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A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems. (English) Zbl 1226.65095
The author proposes a stabilized finite element method for the approximation of the biharmonic equation with a clamped boundary condition. The formulation is obtained by introducing the gradient of the solution of the biharmonic equation as a new unknown and writing an additional variational equation in terms of a Lagrange multiplier. An optimal a priori error estimate for the finite element solution when the mesh is uniformly regular is also given.
MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65N15Error bounds (BVP of PDE)
65N12Stability and convergence of numerical methods (BVP of PDE)
35J40Higher order elliptic equations, boundary value problems
74K20Plates (solid mechanics)
74S05Finite element methods in solid mechanics
References:
[1]Girault, V.; Raviart, P. -A.: Finite element methods for Navier–Stokes equations, (1986)
[2]Ciarlet, P. G.: The finite element method for elliptic problems, (1978)
[3]Engel, G.; Garikipati, K.; Hughes, T. J. R.; Larson, M. G.; Mazzei, L.; Taylor, R. L.: 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, Computer methods in applied mechanics and engineering 191, 3669-3750 (2002) · Zbl 1086.74038 · doi:10.1016/S0045-7825(02)00286-4
[4]Wells, G. N.; Kuhl, E.; Garikipati, K.: A discontinuous Galerkin method for the Cahn–Hilliard equation, Journal of computational physics 218, 860-877 (2006) · Zbl 1106.65086 · doi:10.1016/j.jcp.2006.03.010
[5]Wahba, G.: Spline models for observational data, Series in applied mathematic 59 (1990) · Zbl 0813.62001
[6]Ciarlet, P.; Raviart, P.: A mixed finite element method for the biharmonic equation, Symposium on mathematical aspects of finite elements in partial differential equations, 125-143 (1974) · Zbl 0337.65058
[7]Ciarlet, P. G.; Glowinski, R.: Dual iterative techniques for solving a finite element approximation of the biharmonic euation, Computer methods in applied mechanics and engineering 5, 277-295 (1975) · Zbl 0305.65068 · doi:10.1016/0045-7825(75)90002-X
[8]Falk, R. S.: Approximation of the biharmonic equation by a mixed finite element method, SIAM journal of numerical analysis 15, 556-567 (1978) · Zbl 0383.65059 · doi:10.1137/0715036
[9]Falk, R.; Osborn, J. E.: Error estimates for mixed methods, RAIRO analyse numérique 14, 249-277 (1980) · Zbl 0467.65062
[10]Babuška, I.; Osborn, J.; Pitkäranta, J.: Analysis of mixed methods using mesh dependent norms, Mathematics of computation 35, 1039-1062 (1980) · Zbl 0472.65083 · doi:10.2307/2006374
[11]Johnson, C.; Pitkäranta, J.: Some mixed finite element methods related to reduced integration, Mathematics of computation 38, 375-400 (1982) · Zbl 0482.65058 · doi:10.2307/2007276
[12]Monk, P.: A mixed finite element method for the biharmonic equation, SIAM journal of numerical analysis 24, 737-749 (1987) · Zbl 0632.65112 · doi:10.1137/0724048
[13]Cheng, X.; Han, W.; Huang, H.: Some mixed finite element methods for biharmonic equation, Journal of computational and applied mathematics 126, 91-109 (2000) · Zbl 0973.65098 · doi:10.1016/S0377-0427(99)00342-8
[14]Brenner, S. C.; Sung, L. -Y.: C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains, Journal of scientific computing 22-23, 83-118 (2005) · Zbl 1071.65151 · doi:10.1007/s10915-004-4135-7
[15]Lamichhane, B. P.: A mixed finite element method for the biharmonic problem using biorthogonal or quasi-biorthogonal systems, Journal of scientific computing 46, 379-396 (2011)
[16]Arnold, D. N.; Brezzi, F.: Some new elements for the Reissner–Mindlin plate model, , 287-292 (1993) · Zbl 0817.73058
