Summary
In this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems. For the rectangular finite elements of Raviart and Thomas [19] and for those of Brezzi et al. [4] we prove that the distance inL 2 between the approximate solution and a projection of the exact one is of higher order than the error itself.
This result is exploited to obtain superconvergence at Gaussian points and to construct higher order approximations by a local postprocessing.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ainsworth, M., Craig, A.W.: A posteriori error estimates in the finite element method. Numer. Anal. Rep. NA88/03. Durham, England 1988
Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO, Modél. Math. Anal. Numér.19, 7–32 (1985)
Brezzi, F.: On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO, Anal. Numer.2, 129–151 (1974)
Brezzi, F., Douglas Jr., J., Fortin M., Marini, L.D.: Efficient rectangular mixed finite elements in two and three space variables. RAIRO, Modél. Math. Anal. Numér.21, 581–604 (1987)
Brezzi, F., Douglas Jr., J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math.47, 217–235 (1985)
Clement, P.: Approximation by finite element functions using local regularization. RAIRO, Anal. Numer.9, 33–76 (1975)
Douglas Jr. J.: Superconvergence in the pressure in the simulation of miscible displacement. SIAM J. Numer. Anal.22, 962–969 (1985)
Douglas Jr., J., Milner, F.A.: Interior and superconvergence estimates for mixed methods for second order elliptic equations. RAIRO, Modél. Math. Anal. Numér.19, 297–328 (1985)
Douglas Jr., J., Roberts, J.E.: Global estimates for mixed methods for second order elliptic equations. Math. Comput.44, 39–52 (1985)
Dupont, T., Scott, L.R.: Polynomial approximation of functions in Sobolev spaces. Math. Comput.34, 441–464 (1980)
Durán, R.: Error analysis inL P, 1≦p≦∞, for mixed finite element methods for linear and quasi-linear elliptic problems. RAIRO, Modél. Math. Anal. Numér.22, 371–387 (1988)
Durán, R., Muschietti, M.A., Rodríguez, R.: Asymptotically exact error estimators for rectangular finite elements. Notas de Mathemática No 50, Universidad Nacional de La Plata, Argentina 1989
Ewing, R.E., Lazarov, R.D., Wang, J.: Superconvergence of the velocities along the Gaussian lines of the mixed finite element solutions to second order elliptic equations in 2D rectangular domains (to appear).
Falk, R.S., Osborn, J.E.: Error estimates for mixed methods. RAIRO, Anal. Numér.14, 249–277 (1980)
Fortin, M.: An analysis of the convergence of mixed finite element methods. RAIRO, Anal. Numér.11, 341–354 (1977)
Gastaldi, L., Nochetto, R.H.: Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations. RAIRO, Modél. Math. Anal. Numér.23, 103–128 (1989)
Krîzêk, M., Neittaanmäki, P.: On superconvergence techniques. Acta Aplicandae Mathematicae9, 175–198 (1987)
Nedelec, J.: Mixed finite elements in R3. Numer.Math.35, 315–341 (1980)
Raviart, P.A., Thomas, J.M.: A mixed finite element method for second order elliptic problems. Mathematical Aspects of the finite element method. Lecture Notes Math.606, 292–315 (1977)
Zlámal, M.: Some superconvergence results in the finite element method. Mathematical Aspects of the finite element method. Lecture Notes in Math.606, 353–362 (1977)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Durán, R. Superconvergence for rectangular mixed finite elements. Numer. Math. 58, 287–298 (1990). https://doi.org/10.1007/BF01385626
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01385626