Abstract
A Morley–Wang–Xu (MWX) element method with a simply modified right hand side is proposed for a fourth order elliptic singular perturbation problem, in which the discrete bilinear form is standard as usual nonconforming finite element methods. The sharp error analysis is given for this MWX element method. And the Nitsche’s technique is applied to the MXW element method to achieve the optimal convergence rate in the case of the boundary layers. An important feature of the MWX element method is solver-friendly. Based on a discrete Stokes complex in two dimensions, the MWX element method is decoupled into one Lagrange element method of Poisson equation, two Morley element methods of Poisson equation and one nonconforming \(P_1\)–\(P_0\) element method of Brinkman problem, which implies efficient and robust solvers for the MWX element method. Some numerical examples are provided to verify the theoretical results.
Similar content being viewed by others
References
Argyris, J., Fried, I., Scharpf, D.: The TUBA family of plate elements for the matrix displacement method. Aeronaut. J. R. Aeronaut. Soc. 72, 701–709 (1968)
Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)
Brenner, S.C., Neilan, M.: A \(C^0\) interior penalty method for a fourth order elliptic singular perturbation problem. SIAM J. Numer. Anal. 49(2), 869–892 (2011)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)
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)
Burman, E., Ern, A.: Continuous interior penalty \(hp\)-finite element methods for advection and advection-diffusion equations. Math. Comput. 76(259), 1119–1140 (2007)
Cahouet, J., Chabard, J.-P.: Some fast 3D finite element solvers for the generalized Stokes problem. Internat. J. Numer. Methods Fluids 8(8), 869–895 (1988)
Chen, H., Chen, S.: Uniformly convergent nonconforming element for 3-D fourth order elliptic singular perturbation problem. J. Comput. Math. 32(6), 687–695 (2014)
Chen, H., Chen, S., Qiao, Z.: \(C^0\)-nonconforming tetrahedral and cuboid elements for the three-dimensional fourth order elliptic problem. Numer. Math. 124(1), 99–119 (2013)
Chen, H., Chen, S., Xiao, L.: Uniformly convergent \(C^0\)-nonconforming triangular prism element for fourth-order elliptic singular perturbation problem. Numer. Methods Partial Differ. Equ. 30(6), 1785–1796 (2014)
Chen, L., Hu, J., Huang, X.: Fast auxiliary space preconditioners for linear elasticity in mixed form. Math. Comput. 87(312), 1601–1633 (2018)
Chen, L., Huang, X.: Decoupling of mixed methods based on generalized Helmholtz decompositions. SIAM J. Numer. Anal. 56(5), 2796–2825 (2018)
Chen, L., Huang, X.: Nonconforming virtual element method for \(2m\)th order partial differential equations in \(\mathbb{R}^n\). Math. Comput. 89(324), 1711–1744 (2020)
Chen, S., Liu, M., Qiao, Z.: An anisotropic nonconforming element for fourth order elliptic singular perturbation problem. Int. J. Numer. Anal. Model. 7(4), 766–784 (2010)
Chen, S.-C., Zhao, Y.-C., Shi, D.-Y.: Non \(C^0\) nonconforming elements for elliptic fourth order singular perturbation problem. J. Comput. Math. 23(2), 185–198 (2005)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland Publishing Co., Amsterdam (1978)
Epshteyn, Y., Rivière, B.: Estimation of penalty parameters for symmetric interior penalty Galerkin methods. J. Comput. Appl. Math. 206(2), 843–872 (2007)
Falk, R.S., Morley, M.E.: Equivalence of finite element methods for problems in elasticity. SIAM J. Numer. Anal. 27(6), 1486–1505 (1990)
Franz, S., Roos, H.-G., Wachtel, A.: A \(C^0\) interior penalty method for a singularly-perturbed fourth-order elliptic problem on a layer-adapted mesh. Numer. Methods Partial Differ. Equ. 30(3), 838–861 (2014)
Gallistl, D.: Stable splitting of polyharmonic operators by generalized Stokes systems. Math. Comput. 86(308), 2555–2577 (2017)
Georgoulis, E.H., Houston, P.: Discontinuous Galerkin methods for the biharmonic problem. IMA J. Numer. Anal. 29(3), 573–594 (2009)
Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman (Advanced Publishing Program), Boston (1985)
Gustafsson, T., McBain, G.D.: scikit-fem: A Python package for finite element assembly. J. Open Source Softw. 5(52), 2369 (2020)
