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A superconvergent local discontinuous Galerkin method for nonlinear two-point boundary-value problems

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Abstract

In this paper, we present and analyze a superconvergent and high order accurate local discontinuous Galerkin (LDG) method for nonlinear two-point boundary-value problems (BVPs) of the form u ″ = f (t, u), which arise in a wide variety of engineering applications. We prove the L 2 stability of the LDG scheme and optimal L 2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p + 1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal rates of convergence. Moreover, we show that the derivatives of the LDG solutions are superconvergent with order p + 1 toward the derivatives of Gausss-Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order p + 3/2 toward Gauss-Radau projections of the exact solutions. Our computational results indicate that the observed numerical superconvergence rate is p + 2. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p ≥ 1 and for the periodic, Dirichlet, and mixed boundary conditions. All proofs are valid under the hypotheses of the existence and uniqueness theorem for BVPs. Several numerical results are presented to validate the theoretical results.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965)

    MATH  Google Scholar 

  2. Adjerid, S., Baccouch, M.: The discontinuous Galerkin method for two-dimensional hyperbolic problems. Part I: Superconvergence error analysis. J. Sci. Comput. 33, 75–113 (2007)

    Article  MathSciNet  Google Scholar 

  3. Adjerid, S., Baccouch, M.: The discontinuous Galerkin method for two-dimensional hyperbolic problems. Part II: A posteriori error estimation. J. Sci. Comput. 38, 15–49 (2009)

    Article  MathSciNet  Google Scholar 

  4. Adjerid, S., Baccouch, M.: Asymptotically exact a posteriori error estimates for a one-dimensional linear hyperbolic problem. Appl. Numer. Math. 60, 903–914 (2010)

    Article  MathSciNet  Google Scholar 

  5. Adjerid, S., Baccouch, M.: Adaptivity and error estimation for discontinuous Galerkin methods. 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. 63–96. Springer International Publishing, Switzerland (2014)

    MATH  Google Scholar 

  6. Adjerid, S., Devine, K.D., Flaherty, J.E., Krivodonova, L.: A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput. Methods Appl. Mech. Eng. 191, 1097–1112 (2002)

    Article  MathSciNet  Google Scholar 

  7. Ascher, U.M., Matheij, R.M., Russel, R.D.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Society for Industrial and Applied Mathematics, Philadelphia (1995)

    Book  Google Scholar 

  8. Baccouch, M.: A local discontinuous Galerkin method for the second-order wave equation. Comput. Methods Appl. Mech. Eng. 209–212, 129–143 (2012)

    Article  MathSciNet  Google Scholar 

  9. Baccouch, M.: A posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional nonlinear scalar conservation laws. Appl. Numer. Math. 84, 1–21 (2014)

    Article  MathSciNet  Google Scholar 

  10. Baccouch, M.: Asymptotically exact a posteriori LDG error estimates for one-dimensional transient convection-diffusion problems. Appl. Math. Comput. 226, 455–483 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Baccouch, M., Adjerid, S.: Discontinuous Galerkin error estimation for hyperbolic problems on unstructured triangular meshes. Comput. Methods Appl. Mech. Eng. 200, 162–177 (2010)

    Article  MathSciNet  Google Scholar 

  12. Birkhoff, G., Rota, G.-C.: Ordinary Differential Equations. Wiley, New York (1989)

    MATH  Google Scholar 

  13. Burden, R.L., Faires, J.D., Burden, A.M.: Numerical Analysis. Cengage Learning, Boston (2016)

    MATH  Google Scholar 

  14. Castillo, P.: A superconvergence result for discontinuous Galerkin methods applied to elliptic problems. Comput. Methods Appl. Mech. Eng. 192, 4675–4685 (2003)

    Article  MathSciNet  Google Scholar 

  15. 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)

    Article  MathSciNet  Google Scholar 

  16. Cheng, Y., Shu, C.-W.: Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension. SIAM J. Numer. Anal. 47, 4044–4072 (2010)

    Article  MathSciNet  Google Scholar 

  17. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland Pub. Co. , Amsterdam (1978)

    MATH  Google Scholar 

  18. 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)

    Google Scholar 

  19. Cockburn, B., Shu, C.W.: TVB Runge-Kutta local projection discontinuous Galerkin methods for scalar conservation laws II: General framework. Math. Comput. 52, 411–435 (1989)

