Positivity preserving finite element approximation. (English) Zbl 1001.41011

In this paper, finite elements on quasi-uniform partitions into simplices are studied. In two dimensions, the partitions are the usual triangulations, but the results in this paper work in higher dimensions, where general simplices are used, as well. The particular question addressed is whether there are higher order accurate approximations by finite elements on such partitions if positivity preservation is required. The approximation is by a Lagrange finite element reproducing a finite element operator which is positive and bounded. There is an impossibility result on the approximation of general functions to higher order accuracy at extreme points of a domain.


41A25 Rate of convergence, degree of approximation
41A36 Approximation by positive operators
65D05 Numerical interpolation
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text: DOI


[1] Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. · Zbl 0804.65101
[2] Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. · Zbl 0788.73002
[3] Zhiming Chen and Ricardo H. Nochetto, Residual type a posteriori error estimates for elliptic obstacle problems, Numer. Math. 84 (2000), no. 4, 527 – 548. · Zbl 0943.65075
[4] P.G. Ciarlet, Basic Error Estimates for Elliptic Problems, in Handbook of Numerical Analysis, Vol II (Finite Element Methods (Part 1)), P.G. Ciarlet and J.L. Lions eds, North-Holland, 1991, 17-351. CMP 91:14
[5] Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. \jname RAIRO Analyse Numérique 9 (1975), no. R-2, 77 – 84 (English, with Loose French summary). · Zbl 0368.65008
[6] Stephen Hilbert, A mollifier useful for approximations in Sobolev spaces and some applications to approximating solutions of differential equations, Math. Comp. 27 (1973), 81 – 89. · Zbl 0257.65087
[7] P. P. Korovkin, Linear operators and approximation theory, Translated from the Russian ed. (1959). Russian Monographs and Texts on Advanced Mathematics and Physics, Vol. III, Gordon and Breach Publishers, Inc., New York; Hindustan Publishing Corp. (India), Delhi, 1960. · Zbl 0094.10201
[8] G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. · Zbl 0153.38901
[9] R. H. Nochetto, K. Siebert, and A. Veeser, Pointwise a posteriori error control of elliptic obstacle problems (to appear). · Zbl 1027.65089
[10] L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483 – 493. · Zbl 0696.65007
[11] Gilbert Strang, Approximation in the finite element method, Numer. Math. 19 (1972), 81 – 98. · Zbl 0221.65174
[12] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, 1996. · Zbl 0853.65108
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.