×

zbMATH — the first resource for mathematics

Asymptotic expansions and \(L^{\infty}\)-error estimates for mixed finite element methods for second order elliptic problems. (English) Zbl 0676.65109
A mixed formation of a second order elliptic problem is considered and asymptotic properties of mixed finite element approximations which are obtained by using new families of finite element spaces (BDM spaces) [cf. F. Brezzi, J. Douglas jun. and L. D. Marini, ibid. 47, 217-235 (1985; Zbl 0599.65072)] are studied. Assuming the only geometric requirement of quasi-uniform regularity on the triangulation of the domain, asymptotic expansions for the first order approximation of the scalar field are desired. It is shown that Richardson extrapolation can be applied to increase the accuracy of the approximations. Another application of the asymptotic expansion which is called error corrected method is also considered.
Reviewer: H.P.Dikshit

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Babu?ka, I.: The finite element method with Lagrangian multipliers. Numer. Math.20, 179-192 (1973) · Zbl 0258.65108
[2] Blum, H., Lin, Q., Rannacher, R.: Asymptotic error expansion and Richardson extrapolation for linear elements. Numer. Math.49, 11-37 (1986) · Zbl 0594.65082
[3] Brezzi, F.: On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO, Anal. Numér.2, 129-151 (1974) · Zbl 0338.90047
[4] 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) · Zbl 0599.65072
[5] Ciarlet, P.: The finite element method for elliptic problem. Amsterdam: North Holland 1978 · Zbl 0383.65058
[6] Douglas Jr., J., Dupont, T., Wahlbin, L.B.: OptimalL ? error estimates for Galerkin approximations to solutions of two-point boundary value problems. Math. Comput29, 475-483 (1975) · Zbl 0306.65053
[7] Douglas Jr., J., Dupont, T., Wheeler, M.F.: AnL ? estimate and a superconvergence result for a Galerkin method for elliptic equations based on tensor products of piecewise polynomials. RAIRO, Anal. Numér.8, 61-66 (1974) · Zbl 0315.65062
[8] Douglas Jr., J., Roberts, J.E.: Global estimates for mixed finite element methods for second order elliptic equations. Math. Comput44, 39-52 (1985) · Zbl 0624.65109
[9] Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comput.34, 441-463 (1980) · Zbl 0423.65009
[10] Durán, R.: Error analysis inL p , 1?p??, for mixed finite element methods for linear and quasilinear elliptic problems. RAIRO, Anal. Numér22, 371-387 (1988) · Zbl 0698.65060
[11] Durán, R., Nochetto, R.H., Wang, J.: Sharp maximum norm error estimates for finite element approximations of the Stokes problem in 2-D. Math. Comput.51, 491-506 (1988) · Zbl 0699.76038
[12] Falk, R., Osborn, J.: Error estimates for mixed methods. RAIRO Anal. Numér.14, 249-277 (1980) · Zbl 0467.65062
[13] Fortin, M.: An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numér.11, 341-354 (1977) · Zbl 0373.65055
[14] Frehse, J., Rannacher, R.: EineL 1-Fehlerabschätzung für diskrete Grundlösungen in der Methode der finite Elemente. Bonn. Math. Schrift89, 92-114 (1976) · Zbl 0359.65093
[15] Gastaldi, L., Nochetto, R.H.: Sharp maximum norm error estimates for new families of mixed finite element methods (to appear) · Zbl 0673.65060
[16] Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations. New York: Springer 1986 · Zbl 0585.65077
[17] Grisvard, P.: Elliptic problems in nonsmooth domains, London: Pitman Advanced Publishing Program, 1985 · Zbl 0695.35060
[18] Lin, Q., Lu, T.: Asymptotic expansion for finite element approximation of elliptic problems on polygonal domains. Sixth Int. Conf. Comp. Math. Appl. Sci. Eng. (1983) Versailles
[19] Lin, Q., Wang, J.: Some expansions of the finite element approximation, Research Report IMS, 15 (1984), Chengdu Branch of Academia Sinica · Zbl 0553.65071
[20] Marchuk, G.I., Shaidurov, V.V.: Difference methods and their extrapolations, Berlin Heidelberg New York: Springer 1983 · Zbl 0511.65076
[21] Nakata, M., Weiser, A., Wheeler, M.F.: Some superconvergence results for mixed finite element methods for elliptic problems on rectangular domains. In: Whiteman, J.R. (ed.) The mathematics of finite elements and applications V, pp. 59-81. London: Academic Press, 1985 · Zbl 0583.65078
[22] Natterer, F.: Über die punktweise Konvergenz finiter Elemente. Numer. Math.25, 67-77 (1975) · Zbl 0331.65073
[23] Nitsche, J.A.:L ?-Convergence of finite element approximations, Mathematical Aspects of Finite Element methods. Lect. Notes Math.606, 261-274 (1977)
[24] Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximations. Math. Comput.38, 437-445 (1982) · Zbl 0483.65007
[25] Raviart, P.-A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of Finite Element Methods. Lect. Notes Math.606, 292-315 (1977) · Zbl 0362.65089
[26] Schatz, A.H., Wahlbin, L.B.: Interior maximum norm estimates for finite element methods. Math. Comput.31, 414-442 (1976) · Zbl 0364.65083
[27] Scholz, R.:L ?-convergence of saddle-point approximations for second order problems. RAIRO Anal. Numér.11, 209-216 (1977) · Zbl 0356.35026
[28] Scott, R.: OptimalL ?-estimates for the finite element method on irregular meshes. Math. Comput.30, 681-697 (1976) · Zbl 0349.65060
[29] Wheeler, M.F.: An optimalL ? error estimate for Galerkin approximations to solutions of two-point boundary value problems. SIAM J. Numer. Anal.10, 914-917 (1973) · Zbl 0266.65061
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.