Guzmán, J., Leykekhman, D., Neilan, M.: A family of non-conforming elements and the analysis of Nitsche’s method for a singularly perturbed fourth order problem. Calcolo 49(2), 95–125 (2012)
Huang, J., Huang, X., Han, W.: A new \(C^0\) discontinuous Galerkin method for Kirchhoff plates. Comput. Methods Appl. Mech. Eng. 199(23–24), 1446–1454 (2010)
Huang, X.: New Finite Element Methods and Efficient Algorithms for Fourth Order Elliptic Equations. Ph.D. Thesis, Shanghai Jiao Tong University (2010)
Mardal, K.-A., Winther, R.: Uniform preconditioners for the time dependent Stokes problem. Numer. Math. 98(2), 305–327 (2004)
Morley, L.S.D.: The triangular equilibrium element in the solution of plate bending problems. Aeronaut. Q. 19, 149–169 (1968)
Mozolevski, I., Bösing, P.R.: Sharp expressions for the stabilization parameters in symmetric interior-penalty discontinuous Galerkin finite element approximations of fourth-order elliptic problems. Comput. Methods Appl. Math. 7(4), 365–375 (2007)
Nilssen, T.K., Tai, X.-C., Winther, R.: A robust nonconforming \(H^2\)-element. Math. Comput. 70(234), 489–505 (2001)
Olshanskii, M.A., Peters, J., Reusken, A.: Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations. Numer. Math. 105(1), 159–191 (2006)
Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)
Semper, B.: Conforming finite element approximations for a fourth-order singular perturbation problem. SIAM J. Numer. Anal. 29(4), 1043–1058 (1992)
Tai, X.-C., Winther, R.: A discrete de Rham complex with enhanced smoothness. Calcolo 43(4), 287–306 (2006)
Wang, L., Wu, Y., Xie, X.: Uniformly stable rectangular elements for fourth order elliptic singular perturbation problems. Numer. Methods Partial Differ. Equ. 29(3), 721–737 (2013)
Wang, M.: On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements. SIAM J. Numer. Anal. 39(2), 363–384 (2001)
Wang, M., Meng, X.: A robust finite element method for a 3-D elliptic singular perturbation problem. J. Comput. Math. 25(6), 631–644 (2007)
Wang, M., Shi, Z.-C., Xu, J.: A new class of Zienkiewicz-type non-conforming element in any dimensions. Numer. Math. 106(2), 335–347 (2007)
Wang, M., Shi, Z.-C., Xu, J.: Some \(n\)-rectangle nonconforming elements for fourth order elliptic equations. J. Comput. Math. 25(4), 408–420 (2007)
Wang, M., Xu, J.: The Morley element for fourth order elliptic equations in any dimensions. Numer. Math. 103(1), 155–169 (2006)
Wang, M., Xu, J.: Minimal finite element spaces for \(2m\)-th-order partial differential equations in \(R^n\). Math. Comput. 82(281), 25–43 (2013)
Wang, M., Xu, J.-C., Hu, Y.-C.: Modified Morley element method for a fourth order elliptic singular perturbation problem. J. Comput. Math. 24(2), 113–120 (2006)
Wang, W., Huang, X., Tang, K., Zhou, R.: Morley–Wang–Xu element methods with penalty for a fourth order elliptic singular perturbation problem. Adv. Comput. Math. 44(4), 1041–1061 (2018)
Warburton, T., Hesthaven, J.S.: On the constants in \(hp\)-finite element trace inverse inequalities. Comput. Methods Appl. Mech. Eng. 192(25), 2765–2773 (2003)
Xie, P., Shi, D., Li, H.: A new robust \(C^0\)-type nonconforming triangular element for singular perturbation problems. Appl. Math. Comput. 217(8), 3832–3843 (2010)
Xu, J.: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56(3), 215–235 (1996)
Zhang, S.: A family of 3D continuously differentiable finite elements on tetrahedral grids. Appl. Numer. Math. 59(1), 219–233 (2009)
Acknowledgements
The authors would like to thank the anonymous referees for their valuable suggestions and comments. The authors would also like to thank Dr. Tom Gustafsson for helpful suggestions in programing.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by the National Natural Science Foundation of China Project (Grant Nos. 11771338 and 12071289) and the Fundamental Research Funds for the Central Universities (Grant No. 2019110066)
Rights and permissions
About this article
Cite this article
Huang, X., Shi, Y. & Wang, W. A Morley–Wang–Xu Element Method for a Fourth Order Elliptic Singular Perturbation Problem. J Sci Comput 87, 84 (2021). https://doi.org/10.1007/s10915-021-01483-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-021-01483-2
Keywords
- Fourth order elliptic singular perturbation problem
- Morley–Wang–Xu element
- Decoupling
- Boundary layers
- Fast solver