Stable splitting of polyharmonic operators by generalized Stokes systems

Gallistl, Dietmar

doi:10.1090/mcom/3208

Stable splitting of polyharmonic operators by generalized Stokes systems
HTML articles powered by AMS MathViewer

by Dietmar Gallistl PDF

Math. Comp. 86 (2017), 2555-2577 Request permission

Abstract:

A stable splitting of $2m$-th order elliptic partial differential equations into $2(m-1)$ problems of Poisson type and one generalized Stokes problem is established for any space dimension $d\geq 2$ and any integer $m\geq 1$. This allows a numerical approximation with standard finite elements that are suited for the Poisson equation and the Stokes system, respectively. For some fourth- and sixth-order problems in two and three space dimensions, precise finite element formulations along with a priori error estimates and numerical experiments are presented.

References

Chérif Amrouche, Philippe G. Ciarlet, and Patrick Ciarlet Jr., Vector and scalar potentials, Poincaré’s theorem and Korn’s inequality, C. R. Math. Acad. Sci. Paris 345 (2007), no. 11, 603–608 (English, with English and French summaries). MR 2371475, DOI 10.1016/j.crma.2007.10.020
Chérif Amrouche and Vivette Girault, Problèmes généralisés de Stokes, Portugal. Math. 49 (1992), no. 4, 463–503 (French, with English and French summaries). MR 1300580
D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, 7–32 (English, with French summary). MR 813687, DOI 10.1051/m2an/1985190100071
Garth A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp. 31 (1977), no. 137, 45–59. MR 431742, DOI 10.1090/S0025-5718-1977-0431742-5
John W. Barrett, Stephen Langdon, and Robert Nürnberg, Finite element approximation of a sixth order nonlinear degenerate parabolic equation, Numer. Math. 96 (2004), no. 3, 401–434. MR 2028722, DOI 10.1007/s00211-003-0479-4
H. Blum and R. Rannacher, On the boundary value problem of the biharmonic operator on domains with angular corners, Math. Methods Appl. Sci. 2 (1980), no. 4, 556–581. MR 595625, DOI 10.1002/mma.1670020416
Daniele Boffi, Finite element approximation of eigenvalue problems, Acta Numer. 19 (2010), 1–120. MR 2652780, DOI 10.1017/S0962492910000012
Daniele Boffi, Franco Brezzi, and Michel Fortin, Mixed finite element methods and applications, Springer Series in Computational Mathematics, vol. 44, Springer, Heidelberg, 2013. MR 3097958, DOI 10.1007/978-3-642-36519-5
Dietrich Braess, Finite elements, 3rd ed., Cambridge University Press, Cambridge, 2007. Theory, fast solvers, and applications in elasticity theory; Translated from the German by Larry L. Schumaker. MR 2322235, DOI 10.1017/CBO9780511618635
Susanne C. Brenner, $C^0$ interior penalty methods, Frontiers in numerical analysis—Durham 2010, Lect. Notes Comput. Sci. Eng., vol. 85, Springer, Heidelberg, 2012, pp. 79–147. MR 3051409, DOI 10.1007/978-3-642-23914-4_{2}
S. C. Brenner, P. Monk, and J. Sun, $C^0$ interior penalty Galerkin method for biharmonic eigenvalue problems, Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014 (R. M. Kirby, M. Berzins, and J. S. Hesthaven, eds.), Lecture Notes in Computational Science and Engineering, vol. 106, Springer International Publishing, 2015, pp. 3–15.
F. Brezzi and M. Fortin, Numerical approximation of Mindlin-Reissner plates, Math. Comp. 47 (1986), no. 175, 151–158. MR 842127, DOI 10.1090/S0025-5718-1986-0842127-7
Franco Brezzi, Michel Fortin, and Rolf Stenberg, Error analysis of mixed-interpolated elements for Reissner-Mindlin plates, Math. Models Methods Appl. Sci. 1 (1991), no. 2, 125–151. MR 1115287, DOI 10.1142/S0218202591000083
C. Carstensen, K. Köhler, D. Peterseim, and M. Schedensack, Comparison results for the Stokes equations, Appl. Numer. Math. 95 (2015), 118–129. MR 3349689, DOI 10.1016/j.apnum.2013.12.005
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33–75. MR 343661
Philippe Destuynder and Thierry Nevers, Une modification du modèle de Mindlin pour les plaques minces en flexion présentant un bord libre, RAIRO Modél. Math. Anal. Numér. 22 (1988), no. 2, 217–242 (French, with English summary). MR 945123, DOI 10.1051/m2an/1988220202171
G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. Mazzei, and R. L. Taylor, 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. Engrg. 191 (2002), no. 34, 3669–3750. MR 1915664, DOI 10.1016/S0045-7825(02)00286-4
Dietmar Gallistl, Adaptive nonconforming finite element approximation of eigenvalue clusters, Comput. Methods Appl. Math. 14 (2014), no. 4, 509–535. MR 3259027, DOI 10.1515/cmam-2014-0020
Dietmar Gallistl, Morley finite element method for the eigenvalues of the biharmonic operator, IMA J. Numer. Anal. 35 (2015), no. 4, 1779–1811. MR 3407244, DOI 10.1093/imanum/dru054
P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR 1173209
Thirupathi Gudi and Michael Neilan, An interior penalty method for a sixth-order elliptic equation, IMA J. Numer. Anal. 31 (2011), no. 4, 1734–1753. MR 2846773, DOI 10.1093/imanum/drq031
J. R. King, The isolation oxidation of silicon: the reaction-controlled case, SIAM J. Appl. Math. 49 (1989), no. 4, 1064–1080. MR 1005497, DOI 10.1137/0149064
P. Lascaux and P. Lesaint, Some nonconforming finite elements for the plate bending problem, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. 9 (1975), no. R-1, 9–53 (English, with French summary). MR 423968
Zengjian Lou and Alan McIntosh, Hardy space of exact forms on ${\Bbb R}^N$, Trans. Amer. Math. Soc. 357 (2005), no. 4, 1469–1496. MR 2115373, DOI 10.1090/S0002-9947-04-03535-4
H. Melzer and R. Rannacher, Spannungskonzentrationen in Eckpunkten der Kirchhoffschen Platte, Bauingenieur 55 (1980), 181–184.
M. Schedensack, A class of mixed finite element methods based on the Helmholtz decomposition, Oberwolfach Reports 12 (2015), no. 3, 2555–2556.
M. Schedensack, A class of mixed finite element methods based on the Helmholtz decomposition in computational mechanics, Doctoral dissertation, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2015.
M. Schedensack, A new discretization for $m$th-Laplace equations with arbitrary polynomial degrees, SIAM J. Numer. Anal. 54 (2016), no. 4, 2138–2162. MR 3519555, DOI 10.1137/15M1013651
Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
Ming Wang and Jinchao Xu, Minimal finite element spaces for $2m$-th-order partial differential equations in $R^n$, Math. Comp. 82 (2013), no. 281, 25–43. MR 2983014, DOI 10.1090/S0025-5718-2012-02611-1
Alexander Ženíšek, Interpolation polynomials on the triangle, Numer. Math. 15 (1970), 283–296. MR 275014, DOI 10.1007/BF02165119