We prove regularity results for divergence form periodic second order elliptic difference operators on the space of functions of mean value zero, valid in maximum norm. The estimates obtained are discrete analogues of the regularity results for continuous operators. The maximum norms of the inverse of such an elliptic operator and of its first spatial differences are uniformly bounded in the grid spacing, and second spatial differences are uniformly bounded except for a logarithmic factor in the grid spacing.
DOI : 10.1051/m2an/2015018
Mots-clés : Elliptic, finite difference, variable coefficients, periodic boundary conditions
@article{M2AN_2015__49_5_1451_0,
author = {Pruitt, Michael},
title = {Maximum {Norm} {Regularity} of {Periodic} {Elliptic} {Difference} {Operators} {With} {Variable} {Coefficients}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1451--1461},
publisher = {EDP-Sciences},
volume = {49},
number = {5},
year = {2015},
doi = {10.1051/m2an/2015018},
zbl = {1360.65262},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2015018/}
}
TY - JOUR AU - Pruitt, Michael TI - Maximum Norm Regularity of Periodic Elliptic Difference Operators With Variable Coefficients JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 1451 EP - 1461 VL - 49 IS - 5 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2015018/ DO - 10.1051/m2an/2015018 LA - en ID - M2AN_2015__49_5_1451_0 ER -
%0 Journal Article %A Pruitt, Michael %T Maximum Norm Regularity of Periodic Elliptic Difference Operators With Variable Coefficients %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 1451-1461 %V 49 %N 5 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2015018/ %R 10.1051/m2an/2015018 %G en %F M2AN_2015__49_5_1451_0
Pruitt, Michael. Maximum Norm Regularity of Periodic Elliptic Difference Operators With Variable Coefficients. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1451-1461. doi : 10.1051/m2an/2015018. http://archive.numdam.org/articles/10.1051/m2an/2015018/
, Smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm. SIAM J. Numer. Anal. 47 (2009) 2476–2495. | DOI | Zbl
and , On the accuracy of finite difference methods for elliptic problems with interfaces. Commun. Appl. Math. Comput. Sci. 1 (2006) 91–119. | DOI | Zbl
, Interior a priori estimates in discrete norms for solutions of parabolic and elliptic difference equations. Ann. Mat. Pura Appl. 95 (1973) 1–43. | DOI | Zbl
, On the regularity of difference schemes. Ark. Mat. 19 (1981) 71–95. | DOI | Zbl
, On the regularity of difference schemes part ii. regularity estimates for linear and nonlinear problems, Ark. Mat. 21 (1983) 3–28. | DOI | Zbl
M. Pruitt, Maximum Norm Regularity of Implicit Difference Methods for Parabolic Equations. Ph.D. Thesis, Duke University (2011).
M. Pruitt, Large time step maximum norm regularity of l-stable difference methods for parabolic equations. Numer. Math. (2014) 1–37. | Zbl
M. Renardy and R. Rogers, An Introduction to Partial Differential Equations. Texts Appl. Math. Springer (2004). | Zbl
A. Samarskii, The Theory of Difference Schemes. Pure Appl. Math., Marcel Decker (2001). | Zbl
, The finite fourier series and elementary geometry. Amer. Math. Mont. 57 (1950) 390–404. | DOI | Zbl
, Interior Estimates in for Elliptic Difference Operators. SIAM J. Numer. Anal. 10 (1973) 69–80. | DOI | Zbl
, On estimates for certain sums for functions defined on a grid, Izv. Akad. Nauk SSSR, Ser. Mat. 4 (1940) 5–16.
, and , Regularity estimates up to the boundary for elliptic systems of difference equations. SIAM J. Numer. Anal. 27 (1990) 292–322. | DOI | Zbl
, Discrete interior schauder estimates for elliptic difference operators. SIAM J. Numer. Anal. 5 (1968) 626–645. | DOI | Zbl
and , Elliptic difference equations and interior regularity. Numer. Math. 11 (1968) 196–210. | DOI | Zbl
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