Abstract
This paper is concerned with the convergence and superconvergence of the local discontinuous Galerkin (LDG) finite element method for nonlinear fourth-order boundary value problems of the type \(u^{(4)}=f(x,u,u^{\prime },u^{\prime \prime },u^{\prime \prime \prime })\), x ∈ [a,b] with classical boundary conditions at the endpoints. Convergence properties for the solution and for all three auxiliary variables are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L2 norm achieve optimal (p + 1)th-order convergence, when polynomials of degree p are used. We also prove that the derivatives of the errors between the LDG solutions and Gauss-Radau projections of the exact solutions in the L2 norm are superconvergent with order p + 1. Furthermore, a (2p + 1)th-order superconvergent for the errors of the numerical fluxes at mesh nodes as well as for the cell averages is also obtained under quasi-uniform meshes. Finally, we prove that the LDG solutions are superconvergent with an order of p + 2 toward particular projections of the exact solutions. The error analysis presented in this paper is valid for p ≥ 1. Numerical experiments indicate that our theoretical findings are optimal.
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References
Aftabizadeh, A.: Existence and uniqueness theorems for fourth-order boundary value problems. J. Math. Anal. Appl. 116(2), 415–426 (1986)
Agarwall, R.P., Akrivis, G.: Boundary value problems occurring in plate deflection theory. J. Comput. Appl. Math. 8(3), 145–154 (1982)
Al-Hayani, W., Casasús, L.: Approximate analytical solution of fourth order boundary value problems. Numer. Algor. 40(1), 67–78 (2005)
Baccouch, M.: The local discontinuous Galerkin method for the fourth-order Euler-Bernoulli partial differential equation in one space dimension. Part I: Superconvergence error analysis. J. Sci. Comput. 59, 795–840 (2014)
Baccouch, M.: The local discontinuous Galerkin method for the fourth-order Euler-Bernoulli partial differential equation in one space dimension. Part II: A posteriori error estimation. J. Sci. Comput. 60, 1–34 (2014)
Baccouch, M.: Superconvergence and a posteriori error estimates of a local discontinuous Galerkin method for the fourth-order initial-boundary value problems arising in beam theory. Int. J. Numer. Anal. Model. Series B 5, 188–216 (2014)
Baccouch, M.: A superconvergent local discontinuous Galerkin method for nonlinear two-point boundary-value problems. Numer. Algor. 79(3), 697–718 (2018)
Baccouch, M.: Analysis of optimal superconvergence of a local discontinuous Galerkin method for nonlinear second-order two-point boundary-value problems. Appl. Numer. Math. 145, 361–383 (2019)
Baccouch, M.: A superconvergent local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems. Int. J. Comput. Methods 5(56), 1950035 (2019)
Castillo, P.: A review of the Local discontinuous Galerkin (LDG) method applied to elliptic problems. Appl. Numer. Math. 56, 1307–1313 (2006)
Castillo, P., Cockburn, B., Schötzau, D., Schwab, C.: Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems. Math. Comput. 71, 455–478 (2002)
Celiker, F., Cockburn, B.: Superconvergence of the numerical traces for discontinuous Galerkin and hybridized methods for convection-diffusion problems in one space dimension. Math. Comput. 76, 67–96 (2007)
Chang, M.P.S.K.: Existence of solutions for a fully nonlinear fourth-order two-point boundary value problem. J. Appl. Math. Comput. 37, 287–295 (2011)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland Pub. Co., Amsterdam (1978)
Cockburn, B., Kanschat, G., Schötzau, D.: A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comput. 74, 1067–1095 (2004)
Cockburn, B., Kanschat, G., Schötzau, D.: The local discontinuous Galerkin method for linearized incompressible fluid flow: A review. Comput.Fluids 34(4-5), 491–506 (2005)
Cockburn, B., Karniadakis, G.E., Shu, C.W.: Discontinuous Galerkin Methods Theory, Computation and Applications Lecture Notes in Computational Science and Engineering, vol. 11. Springer, Berlin (2000)
Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)
Delfour, M., Hager, W., Trochu, F.: Discontinuous Galerkin methods for ordinary differential equation. Math. Comput. 154, 455–473 (1981)
Geng, F.: A new reproducing kernel hilbert space method for solving nonlinear fourth-order boundary value problems. Appl. Math. Comput. 213(1), 163–169 (2009)
Gupta, C.P.: Existence and uniqueness theorems for some fourth order fully quasilinear boundary value problems. Appl. Anal. 36(3-4), 157–169 (1990)
Liang, S., Jeffrey, D.J.: An efficient analytical approach for solving fourth order boundary value problems. Comput. Phys. Commun. 180(11), 2034–2040 (2009)
Lin, R.: Discontinuous discretization for least-squares formulation of singularly perturbed reaction-diffusion problems in one and two dimensions. SIAM J. Numer. Anal. 47(1), 89–108 (2009)
Momani, S., Noor, M.A.: Numerical comparison of methods for solving a special fourth-order boundary value problem. Appl. Math. Comput. 191(1), 218–224 (2007)
Na, T.: Computational Methods in Engineering Boundary Value Problems, Mathematics in Science and Engineering: A Series of Monographs and Textbooks. Academic Press (1979)
Reed, W.H., Hill, T.R.: Triangular Mesh Methods for the Neutron Transport Equation, Tech. Rep. LA-UR-73-479. Los Alamos Scientific Laboratory, Los Alamos (1991)
Shu, C.-W.: Discontinuous Galerkin method for time-dependent problems: Survey and recent developments. In: Feng, X., Karakashian, O., Xing, Y. (eds.) Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, vol. 157 of The IMA Volumes in Mathematics and its Applications, pp 25–62. Springer International Publishing (2014)
Soedel, W.: Vibrations of Shells and Plates. Marcel Dekker, New York (2004)
Timoshenko, S.: Theory of Elastic Stability. Dover Publications, Mineola (2009)
Xie, Z., Zhang, Z.: Superconvergence of DG method for one-dimensional singularly perturbed problems. J. Comput. Math. 25(2), 185–200 (2007)
Xie, Z., Zhang, Z.: Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D. Math. Comput. 79(269), 35–45 (2010)
Xie, Z., Zhang, Z., Zhang, Z.: A numerical study of uniform superconvergence of LDG method for solving singularity perturbed problems. J. Comput. Math. 27, 280–298 (2009)
Zhang, Z., Xie, Z., Zhang, Z.: Superconvergence of discontinuous Galerkin methods for convection-diffusion problems. J. Sci. Comput. 41, 70–93 (2009)
Zhu, H., Zhang, H.T.Z.: Convergence analysis of the LDG method for singularly perturbed two-point boundary value problems. Commun. Math. Sci. 9 (4), 1013–1032 (2011)
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The author would like to thank the two anonymous reviewers for the valuable comments and suggestions which improved the quality of the paper.
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Baccouch, M. Analysis of optimal superconvergence of the local discontinuous Galerkin method for nonlinear fourth-order boundary value problems. Numer Algor 86, 1615–1650 (2021). https://doi.org/10.1007/s11075-020-00947-0
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DOI: https://doi.org/10.1007/s11075-020-00947-0
Keywords
- Nonlinear fourth-order boundary value problems
- Local discontinuous Galerkin method
- a priori error estimates
- Superconvergence