Local energy estimates for the finite element method on sharply varying grids

Demlow, Alan; Guzmán, Johnny; Schatz, Alfred

doi:10.1090/S0025-5718-2010-02353-1

Local energy estimates for the finite element method on sharply varying grids
HTML articles powered by AMS MathViewer

by Alan Demlow, Johnny Guzmán and Alfred H. Schatz PDF

Math. Comp. 80 (2011), 1-9 Request permission

Abstract:

Local energy error estimates for the finite element method for elliptic problems were originally proved in 1974 by Nitsche and Schatz. These estimates show that the local energy error may be bounded by a local approximation term, plus a global “pollution” term that measures the influence of solution quality from outside the domain of interest and is heuristically of higher order. However, the original analysis of Nitsche and Schatz is restricted to quasi-uniform grids. We present local a priori energy estimates that are valid on shape regular grids, an assumption which allows for highly graded meshes and which matches much more closely the typical practical situation. Our chief technical innovation is an improved superapproximation result.

References

Douglas N. Arnold and Xiao Bo Liu, Local error estimates for finite element discretizations of the Stokes equations, RAIRO Modél. Math. Anal. Numér. 29 (1995), no. 3, 367–389 (English, with English and French summaries). MR 1342712, DOI 10.1051/m2an/1995290303671
Randolph E. Bank and Michael Holst, A new paradigm for parallel adaptive meshing algorithms, SIAM J. Sci. Comput. 22 (2000), no. 4, 1411–1443. MR 1797889, DOI 10.1137/S1064827599353701
Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376, DOI 10.1007/978-1-4757-3658-8
Alan Demlow, Localized pointwise error estimates for mixed finite element methods, Math. Comp. 73 (2004), no. 248, 1623–1653. MR 2059729, DOI 10.1090/S0025-5718-04-01650-3
Alan Demlow, Local a posteriori estimates for pointwise gradient errors in finite element methods for elliptic problems, Math. Comp. 76 (2007), no. 257, 19–42. MR 2261010, DOI 10.1090/S0025-5718-06-01879-5
Johnny Guzmán, Pointwise error estimates for discontinuous Galerkin methods with lifting operators for elliptic problems, Math. Comp. 75 (2006), no. 255, 1067–1085. MR 2219019, DOI 10.1090/S0025-5718-06-01823-0
Xiaohai Liao and Ricardo H. Nochetto, Local a posteriori error estimates and adaptive control of pollution effects, Numer. Methods Partial Differential Equations 19 (2003), no. 4, 421–442. MR 1980188, DOI 10.1002/num.10053
R. H. Nochetto, M. Paolini, and C. Verdi, An adaptive finite element method for two-phase Stefan problems in two space dimensions. I. Stability and error estimates, Math. Comp. 57 (1991), no. 195, 73–108, S1–S11. MR 1079028, DOI 10.1090/S0025-5718-1991-1079028-X
M. S. Robertson, The variation of the sign of $V$ for an analytic function $U+iV$, Duke Math. J. 5 (1939), 512–519. MR 51
Albert H. Schatz, Vidar Thomée, and Wolfgang L. Wendland, Mathematical theory of finite and boundary element methods, DMV Seminar, vol. 15, Birkhäuser Verlag, Basel, 1990. MR 1116555, DOI 10.1007/978-3-0348-7630-8
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
A. H. Schatz and L. B. Wahlbin, Interior maximum-norm estimates for finite element methods. II, Math. Comp. 64 (1995), no. 211, 907–928. MR 1297478, DOI 10.1090/S0025-5718-1995-1297478-7
Jinchao Xu and Aihui Zhou, Local and parallel finite element algorithms based on two-grid discretizations, Math. Comp. 69 (2000), no. 231, 881–909. MR 1654026, DOI 10.1090/S0025-5718-99-01149-7