Logarithm cannot be removed in maximum norm error estimates for linear finite elements in 3D

Kopteva, Natalia

doi:10.1090/mcom/3384

Logarithm cannot be removed in maximum norm error estimates for linear finite elements in 3D
HTML articles powered by AMS MathViewer

by Natalia Kopteva HTML | PDF

Math. Comp. 88 (2019), 1527-1532 Request permission

Abstract:

For linear finite element discretizations of the Laplace equation in three dimensions, we give an example of a tetrahedral mesh in the cubic domain for which the logarithmic factor cannot be removed from the standard upper bounds on the error in the maximum norm.

References

V. B. Andreev, The logarithmic factor in the $L_\infty$-norm error estimate for linear finite elements is sharp, Semin. I. Vekua Inst. Appl. Math. Rep. (Dokl. Semin. Inst. Prikl. Mat. im. I. N. Vekua), 4 (1989), no. 3, 17–20 (in Russian).
H. Blum, Q. Lin, and R. Rannacher, Asymptotic error expansion and Richardson extrapolation for linear finite elements, Numer. Math. 49 (1986), no. 1, 11–37. MR 847015, DOI 10.1007/BF01389427
Richard Haverkamp, Eine Aussage zur $L_{\infty }$-Stabilität und zur genauen Konvergenzordnung der $H^{1}_{0}$-Projektionen, Numer. Math. 44 (1984), no. 3, 393–405 (German, with English summary). MR 757494, DOI 10.1007/BF01405570
N. Kopteva, Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions, submitted for publication (September 2017). arXiv:1709.09136.
Pentti Laasonen, On the solution of Poisson’s difference equation, J. Assoc. Comput. Mach. 5 (1958), 370–382. MR 121999, DOI 10.1145/320941.320951
Dmitriy Leykekhman and Boris Vexler, Finite element pointwise results on convex polyhedral domains, SIAM J. Numer. Anal. 54 (2016), no. 2, 561–587. MR 3470741, DOI 10.1137/15M1013912
A. A. Samarskiĭ, Teoriya raznostnykh skhem, 3rd ed., “Nauka”, Moscow, 1989 (Russian, with Russian summary). MR 1196231
Alfred H. Schatz, A weak discrete maximum principle and stability of the finite element method in $L_{\infty }$ on plane polygonal domains. I, Math. Comp. 34 (1980), no. 149, 77–91. MR 551291, DOI 10.1090/S0025-5718-1980-0551291-3
A. H. Schatz and L. B. Wahlbin, On the quasi-optimality in $L_{\infty }$ of the $\dot H^{1}$-projection into finite element spaces, Math. Comp. 38 (1982), no. 157, 1–22. MR 637283, DOI 10.1090/S0025-5718-1982-0637283-6
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 10.1090/S0025-5718-1976-0436617-2
E. A. Volkov, On differential properties of solutions of boundary value problems for the Laplace and Poisson equations on a rectangle, Trudy Mat. Inst. Steklov 77 (1965), 89–112 (Russian). MR 0192077