×

zbMATH — the first resource for mathematics

A unifying theory of a posteriori error control for nonconforming finite element methods. (English) Zbl 1127.65083
Residual-based a posteriori error estimates for nonconforming elements contain extra terms in comparison to the conforming case, e.g., jumps of tangential components. A unified framework is the topic. The paper starts with mixed methods and assumes that there is an associated space \(V_h^{nc}\) of nonconforming elements. Another hypothesis is the existence of a space \(V_h^c\) of conforming elements for which an interpolation operator of Clément type and a mapping \(V_h^c\to V_h^{nc}\) exist. Three tables contain lists of examples to which the theory applies. The efficiency of the estimators is not treated in this general framework.

MSC:
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ainsworth M. and Oden J.T. (2000). A posteriori error estimation in finite element analysis. Wiley, New York · Zbl 1008.65076
[2] Ainsworth M. (2005). A posteriori error estimation for non-conforming quadrilateral finite elements. Int. J. Numer. Anal. Model 2: 1–18 · Zbl 1071.65141
[3] Arnold D.N. (1982). An interior penalty finite element method with discontinuous elements. IAM J. Numer. Anal. 19: 742–760 · Zbl 0482.65060 · doi:10.1137/0719052
[4] Arnold D.N., Brezzi F., Cockburn B. and Marini D. (2000). Discontinuous Galerkin methods for elliptic problems. In: Cockburn, B., Karniadakis, G., and Shu, C.W. (eds) Discontinuous Galerkin Methods: Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering, vol. 11., pp 89–101. Springer, New York · Zbl 0948.65127
[5] Baker G. (1977). Finite element methods for elliptic equations using nonconforming elements. Math. Comp. 31: 45–59 · Zbl 0364.65085 · doi:10.1090/S0025-5718-1977-0431742-5
[6] Baker G.A., Jureidini W.N. and Karakashian O.A. (1990). Piecewise solenoidal vector fields and the Stokes problem. SIAM J. Numer. Anal. 27: 1466–1485 · Zbl 0719.76047 · doi:10.1137/0727085
[7] Becker R., Hansbo P. and Larson M. (2003). Energy norm a posteriori error estimation for. discontinuous Galerkin methods. Comput. Methods Appl. Mech. Engrg. 192: 723–733 · Zbl 1042.65083 · doi:10.1016/S0045-7825(02)00593-5
[8] Bernardi C. and Girault V. (1998). A local regularisation operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35: 1893–1916 · Zbl 0913.65007 · doi:10.1137/S0036142995293766
[9] Bernardi C. and Hecht F. (2002). Error indicators for the mortar finite element discretization of the Laplace equation. Math. Comp. 71: 1371–1403 · Zbl 1012.65108 · doi:10.1090/S0025-5718-01-01401-6
[10] Bernardi C., Maday Y. and Patera A.T. (1993). Domain decomposition by the mortar element method. In: Kaper, H. (eds) Asymptotic and Numerical Methods for Partial Differential Equations and Their Applications., pp 269–286. Reidel, Dordrecht · Zbl 0799.65124
[11] Bernardi, C., Maday, Y., Patera, A.T.: A new nonconforming approach to domain decomposition: the mortar element method. In: Nonlinear Partial Differential Equations and Their Applications, Paris, pp. 13–51 (1994) · Zbl 0797.65094
[12] Bernardi, C., Owens, R.G. Valenciano, J.: An error indicator for mortar element solutions to the Stokes problem. In: Internal Report 99030, Laboratoire d’Analyse Numrique, Université Pierre et Marie Curie, Paris (1999)
[13] Braess D. (1997). Finite Elements. Cambridge University Press, London · Zbl 0870.65097
[14] Braess D., Carstensen C. and Reddy B.D. (2004). Uniform convergence and a posteriori error estimators for the enhanced strain finite element method. Numer. Math. 96: 461–479 · Zbl 1050.65097 · doi:10.1007/s00211-003-0486-5
[15] Brenner S.C. and Scott L.R. (2002). The Mathematical Theory of Finite Element Methods. Springer, Berlin · Zbl 1012.65115
[16] Brenner S.C. and Sung L.Y. (1992). Linear finite element methods for planar linear elasticity. Math. Comp. 59: 321–338 · Zbl 0766.73060 · doi:10.1090/S0025-5718-1992-1140646-2
[17] Brezzi F. and Fortin M. (1991). Mixed and Hybrid Finite Element Methods. Springer, Berlin · Zbl 0788.73002
