×

Residual-based a posteriori error estimator for the mixed finite element approximation of the biharmonic equation. (English) Zbl 1223.65087

The author derives a reliable and efficient residual-based a posteriori error for the Ciarlet-Raviart mixed finite element method for the biharmonic equation on polygonal domains. Numerical tests are done to illustrate the performance of the estimator.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J40 Boundary value problems for higher-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ciarlet, Mathematical aspects of finite elements in partial differential equations (Proc Sympos, Math Res Center, Univ Wisconsin, Madison, Wis) pp 125– (1974) · doi:10.1016/B978-0-12-208350-1.50009-1
[2] Ciarlet, The finite element method for elliptic problems (1978)
[3] Morley, The triangular equilibrium problem in the solution of plate bending problems, Aero Quart 19 pp 149– (1968)
[4] Engel, Continuous/discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Comput Methods Appl Mech Eng 191 pp 3669– (2002) · Zbl 1086.74038 · doi:10.1016/S0045-7825(02)00286-4
[5] Brenner, C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains, J Sci Comput 22/23 pp 83– (2005) · Zbl 1071.65151 · doi:10.1007/s10915-004-4135-7
[6] Mozolevski, A priori error analysis for the hp-version of the discontinuous Galerkin finite element method for the biharmonic equation, Comput Methods Appl Math 3 pp 596– (2003) · Zbl 1048.65100 · doi:10.2478/cmam-2003-0037
[7] Feng, Two-level nonoverlapping additive Schwarz methods for a discontinuous Galerkin approximation of the biharmonic problem, J Sci Comput 22 pp 299– (2005) · Zbl 1072.65161 · doi:10.1007/s10915-004-4141-9
[8] Mozolevski, hp-version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation, J Sci Comput 30 pp 465– (2007) · Zbl 1116.65117 · doi:10.1007/s10915-006-9100-1
[9] Süli, hp-version interior penalty DGFEMs for the biharmonic equation, Comput Methods Appl Mech Eng 196 pp 13– (2007) · Zbl 1173.65360 · doi:10.1016/j.cma.2006.06.014
[10] Feng, A posteriori error estimates for finite element approximations of the Cahn-Hilliard equation and Hele-Shaw flow, J Comput Math 26 pp 767– (2008) · Zbl 1174.65035
[11] Gudi, Mixed discontinuous Galerkin methods for the biharmonic equation, J Sci Comput 37 pp 103– (2008) · Zbl 1203.65254 · doi:10.1007/s10915-008-9200-1
[12] Ainsworth, A posteriori error estimation in finite element analysis, Pure and applied mathematics (2000) · Zbl 1008.65076 · doi:10.1002/9781118032824
[13] Babuška, The Finite element method and its reliability (2001)
[14] Bangerth, Adaptive finite element methods for differential equations, Lectures in Mathematics ETH Zürich (2003) · Zbl 1020.65058 · doi:10.1007/978-3-0348-7605-6
[15] Binev, Adaptive finite element methods with convergence rates, Numer Math 97 pp 219– (2004) · Zbl 1063.65120 · doi:10.1007/s00211-003-0492-7
[16] Brenner, The mathematical theory of finite element methods (2008) · Zbl 1135.65042 · doi:10.1007/978-0-387-75934-0
[17] Carstensen, A unifying theory of a posteriori finite element error control, Numer Math 100 pp 617– (2005) · Zbl 1100.65089 · doi:10.1007/s00211-004-0577-y
[18] Carstensen, A posteriori error estimates for nonconforming finite element methods, Numer Math 92 pp 233– (2002) · Zbl 1010.65044 · doi:10.1007/s002110100378
[19] Dari, A posteriori error estimators for nonconforming finite element methods, RAIRO Model Math Anal Numer 30 pp 385– (1996) · Zbl 0853.65110
[20] Karakashian, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J Numer Anal 41 pp 2374– (2003) · Zbl 1058.65120 · doi:10.1137/S0036142902405217
[21] Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques (1995)
[22] Beirão da Veiga, A posteriori error estimates for the Morley plate bending element, Numer Math 106 pp 165– (2007) · Zbl 1110.74050 · doi:10.1007/s00211-007-0066-1
[23] Brenner, An a posteriori error estimator for a quadratic C0 interior penalty method for the biharmonic problem, IMA J Numer Anal (2009)
[24] Georgoulis, An a posteriori error indicator for discontinuous Galerkin approximations of fourth order elliptic problems, IMA J Numer Anal · Zbl 1209.65124
[25] P. Hansbo M. G. Larson A posteriori error estimates for continuous/discontinuous Galerkin approximations of the Kirchoff-Love plate 2008 10
[26] Charbonneau, A residual-based a posteriori error estimator for the Ciarlet-Raviart formulation of the first biharmonic problem, Numer Methods Partial Differential Eq 13 pp 93– (1997) · Zbl 0867.65056 · doi:10.1002/(SICI)1098-2426(199701)13:1<93::AID-NUM7>3.0.CO;2-H
[27] Babuška, Analysis of mixed methods using mesh dependent norms, Math Comput 35 pp 1039– (1980) · doi:10.1090/S0025-5718-1980-0583486-7
[28] Brezzi, Mixed and hybrid finite element methods (1991) · Zbl 0788.73002 · doi:10.1007/978-1-4612-3172-1
[29] R. Verfürth A posteriori error estimation and adaptive mesh-refinement techniques 50 1994 67 83 · Zbl 0811.65089
[30] Grisvard, Singularities in boundary value problems (1992) · Zbl 0766.35001
[31] Carstensen, Error reduction and convergence for an adaptive mixed finite element method, Math Comp 75 pp 1033– (2006) · Zbl 1094.65112 · doi:10.1090/S0025-5718-06-01829-1
[32] Carstensen, Convergence analysis of an adaptive nonconforming finite element method, Numer Math 103 pp 251– (2006) · Zbl 1101.65102 · doi:10.1007/s00211-005-0658-6
[33] J. M. Cascon C. Kreuzer R. H. Nochetto K. G. Siebert Quasi-optimal convergence rate for an adaptive finite element method, Preprint 009, Institut fur Mathematik 2007 · Zbl 1176.65122
[34] Dörfler, A convergent adaptive algorithm for poisson’s equation, SIAM J Numer Anal 33 pp 1106– (1996) · Zbl 0854.65090 · doi:10.1137/0733054
[35] Karakashian, Convergence of adaptive discontinuous galerkin approximations of second-order elliptic problems, SIAM J Numer Anal 45 pp 641– (2007) · Zbl 1140.65083 · doi:10.1137/05063979X
[36] Mekchay, Convergence of adaptive finite element methods for general second order linear elliptic PDEs, SIAM J Numer Anal 43 pp 1803– (2005) · Zbl 1104.65103 · doi:10.1137/04060929X
[37] Morin, Data oscillation and convergence of adaptive FEM, SIAM J Numer Anal 38 pp 466– (2000) · Zbl 0970.65113 · doi:10.1137/S0036142999360044
[38] Stevenson, Optimality of a standard adaptive finite element method, FoCM 7 pp 245– (2007) · Zbl 1136.65109
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.