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Mixed discontinuous Galerkin finite element method for the biharmonic equation. (English) Zbl 1203.65254
Summary: We first split the biharmonic equation ${{\Delta }}^{2}u=f$ with nonhomogeneous essential boundary conditions into a system of two second-order equations by introducing an auxiliary variable $v={\Delta }u$ and then apply an $hp$-mixed discontinuous Galerkin method to the resulting system. The unknown approximation ${v}_{h}$ of $v$ can easily be eliminated to reduce the discrete problem to a Schur complement system in ${u}_{h}$, which is an approximation of $u$. A direct approximation ${v}_{h}$ of $v$ can be obtained from the approximation ${u}_{h}$ of $u$. Using piecewise polynomials of degree $p\ge 3$, a priori error estimates of $u-{u}_{h}$ in the broken ${H}^{1}$ norm as well as in the ${L}^{2}$ norm which are optimal in $h$ and suboptimal in $p$ are derived. Moreover, an a priori error bound for $v-{v}_{h}$ in ${L}^{2}$ norm which is suboptimal in $h$ and $p$ is also discussed. When $p=2$, the method also converges, but with suboptimal convergence rate. Finally, numerical experiments are presented to illustrate the theoretical results.
##### MSC:
 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 65N15 Error bounds (BVP of PDE)
##### References:
 [1] Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002) · Zbl 1008.65080 · doi:10.1137/S0036142901384162 [2] Babuska, I., Suri, M.: The h-p version of the finite element method with quasiuniform meshes. RAIRO Model. Math. Anal. Numer. 21, 199–238 (1987) [3] Baker, G.: Finite element methods for elliptic equations using nonconforming elements. Math. Comput. 31, 44–59 (1977) [4] Bernardi, C., Dauge, M., Maday, Y.: Polynomials in the Sobolev world. Preprint of the Laboratoire Jacques-Louis Lions, No. R03038 (2003) [5] Brenner, S.C., Sung, L.Y.: C 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22–23, 83–118 (2005) · Zbl 1071.65151 · doi:10.1007/s10915-004-4135-7 [6] Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38, 1676–1706 (2000) · Zbl 0987.65111 · doi:10.1137/S0036142900371003 [7] Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978) [8] Ciarlet, P.G., Raviart, P.A.: A mixed finite element method for biharmonic equation. In: Proceedings of the Symposium on Mathematical Aspects in Finite Elements in Partial Differential Equations. Math. Research Center, Univ. of Wiscosin, Madison, April 1974, pp. 125–245 [9] Cockburn, B., Karniadakis, G., Shu, C.W.: The development of discontinuous Galerkin methods. In: Cockburn, B., Karniadakis, G., Shu, C.W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications. Part I: Overview. Lecture Notes in Computational Science and Engineering, vol. 11, pp. 3–50. Springer, Berlin (2000) [10] Engel, G., Garikipati, K., Hughes, T.J.R., Larson, M.G., Mazzei, L., Taylor, R.L.: Continuous/discontinuous finite element approximation 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, 3669–3750 (2002) · Zbl 1086.74038 · doi:10.1016/S0045-7825(02)00286-4 [11] Falk, R.S.: Approximation of the biharmonic equation by a mixed finite element method. SIAM J. Numer. Anal. 15, 556–567 (1978) · Zbl 0383.65059 · doi:10.1137/0715036 [12] Glowinski, R.: Pironneau, O. Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problems. SIAM Rev. 21, 167–212 (1979) · Zbl 0427.65073 · doi:10.1137/1021028 [13] Grisvard, P.: Singularities in Boundary Value Problems. Research Notes in Applied Mathematics. Masson, Paris (1992) [14] Gudi, T., Pani, A.K.: Discontinuous Galerkin methods for quasilinear elliptic problems of nonmonotone-type. SIAM J. Numer. Anal. 45, 163–192 (2007) · Zbl 1140.65082 · doi:10.1137/050643362 [15] Houston, P., Robson, J.A., Süli, E.: Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case. IMA J. Numer. Anal. 25, 726–749 (2005) · Zbl 1084.65116 · doi:10.1093/imanum/dri014 [16] Monk, P.: A mixed finite element methods for the biharmonic equation. SIAM J. Numer. Anal. 24, 737–749 (1987) · Zbl 0632.65112 · doi:10.1137/0724048 [17] Mozolevski, I., Süli, E.: A priori error analysis for the hp-version of the discontinuous Galerkin finite element method for the biharmonic equation. Comput. Methods Appl. Math. 3, 1–12 (2003) [18] Mozolevski, I., Süli, E., Bösing, P.R.: hp-Version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation. J. Sci. Comput. 30, 465–491 (2007) · Zbl 1116.65117 · doi:10.1007/s10915-006-9100-1 [19] Nitsche, J.A.: Über ein Variationprinzip zur Lösung Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36, 9–15 (1971) · Zbl 0229.65079 · doi:10.1007/BF02995904 [20] Prudhomme, S., Pascal, F., Oden, J.T.: Review of error estimation for discontinuous Galerkin methods. TICAM Report, pp. 0–27 (17 October 2000) [21] Rivière, B., Wheeler, M.F., Girault, V.: A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39, 902–931 (2001) · Zbl 1010.65045 · doi:10.1137/S003614290037174X [22] Scholz, R.: A mixed method for 4th order problems using linear finite elements. RAIRO Model. Math. Anal. Numer. 12, 85–90 (1978) [23] Schwab, C.: p- and hp-Finite Element Methods. In: Theory and Applications to Solid and Fluid Mechanics. Oxford University Press, Oxford (1998) [24] Süli, E., Mozolevski, I.: hp-Version interior DGFEMs for the biharmonic equation. Comput. Methods Appl. Mech. Eng. 196, 1851–1863 (2007) · Zbl 1173.65360 · doi:10.1016/j.cma.2006.06.014