[17]Braess, D.: Finite elements. Theory, fast solver, and applications in solid mechanics, (2001)
[18]Xu, J.; Zhang, Z.: Analysis of recovery type a posteriori error estimators for mildly structured grids, Mathematics of computation 73, 1139-1152 (2004) · Zbl 1050.65103 · doi:10.1090/S0025-5718-03-01600-4
[19]Scholz, R.: A mixed method for 4th order problems using linear finite elements, RAIRO analyse numérique 12, 85-90 (1978) · Zbl 0382.65059
[20]Davini, C.; Pitacco, I.: An uncontrained mixed method for the biharmonic problem, SIAM journal of numerical analysis 38, 820-836 (2001) · Zbl 0982.65124 · doi:10.1137/S0036142999359773
[21]Brenner, S. C.; Scott, L. R.: The mathematical theory of finite element methods, (1994)
[22]Lions, J. -L.; Magenes, E.: Non-homogeneous boundary value problems and applications, Non-homogeneous boundary value problems and applications (1972)
[23]Adams, R. A.: Sobolev spaces, (1975)
[24]Grisvard, P.: Elliptic problems in nonsmooth domains, Monographs and studies in mathematics 24 (1985) · Zbl 0695.35060
[25]Brenner, S. C.; Sung, L.: Linear finite element methods for planar linear elasticity, Mathematics of computation 59, 321-338 (1992) · Zbl 0766.73060 · doi:10.2307/2153060
[26]Brezzi, F.; Fortin, M.: Mixed and hybrid finite element methods, (1991) · Zbl 0788.73002
[27]Braess, D.: Stability of saddle point problems with penalty, M2an 30, 731-742 (1996) · Zbl 0860.65054
[28]Arnold, D. N.; Falk, R. S.: Analysis of a linear-linear finite element for the Reissner–Mindlin plate model, Mathemattical models and methods in applied science, 217-238 (1997) · Zbl 0869.73068 · doi:10.1142/S0218202597000141
[29]Boffi, D.; Lovadina, C.: Analysis of new augmented Lagrangian formulations for mixed finite element schemes, Numerische Mathematik 75, 405-419 (1997) · Zbl 0874.65085 · doi:10.1007/s002110050246
[30]B.P. Lamichhane, Higher Order Mortar Finite Elements with Dual Lagrange Multiplier Spaces and Applications. Ph.D. Thesis, Universität Stuttgart, 2006. · Zbl 1196.65012 · doi:http://elib.uni-stuttgart.de/opus/volltexte/2006/2621/
[31]Kim, C.; Lazarov, R. D.; Pasciak, J. E.; Vassilevski, P. S.: Multiplier spaces for the mortar finite element method in three dimensions, SIAM journal of numerical analysis 39, 519-538 (2001) · Zbl 1006.65129 · doi:10.1137/S0036142900367065
[32]Lamichhane, B. P.: A mixed finite element method for nonlinear and nearly incompressible elasticity based on biorthogonal systems, International journal for numerical methods in engineering 79, 870-886 (2009) · Zbl 1171.74446 · doi:10.1002/nme.2594
[33]Scott, L. R.; Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions, Mathematics of computation 54, No. 190, 483-493 (1990) · Zbl 0696.65007 · doi:10.2307/2008497
[34]C. Bernardi, Y. Maday, A.T. Patera, A new nonconforming approach to domain decomposition: the mortar element method. in: H. Brezzi et al., (Eds.), Nonlinear Partial Differential Equations and their Applications, Paris, 1994, pp. 13–51. · Zbl 0797.65094
[35]Galántai, A.: Projectors and projection methods, (2003)
[36]Szyld, D. B.: The many proofs of an identity on the norm of oblique projections, Numerical algorithms 42, 309-323 (2006) · Zbl 1102.47002 · doi:10.1007/s11075-006-9046-2
[37]Ainsworth, M.; Oden, J. T.: A posteriori error estimation in finite element analysis, (2000) · Zbl 1008.65076
[38]Bank, R.; Xu, J.: Asymptotically exact a posteriori error estimators, part I: Grids with superconvergence, SIAM journal on numerical analysis 41, 2294-2312 (2003) · Zbl 1058.65116 · doi:10.1137/S003614290139874X