×

Anisotropic interpolation with applications to the finite element method. (English) Zbl 0746.65077

For the approximation of anisotropic structures such as edges, boundary or interior layers, it is natural to use a finite element mesh with different mesh sizes in different directions. The authors extend the usual Bramble-Hilbert theory for proving more refined estimates of the interpolation error. These results are applied to common finite elements, and it is observed that elements with a large angle may be useful for approximating anisotropic structures.
The difference between the interpolation and the finite element approximation error is discussed and some results for rectangular elements are derived. The results are applied to the finite element approximation of elliptic equations on domains with edges.
The authors also suggest that it may be interesting to extend their results for other approximation operators [see P. Clément, Revue Franc. Automat. Inform. Rech. Opérat. 9, R-2, 77-84 (1975; Zbl 0368.65008); L. R. Scott and S. Zhang, Math. Comput. 54, No. 190, 483-493 (1990; Zbl 0696.65007)].

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
41A05 Interpolation in approximation theory
65N15 Error bounds for boundary value problems involving PDEs
41A63 Multidimensional problems
35J25 Boundary value problems for second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams, R. A.: Sobolev spaces. New York: Academic Press 1975. · Zbl 0314.46030
[2] Apel, Th.: Finite-Elemente-Methoden über lokal verfeinerten Netzen für elliptische Probleme in Gebieten mit Kanten. Thesis, University of Technology Chemnitz, 1991. · Zbl 0745.65059
[3] Arbenz, P.: Computable finite element error bounds for poissons equation. IMA J. Number. Anal.29, 475–479 (1982). · Zbl 0514.65078 · doi:10.1093/imanum/2.4.475
[4] Arnold, D. N., Breezi, F., Fortin, M.: A stable finite element for the Stokes equations. Calcolo21, 337–344 (1984). · Zbl 0593.76039 · doi:10.1007/BF02576171
[5] Barnhill, R. E., Brown, J. H., Mitchell, A. R.: A comparison of finite element error bounds for Poissons equation. IMA J. Numer. Anal.28, 95–103 (1981). · Zbl 0452.65065 · doi:10.1093/imanum/1.1.95
[6] Bramble, J. H., Hilbert, S. R.: Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation, SIAM. J. Numer. Anal.7, 112–124 (1970). · Zbl 0201.07803 · doi:10.1137/0707006
[7] Bramble, J. H., Hilbert, S. R.: Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. Math.16, 362–369 (1971). · Zbl 0214.41405 · doi:10.1007/BF02165007
[8] Ciarlet, P. G.: The finite element method for elliptic problems. Amsterdam: North-Holland 1978. · Zbl 0383.65058
[9] Ciarlet, P. G., Wagschal, C.: Multipoint Taylor formulas and applications to the finite element method. Numer. Math.17, 84–100 (1971). · Zbl 0199.50104 · doi:10.1007/BF01395869
[10] Clement, P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numer.R-2, 77–84 (1975). · Zbl 0368.65008
[11] Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. RAIRO Anal. Numer.R-3, 33–76 (1973). · Zbl 0302.65087
[12] Dobrowolski, M.: Wie groß ist der Diskretisierungsfehler beim finite Elemente Verfahren? ZAMM70, T667-T668 (1990).
[13] Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comp.34, 441–463 (1980). · Zbl 0423.65009 · doi:10.1090/S0025-5718-1980-0559195-7
[14] Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order. Grundlehren der math. Wiss. 224, Springer: Berlin, 1977. · Zbl 0361.35003
[15] Gout, J. L.: Estimation de l’erreur d’interpolation d’Hermite dans\(\mathbb{R}\) n. Numer. Math.28, 407–429 (1977). · Zbl 0365.65007 · doi:10.1007/BF01404344
[16] Hughes, T. J. R., Franca, L. P., Balestra, M.: A new finite element formulation for computationable fluid mechanics. Comp. Appl. Mech. Eng.59, 85–99 (1986). · Zbl 0622.76077 · doi:10.1016/0045-7825(86)90025-3
[17] Jamet, P.: Estimation de l’erreur d’interpolation dans un domaine variable et application aux éléments finis quadrilatéraux dégénérés, in: Méthodes Numériques en Mathématiques Appliquées, 55–100, Presses de l’Université de Montréal 1976.
[18] Křížek, M.: On semiregular families of triagulations and linear interpolation, to eppear in Proc. EQUADIFF VII, 1989.
[19] Kufner, A., Sändig, A.-M.: Some applications of weighted sobolev spaces. Leipzig: BSB B.G. Teubner Verlassgesellschaft, 1987. · Zbl 0662.46034
[20] Nikolski, S. M.: Inequalities for entire functions of exponential type and their application to the theory of differentiable functions of several variables, Trudy Mat. Inst. Steklov38, 244–278 (1951). [English transl.: Amer. Math. Soc. Transl. (2),80, 1–38 (1969)].
[21] Oganesjan, L. A., Rukhovets, L. A.: Variational difference methods for solving elliptic equations (Russian). Isdatelstvo Akad. Nauk Arm. SSR, Jerevan 1979.
[22] Scott, L. R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp.54, 483–493 (1990). · Zbl 0696.65007 · doi:10.1090/S0025-5718-1990-1011446-7
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.