Previous |  Up |  Next


post-processing; averaged gradient; elliptic systems; second order elliptic systems; linear finite elements; regular uniform triangulations; error estimats; optimal order; superconvergence
Second order elliptic systems with boundary conditions of Dirichlet, Neumann's or Newton's type are solved by means of linear finite elements on regular uniform triangulations. Error estimates of the optimal order $O(h^2)$ are proved for the averaged gradient on any fixed interior subdomain, provided the problem under consideration is regular in a certain sense.
[1] J. H. Bramble J. A. Nitsche A. H. Schatz: Maximum-norm interior estimates for Ritz-Galerkin methods. Math. Соmр. 29 (1975), 677-688. MR 0398120
[2] P. G. Ciarlet: The finite element method for elliptic problems. North-Holland, Amsterdam, New York, Oxford, 1978. MR 0520174 | Zbl 0383.65058
[3] G. Fichera: Existence theorems in elasticity. Encycl. of Physics, ed. S. Flügge, vol. VIa/2, Springer-Verlag, Berlin, 1972.
[4] G. Geymonat: Sui problemi ai limiti per i sistemi lineari ellittici. Ann. Mat. Рurа Appl. 69 (1965), 207-284. DOI 10.1007/BF02414374 | MR 0196262
[5] P. Grisvard: Behaviour of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain. in: Numerical solution of partial differential equations III, Academic Press, New York, 1976, 207-274. MR 0466912
[6] P. Grisvard: Boundary value problems in nonsmooth domains. Univ. of Maryland, Dep. of Math., College Park, Lecture Notes 19, 1980.
[7] E. Hewitt K. Stromberg: Real and abstract analysis. Springer-Verlag, New York, Heidelberg, Berlin, 1975. MR 0367121
[8] I. Hlaváček M. Křížek: On a superconvergent finite element scheme for elliptic systems. I. Dirichlet boundary conditions. Apl. Mat. 32 (1987), 131-154. MR 0885758
[9] I. Hlaváček M. Křížek: On a superconvergent finite element scheme for elliptic systems. II. Boundary conditions of Newton's or Neumann's type. Apl. Mat. 32 (1987), 200-213. MR 0895878
[10] J. Kadlec: On the regularity of the solution of the Poisson problem on a domain with boundary locally similar to the boundary of a convex open set. Czechoslovak Math. J. 14 (1964), 386-393. MR 0170088 | Zbl 0166.37703
[11] V. A. Kondratěv: Boundary problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč. 16 (1967), 209-292. MR 0226187
[12] M. Moussaoui: Régularité de la solution d'un problème à derivée oblique. Comptes Rendus Acad. Sci. Paris, 279, sér. A, 25 (1974), 869-872. MR 0358062 | Zbl 0293.35014
[13] J. Nečas: Les méthodes directes en theorie des équations elliptiques. Masson, Paris, or Academia, Prague, 1967. MR 0227584
[14] J. A. Nitsche A. H. Schatz: Interior estimate for Ritz-Galerkin methods. Math. Соmр. 28 (1974), 937-958. MR 0373325
[15] B. Westergren: Interior estimates for elliptic systems of difference equations. (Thesis), Univ. of Göteborg, 1982.
[16] Q. D. Zhu: Natural inner superconvergence for the finite element method. Proc. China-France Sympos. on Finite Element Methods, Beijing, 1982, Gordon and Breach, Sci. Publ., Inc., New York, 1983, 935-960. MR 0754041
Partner of
EuDML logo