×

A comparison of a posteriori error estimates for biharmonic problems solved by the FEM. (English) Zbl 1250.65133

Summary: The classical a posteriori error estimates are mostly oriented to the use in the finite element \(h\)-methods (FEMs) while the contemporary higher-order \(hp\)-methods usually require new approaches in a posteriori error estimation. These methods hold a very important position among adaptive numerical procedures for solving ordinary as well as partial differential equations arising from various technical applications.
In the paper, we are concerned with a review and comparison of error estimation procedures for the biharmonic and some more general fourth order partial differential problems with special regards to the needs of the \(hp\)-method. We point out some advantages and drawbacks of analytical and computational a posteriori error estimates.

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] Szabó, B.; Babuška, I., Introduction to Finite Element Analysis. Formulation, Verification and Validation (2011), John Wiley & Sons: John Wiley & Sons Chichester · Zbl 1410.65003
[2] Demkowicz, L., Computing with \(h p\)-Adaptive Finite Elements, vols. 1, 2 (2007), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL, 2008
[3] Šolín, P.; Segeth, K.; Doležel, I., Higher-Order Finite Element Methods (2004), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL · Zbl 1032.65132
[4] Babuška, I.; Rheinboldt, W. C., A posteriori error estimates for the finite element method, Int. J. Numer. Methods Eng., 12, 1597-1615 (1978) · Zbl 0396.65068
[5] Babuška, I.; Rheinboldt, W. C., Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15, 736-754 (1978) · Zbl 0398.65069
[6] Ainsworth, M.; Oden, J. T., A Posteriori Error Estimation in Finite Element Analysis (2000), John Wiley & Sons: John Wiley & Sons New York · Zbl 1008.65076
[7] Babuška, I.; Strouboulis, T., The Finite Element Method and its Reliability (2001), Clarendon Press: Clarendon Press Oxford · Zbl 0997.74069
[8] Fiedler, M., Special Matrices and their Applications in Numerical Mathematics (1986), Martinus Nijhoff Publishers: Martinus Nijhoff Publishers Dordrecht
[9] Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North Holland: North Holland Amsterdam · Zbl 0445.73043
[10] Verfürth, R., A Review of a Posteriori Error Estimation and Adaptive Mesh Refinement Techniques (1996), John Wiley & Sons: John Wiley & Sons Chichester, B.G. Teubner, Stuttgart · Zbl 0853.65108
[11] Charbonneau, A.; Dossou, K.; Pierre, R., A residual-based a posteriori error estimator for the Ciarlet-Raviart formulation of the first biharmonic problem, Numer. Methods Partial Differential Equations, 13, 93-111 (1997) · Zbl 0867.65056
[12] Ciarlet, P. G.; Raviart, P., A mixed finite element method for the biharmonic equation, (Symposium on Mathematical Aspects of the Finite Element Method in Partial Differential Equations (1974), Academic Press: Academic Press New York), 125-145 · Zbl 0337.65058
[13] Brezzi, F.; Raviart, P. A., Mixed finite element methods for 4th order elliptic equations, (Proceedings of the Royal Irish Academy Conference on Numerical Analysis (1976), Academic Press: Academic Press London) · Zbl 0434.65085
[14] Rannacher, R., On nonconforming and mixed finite element methods for plate bending problems, the linear case, RAIRO Anal. Numér., 13, 369-387 (1979) · Zbl 0425.35042
[15] Liu, K.; Qin, X., A gradient recovery-based a posteriori error estimators for the Ciarlet-Raviart formulation of the second biharmonic equations, Appl. Math. Sci., 1, 997-1007 (2007) · Zbl 1129.65076
[16] Karátson, J.; Korotov, S., Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems, Appl. Math., 54, 297-336 (2009) · Zbl 1212.65249
[17] Neittaanmäki, P.; Repin, S., Reliable Methods for Computer Simulation: Error Control and a Posteriori Estimates (2004), Elsevier: Elsevier Amsterdam · Zbl 1076.65093
[18] Schwab, C., \(p\)- and \(h p\)-Finite Element Methods (1998), Clarendon Press: Clarendon Press Oxford
[19] Repin, S., A Posteriori Estimates for Partial Differential Equations (2008), Walter de Gruyter: Walter de Gruyter Berlin · Zbl 1162.65001
[20] Vejchodský, T., Guaranteed and locally computable a posteriori error estimate, IMA J. Numer. Anal., 26, 525-540 (2006) · Zbl 1096.65112
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.