolic equations, SIAM J. Numer. Anal. 13 (1976), no. 4, 564–576. MR 423836, DOI https://doi.org/10.1137/0713048
Garth A. Baker and James H. Bramble, Semidiscrete and single step fully discrete approximations for second order hyperbolic equations, RAIRO Anal. Numér. 13 (1979), no. 2, 75–100 (English, with French summary). MR 533876, DOI https://doi.org/10.1051/m2an/1979130200751
Garth A. Baker, James H. Bramble, and Vidar Thomée, Single step Galerkin approximations for parabolic problems, Math. Comp. 31 (1977), no. 140, 818–847. MR 448947, DOI https://doi.org/10.1090/S0025-5718-1977-0448947-X
J. H. Bramble, A. H. Schatz, V. Thomée, and L. B. Wahlbin, Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations, SIAM J. Numer. Anal. 14 (1977), no. 2, 218–241. MR 448926, DOI https://doi.org/10.1137/0714015
Felix E. Browder, Asymptotic distribution of eigenvalues and eigenfunctions for non-local elliptic boundary value problems. I, Amer. J. Math. 87 (1965), 175–195. MR 174858, DOI https://doi.org/10.2307/2373230
Colin Clark, The asymptotic distribution of eigenvalues and eigenfunctions for elliptic boundary value problems, SIAM Rev. 9 (1967), 627–646. MR 510064, DOI https://doi.org/10.1137/1009105

M. CROUZEIX, Sur l’Approximation des Equations Différentielles Opérationelles Linéaires par des Méthodes de Runge-Kutta, Thèse, Université Paris-VI, Paris, 1975.

Todd Dupont, $L^{2}$-estimates for Galerkin methods for second order hyperbolic equations, SIAM J. Numer. Anal. 10 (1973), 880–889. MR 349045, DOI https://doi.org/10.1137/0710073
Jens Frehse and Rolf Rannacher, Asymptotic $L^{\infty }$-error estimates for linear finite element approximations of quasilinear boundary value problems, SIAM J. Numer. Anal. 15 (1978), no. 2, 418–431. MR 502037, DOI https://doi.org/10.1137/0715026
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.

J. A. Nitsche, On $L_{\infty }$-convergence of finite element approximations to the solution of a nonlinear boundary value problem, Topics in numerical analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976) Academic Press, London-New York, 1977, pp. 317–325. MR 513215
Rolf Rannacher, Zur $L^{\infty }$-Konvergenz linearer finiter Elemente beim Dirichlet-Problem, Math. Z. 149 (1976), no. 1, 69–77 (German). MR 488859, DOI https://doi.org/10.1007/BF01301633
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
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 https://doi.org/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 https://doi.org/10.1137/0710076