[18] Bustinza R. and Gabriel N. (2005). Gatica and Bernardo Cockburn. An a posteriori error estimate for the local discontinuous Galerkin method applied to linear and nonlinear diffusion problems. J. Sci. Comput. 22: 147–185 · Zbl 1065.76133 · doi:10.1007/s10915-004-4137-5
[19] Cai Z., Ye X. and Douglas J. Jr. (1999). A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier–Stokes equations. CaLcoLo 36: 215–232 · Zbl 0947.76047 · doi:10.1007/s100920050031
[20] Carstensen C. (2005). A unifying theory of a posteriori finite element error control. Numer. Math. 100: 617–637 · Zbl 1100.65089 · doi:10.1007/s00211-004-0577-y
[21] Carstensen C. (1999). Quasi-interpolation and a posteriori error analysis in finite element methods. M2AN Math. Model. Numer. Anal. 33: 1187–1202 · Zbl 0948.65113 · doi:10.1051/m2an:1999140
[22] Carstensen C. and Bartels S. (2002). Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming and mixed FEM. Math. Comp. 71: 945–969 · Zbl 0997.65126
[23] Carstensen C., Bartels S. and Jansche S. (2002). A posteriori error estimates for nonconforming finite element methods. Numer. Math. 92: 233–256 · Zbl 1010.65044 · doi:10.1007/s002110100378
[24] Carstensen C. and Dolzmann G. (1998). A posteriori error estimates for mixed FEM in elasticity. Numer. Math. 81: 187–209 · Zbl 0928.74093 · doi:10.1007/s002110050389
[25] Carstensen, C., Hu, J., Orlando, A.: Framework for the a posteriori error analysis of nonconforming finite elements. Preprint (2005-11), Department of Mathematics, Humboldt University of Berlin (2005). SIAM J. Numer. Anal. 45, 68–82 (2007) · Zbl 1165.65072
[26] Creusé E., Kunert G. and Nicaise S. (2004). A posteriori error estimation for the Stokes problem: anisotropic and isotropic discretications. M3AS 14: 1–48 · Zbl 1071.65142
[27] Crouzeix M. and Raviart P.-A. (1973). Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér. 7: 33–76 · Zbl 0302.65087
[28] Dari E., Duran R. and Padra C. (1995). Error estimators for nonconforming finite element approximations of the Stokes problem. Math. Comp. 64: 1017–1033 · Zbl 0827.76042 · doi:10.1090/S0025-5718-1995-1284666-9
[29] Dari E., Duran R., Padra C. and Vampa V. (1996). A posteriori error estimators for nonconforming finite element methods. Math. Model. Numer. Anal. 30: 385–400 · Zbl 0853.65110
[30] Douglas J. Jr., Dupont T. (1976) Interior penalty procedures for elliptic and parabolic Galerkin methods. In: Lectures Notes in Physics, vol. 58. Springer, Berlin
[31] Santos J.E., Sheen D., Ye X. and Douglas J. Jr. (1999). Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems. Math. Model. Numer. Anal. 33: 747–770 · Zbl 0941.65115 · doi:10.1051/m2an:1999161
[32] Falk R.S. (1991). Nonconforming finite element methods for the equations of linear elasticity. Math. Comp. 57: 529–550 · Zbl 0747.73044 · doi:10.1090/S0025-5718-1991-1094947-6
[33] Grajewski, M., Hron, J., Turek, S.: Numerical analysis for a new non-conforming linear finite element on quadrilaterals. J. Comput. Appl. Math. (in press) · Zbl 1094.65117
[34] Grajewski, M., Hron, J., Turek, S.: Dual Weighted a posteriori error estimation for a new nonconforming linear finite element on quadrilaterals. www.mathematik.uni-dortmund.de/lsiii/static/showpdffile_GrajewskiHronTurek2004.pdf · Zbl 1074.65123
[35] Girault V. and Raviart P.-A. (1986). Finite Element Methods for Navier–Stokes Equations. Springer, Berlin · Zbl 0585.65077
[36] Han H.-D. (1984). Nonconforming elements in the mixed finite element method. J. Comp. Math. 2: 223–233 · Zbl 0573.65083
[37] Houston P., Schotzau D. and Wihler T.P. (2005). Energy norm shape a posteriori error estimation for mixed discontinuous Galerkin approximations of the Stokes problem. J. Sci. Comput. 22: 347–370 · Zbl 1091.78017 · doi:10.1007/s10915-004-4142-8
[38] Houston P., Schotzau D. and Wihler T.P. (2006). An hp-adaptive mixed discontinuous Galerkin FEM for nearly incompressible linear elasticity. Comp. Methods Appl. Mech. Engrg. 195: 224–3246 · Zbl 1118.74049 · doi:10.1016/j.cma.2005.06.012
[39] Houston, P., Schotzau, D., Wihler, T.P.: Energy norm a posteriori error estimation of hp- adaptive discontinuous Galerkin methods for elliptic problems. M3AS (to appear) · Zbl 1116.65115