    MATH  Google Scholar 

  20. Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)

    Article  MathSciNet  Google Scholar 

  21. Cuomo, S., Marasco, A.: A numerical approach to nonlinear two-point boundary value problems for ODEs. Comput. Math. Appl. 55(11), 2476 (2489)

    Article  MathSciNet  Google Scholar 

  22. Delfour, M., Hager, W., Trochu, F.: Discontinuous Galerkin methods for ordinary differential equation. Math. Comput. 154, 455–473 (1981)

    Article  MathSciNet  Google Scholar 

  23. Devine, K.D., Flaherty, J.E.: Parallel adaptive hp-refinement techniques for conservation laws. Comput. Methods Appl. Mech. Eng. 20, 367–386 (1996)

    MathSciNet  MATH  Google Scholar 

  24. Flaherty, J.E., Loy, R., Shephard, M.S., Szymanski, B.K., Teresco, J.D., Ziantz, L.H.: Adaptive local refinement with octree load-balancing for the parallel solution of three-dimensional conservation laws. J. Parallel Distrib. Comput. 47, 139–152 (1997)

    Article  Google Scholar 

  25. Ha, S.N.: A nonlinear shooting method for two-point boundary value problems. Comput. Math. Appl. 42(10), 1411 (1420)

    MathSciNet  MATH  Google Scholar 

  26. Johnson, C.: Error estimates and adaptive time-step control for a class of one-step methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 25, 908–926 (1988)

    Article  MathSciNet  Google Scholar 

  27. Keller, H.B.: Numerical Methods for Two-point Boundary-value Problems, A Blaisdell Book in Numerical Analysis and Computer Science. Blaisdell (1968)

  28. Lasaint, P., Raviart, P.: On a finite element method for solving the neutron transport equations. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations. Academic Press, New York (1974)

    Chapter  Google Scholar 

  29. 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)

    Article  MathSciNet  Google Scholar 

  30. Meng, X., Shu, C.-W., Zhang, Q., Wu, B.: Superconvergence of discontinuous Galerkin methods for scalar nonlinear conservation laws in one space dimension. SIAM J. Numer. Anal. 50(5), 2336–2356 (2012)

    Article  MathSciNet  Google Scholar 

  31. Na, T.: Computational Methods in Engineering Boundary Value Problems, Mathematics in Science and Engineering: A Series of Monographs and Textbooks. Academic Press (1979)

  32. Peterson, T.E.: A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM J. Numer. Anal. 28(1), 133–140 (1991)

    Article  MathSciNet  Google Scholar 

  33. 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 (1973)

  34. 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)

  35. Tirmizi, I., Twizell, E.: Higher-order finite-difference methods for nonlinear second-order two-point boundary-value problems. Appl. Math. Lett. 15(7), 897–902 (2002)

    Article  MathSciNet  Google Scholar 

  36. Xie, Z., Zhang, Z.: Superconvergence of DG method for one-dimensional singularly perturbed problems. J. Comput. Math. 25(2), 185–200 (2007)

    MathSciNet  MATH  Google Scholar 

  37. Xie, Z., Zhang, Z.: Uniform superconvergence analysis of the discontinuous G,alerkin method for a singularly perturbed problem in 1-D. Math. Comput. 79(269), 35–45 (2010)

    Article  Google Scholar 

  38. 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)

    MathSciNet  MATH  Google Scholar 

  39. Zhang, Z., Xie, Z., Zhang, Z.: Superconvergence of discontinuous Galerkin methods for convection-diffusion problems. J. Sci. Comput. 41, 70–93 (2009)

    Article  MathSciNet  Google Scholar 

  40. 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)

    Article  MathSciNet  Google Scholar 

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Funding

This research was supported by the University Committee on Research and Creative Activity (UCRCA Proposal 2016-01-F) at the University of Nebraska at Omaha.

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  1. Department of Mathematics, University of Nebraska at Omaha, Omaha, 68182, USA

    Mahboub Baccouch

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  1. Mahboub Baccouch
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Correspondence to Mahboub Baccouch.

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Baccouch, M. A superconvergent local discontinuous Galerkin method for nonlinear two-point boundary-value problems. Numer Algor 79, 697–718 (2018). https://doi.org/10.1007/s11075-017-0456-0

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