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Two-Grid hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic PDEs | Journal of Scientific Computing Skip to main content
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Two-Grid hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic PDEs

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Abstract

In this article we propose a class of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem of monotone type. The key idea in this setting is to first discretise the underlying nonlinear problem on a coarse finite element space \(V({{\mathcal {T}_{H}}},\boldsymbol {P})\). The resulting ‘coarse’ numerical solution is then exploited to provide the necessary data needed to linearise the underlying discretisation on the finer space \(V({{\mathcal {T}_{h}}},\boldsymbol {p})\); thereby, only a linear system of equations is solved on the richer space \(V({{\mathcal {T}_{h}}},\boldsymbol {p})\). In this article both the a priori and a posteriori error analysis of the two-grid hp-version discontinuous Galerkin finite element method is developed. Moreover, we propose and implement an hp-adaptive two-grid algorithm, which is capable of designing both the coarse and fine finite element spaces \(V({{\mathcal {T}_{H}}},\boldsymbol {P})\) and \(V({{\mathcal {T}_{h}}},\boldsymbol {p})\), respectively, in an automatic fashion. Numerical experiments are presented for both two- and three-dimensional problems; in each case, we demonstrate that the CPU time required to compute the numerical solution to a given accuracy is typically less when the two-grid approach is exploited, when compared to the standard discontinuous Galerkin method.

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References

  1. Amestoy, P.R., Duff, I.S., L’Excellent, J.-Y.: Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Eng. 184, 501–520 (2000)

    Article  MATH  Google Scholar 

  2. Antonietti, P.F., Ayuso, B.: Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case. Math. Model. Numer. Anal. 41(1), 21–54 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Axelsson, O., Layton, W.: A two-level method for the discretization of nonlinear boundary value problems. SIAM J. Numer. Anal. 33(6), 2359–2374 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beilina, L., Korotov, S., Křížek, M.: Nonobtuse tetrahedral partitions that refine locally towards Fichera-like corners. Appl. Math. 50(6), 569–581 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bi, C., Ginting, V.: Two-grid finite volume element method for linear and nonlinear elliptic problems. Numer. Math. 108, 177–198 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bi, C., Ginting, V.: Two-grid discontinuous Galerkin method for quasi-linear elliptic problems. J. Sci. Comput. 49(3), 311–331 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.): Discontinuous Galerkin Methods. Theory, Computation and Applications. Lect. Notes Comput. Sci. Eng., vol. 11. Springer, Berlin (2000)

    MATH  Google Scholar 

  8. Congreve, S.: A posteriori error analysis of hp-adaptive finite element methods for second-order quasi-linear PDEs. Master’s thesis, University of Nottingham (2010)

  9. Congreve, S.: hp–Adaptive discontinuous Galerkin finite element methods for second-order quasi-linear PDEs. PhD thesis, University of Nottingham. In preparation

  10. Congreve, S., Houston, P., Wihler, T.P.: Two-grid hp-version DGFEMs for strongly monotone second–order quasilinear elliptic PDEs. In: Proceedings in Applied Mathematics and Mechanics, 82nd Annual GAMM Scientific Conference, Graz, Austria (2011)

    Google Scholar 

  11. Dawson, C.N., Wheeler, M.F., Woodward, C.S.: A two-grid finite difference scheme for non-linear parabolic equations. SIAM J. Numer. Anal. 35, 435–452 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gudi, T., Nataraj, N., Pani, A.K.: hp-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems. Numer. Math. 109, 233–268 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Houston, P., Robson, J., 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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Houston, P., Schötzau, D., Wihler, T.P.: Energy norm a posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems. Math. Models Methods Appl. Sci. 17(1), 33–62 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Houston, P., Schwab, C., Süli, E.: A note on the design of hp-adaptive finite element methods for elliptic partial differential equations. Comput. Methods Appl. Mech. Eng. 194(2–5), 229–243 (2005)

    Article  MATH  Google Scholar 

  16. Houston, P., Süli, E., Wihler, T.P.: A posteriori error analysis of hp-version discontinuous Galerkin finite-element methods for second-order quasi-linear PDEs. IMA J. Numer. Anal. 28(2), 245–273 (2007)

    Article  Google Scholar 

  17. Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41(6), 2374–2399 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu, W.B., Barrett, J.W.: Quasi-norm error bounds for the finite element approximation of some degenerate quasilinear elliptic equations and variational inequalities. RAIRO Modél. Math. Anal Numér. 28(6), 725–744 (1994)

    MathSciNet  MATH  Google Scholar 

  19. Marion, M., Xu, J.: Error estimates on a new nonlinear Galerkin method based on two-grid finite elements. SIAM J. Numer. Anal. 32(4), 1170–1184 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Metcalf, M., Reid, J., Cohen, M.: Fortran 95/2003 Explained. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  21. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Computer Science and Applied Mathematics. Academic Press, New York (1970)

    MATH  Google Scholar 

  22. Ortner, C., Süli, E.: Discontinuous Galerkin finite element approximation of nonlinear second-order elliptic and hyperbolic systems. SIAM J. Numer. Anal. 45(4), 1370–1397 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  24. Schwab, C.: p- and hp-FEM—Theory and Applications in Solid and Fluid Mechanics. Oxford University Press, Oxford (1998)

    Google Scholar 

  25. Stamm, B., Wihler, T.P.: hp-optimal discontinuous Galerkin methods for linear elliptic problems. Math. Comput. 79(272), 2117–2133 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Utnes, T.: Two-grid finite element formulations of the incompressible Navier–Stokes equations. Commun. Numer. Methods Eng. 13(8), 675–684 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wihler, T.P.: An hp-adaptive strategy based on continuous Sobolev embedding. J. Comput. Appl. Math. 235, 2731–2739 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wihler, T.P., Frauenfelder, P., Schwab, C.: Exponential convergence of the hp-DGFEM for diffusion problems. Comput. Math. Appl. 26, 183–205 (2003)

    Article  MathSciNet  Google Scholar 

  29. Wu, L., Allen, M.B.: Two-grid method for mixed finite-element solution of coupled reaction-diffusion systems. Numer. Methods Partial Differ. Equ. 1999, 589–604 (1999)

    Article  MathSciNet  Google Scholar 

  30. Xu, J.: A new class of iterative methods for nonselfadjoint or indefinite problems. SIAM J. Numer. Anal. 29, 303–319 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. Xu, J.: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15, 231–237 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  32. Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33, 1759–1777 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhu, L., Giani, S., Houston, P., Schötzau, D.: Energy norm a-posteriori error estimation for hp-adaptive discontinuous Galerkin methods for elliptic problems in three dimensions. Math. Model. Methods Appl. Sci. 21(2), 267–306 (2011)

    Article  MATH  Google Scholar 

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Acknowledgements

PH acknowledges the financial support of the EPSRC under the grant EP/H005498. TW acknowledges the financial support of the Swiss National Science Foundation (SNF) under grant No. 200021126594.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    Scott Congreve & Paul Houston

  2. Mathematisches Institut, Universität Bern, Sidlerstrasse 5, 3012, Bern, Switzerland

    Thomas P. Wihler

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  1. Scott Congreve
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  2. Paul Houston
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  3. Thomas P. Wihler
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Correspondence to Paul Houston.

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Congreve, S., Houston, P. & Wihler, T.P. Two-Grid hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic PDEs. J Sci Comput 55, 471–497 (2013). https://doi.org/10.1007/s10915-012-9644-1

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