[40] Hu J., Man H.-Y. and Shi Z.-C. (2005). Constrained nonconforming rotated Q 1 element for Stokes flow and planar elasticity. Math. Numer. Sin. (in Chinese) 27: 311–324
[41] Hu J. and Shi Z.-C. (2005). Constrained quadrilateral nonconforming rotated Q 1-element. J. Comp. Math. 23: 561–586 · Zbl 1086.65111
[42] Kanschat G. and Suttmeier F.-T. (1999). A posteriori error estimates for nonconforming finite element schemes. Calcolo 36: 129–141 · Zbl 0936.65128 · doi:10.1007/s100920050027
[43] Karakashian O.A. and Jureidini W.N. (1998). A nonconforming finite element method for the stationary Navier–Stokes equations. SIAM J. Numer. Anal. 35: 93–120 · Zbl 0933.76047 · doi:10.1137/S0036142996297199
[44] Karakashian O.A. and Pascal F. (2003). A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41: 2374–2399 · Zbl 1058.65120 · doi:10.1137/S0036142902405217
[45] Kouhia R. and Stenberg R. (1995). A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Engrg. 124: 195–212 · Zbl 1067.74578 · doi:10.1016/0045-7825(95)00829-P
[46] Lee C.O., Lee J. and Sheen D.W. (2003). A locking-free nonconforming finite element method for planar linear elasticity. Adv. Comput. Math. 19: 277–291 · Zbl 1064.74165 · doi:10.1023/A:1022838628615
[47] Lin Q., Tobiska L. and Zhou A. (2005). On the superconvergence of nonconforming low order finite elements applied to the Poisson equation. IMA. J. Numer. Anal. 25: 160–181 · Zbl 1068.65122 · doi:10.1093/imanum/drh008
[48] Ming, P.-B.: Nonconforming finite element vs locking problem. Doctorate Dissertation (in Chinese), Institute of Computational Mathematics, Chinese Academy of Science (1999)
[49] Park C. and Sheen D. (2003). P1-nonconforming quadrilateral finite element methods for second-order elliptic problems. SIAM J. Numer. Anal. 41: 624–640 · Zbl 1048.65114 · doi:10.1137/S0036142902404923
[50] Rannacher R. and Turek S. (1992). Simple nonconforming quadrilateral Stokes element. Numer. Methods PDE 8: 97–111 · Zbl 0742.76051 · doi:10.1002/num.1690080202
[51] Riviere B. and Wheeler M.F. (2003). A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems. Comput. Math. Appl. 46: 141–163 · Zbl 1059.65098 · doi:10.1016/S0898-1221(03)90086-1
[52] Riviere B. and Wheeler M.F. (2003). A posteriori error estimates and mesh adaptation strategy for discontinuous Galerkin methods applied to diffusion problems. Comput. Math. Appl. 46: 141–163 · Zbl 1059.65098 · doi:10.1016/S0898-1221(03)90086-1
[53] Simo J.C. and Rifai M.S. (1990). A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Methods Engrg. 29: 1595–1638 · Zbl 0724.73222 · doi:10.1002/nme.1620290802
[54] Shi Z.-C. (1984). A convergence condition for quadrilateral Wilson element. Numer. Math. 44: 349–361 · Zbl 0581.65008 · doi:10.1007/BF01405567
[55] Verfürth R. (1996). A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley–Teubner, Stuttgart · Zbl 0853.65108
[56] Wang L.-H. and Qi H. (2002). On locking-free finite element schemes for the pure displacement boundary value problem in the planar elasticity. Math. Numer. Sin. (in Chinese) 24: 243–256 · Zbl 1081.74557
[57] Wheeler M.F. (1978). An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15: 152–161 · Zbl 0384.65058 · doi:10.1137/0715010
[58] Wihler T.P. (2006). Locking-free adaptive discontinuous Galerkin FEM for linear elasticity problems. Math. Comp. 75: 1087–1102 · Zbl 1088.74047 · doi:10.1090/S0025-5718-06-01815-1
[59] Wilson E.L., Taylor R.L., Doherty W. and Ghaboussi J. (1973). Incompatible displacement models. In: Fenves, S.J., Perrone, N., Robinson, A.R., and Schnobrich, W.C. (eds) Numerical and Computer Methods in Structural Mechanics., pp 43–57. Academic, New York
[60] Wohlmuth B. (1999). A residual based error estimator for mortar finite element discretizations. Numer. Math. 84: 143–171 · Zbl 0962.65090 · doi:10.1007/s002110050467
[61] Zhang Z.-M. (1997). Analysis of some quadrilateral nonconforming elements for incompressible elasticity. SIAM J. Numer. Anal. 34: 640–663 · Zbl 0870.73074 · doi:10.1137/S0036142995282492
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.