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Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities

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Abstract

In this paper, a discontinuous Galerkin least-squares finite element method is developed for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities by recasting the second-order elliptic equations as a system of first-order equations. In a companion paper (Lin in SIAM J Numer Anal 47:89–108, 2008) a similar method has been developed for problems with continuous data and shown to be well-posed, uniformly convergent, and optimal in convergence rate. In this paper the method is modified to take care of conditions that arise at interfaces and boundary singularities. Coercivity and uniform error estimates for the finite element approximation are established in an appropriately scaled norm. Numerical examples confirm the theoretical results.

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References

  1. Adams R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. 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)

    Article  MATH  MathSciNet  Google Scholar 

  3. Babuška I.: The finite element method for elliptic equations with discontinuous coefficients. Computing 5, 207–213 (1970)

    Article  MATH  Google Scholar 

  4. Babuška I., Baumann C.E., Oden J.T.: A discontinuous hp finite element method for diffusion problems: 1-D analysis. Comput. Math. Appl. 37, 103–122 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bensow R.E., Larson M.G.: Discontinuous/continuous least-squares finite element methods for elliptic problems. Math. Models Methods Appl. Sci. 15, 825–842 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bensow R.E., Larson M.G.: Discontinuous least-squares finite element method for the div-curl problem. Numer. Math. 101, 601–617 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bernardi C., Verfürth R.: Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85, 579–608 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Berndt M., Manteuffel T.A., McCormick S.F., Starke G.: Analysis of first-order system least squares (FOSLS) for elliptic problems with discontinuous coefficients. I. SIAM J. Numer. Anal. 43, 386–408 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Berndt M., Manteuffel T.A., McCormick S.F.: Analysis of first-order system least squares (FOSLS) for elliptic problems with discontinuous coefficients. II. SIAM J. Numer. Anal. 43, 409–436 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bjørstad P.E., Dryja M., Rahman T.: Additive Schwarz methods for elliptic mortar finite element problems. Numer. Math. 95, 427–457 (2003)

    Article  MathSciNet  Google Scholar 

  11. Bochev P.B., Gunzburger M.D.: Finite element methods of least-squares type. SIAM Rev. 40, 789–837 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bramble J.H., Lazarov R.D., Pasciak J.E.: A least-squares approach based on a discrete minus one inner product for first order systems. Math. Comput. 66, 935–955 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. Bramble J.H., Lazarov R.D., Pasciak J.E.: Least-squares methods for linear elasticity based on a discrete minus one inner product. Comput. Methods Appl. Mech. Eng. 191, 727–744 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Brayanov I.A.: Numerical solution of a two-dimensional singularly perturbed reaction-diffusion problem with discontinuous coefficients. Appl. Math. Comput. 182, 631–643 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Brenner S.C., Scott L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)

    MATH  Google Scholar 

  16. Cai Z., Lazarov R.D., Manteuffel T.A., McCormick S.F.: First-order system least squares for second-order partial differential equations. I. SIAM J. Numer. Anal. 31, 1785–1799 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Cai Z., Manteuffel T.A., McCormick S.F.: First-order system least squares for second-order partial differential equations. II. SIAM J. Numer. Anal. 34, 425–454 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  18. Cai Z., Manteuffel T.A., McCormick S.F., Ruge J.: First-order system \({\mathcal{LL}^*}\) (FOSLL*): scalar elliptic partial differential equations. SIAM J. Numer. Anal. 39, 1418–1445 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Cai Z., Westphal C.R.: A weighted H(div) least-squares method for second-order elliptic problems. SIAM J. Numer. Anal. 46, 1640–1651 (2008)

    Article  MathSciNet  Google Scholar 

  20. Cao Y., Gunzburger M.D.: Least-squares finite element approximations to solutions of interface problems. SIAM J. Numer. Anal. 35, 393–405 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  21. Chen Z., Dai S.: On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients. SIAM J. Sci. Comput. 24, 443–462 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  23. Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.): Discontinuous galerkin methods. Theory, computation and applications. Lecture Notes in Computational Science and Engineering, vol. 11. Springer, New York (2000)

  24. 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  MATH  MathSciNet  Google Scholar 

  25. Cockburn B., Shu C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  26. Dokeva N., Dryja M., Proskurowski W.: A FETI-DP preconditioner with a special scaling for mortar discretization of elliptic problems with discontinuous coefficients. SIAM J. Numer. Anal. 44, 283–299 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Farrell P.A., Miller J.J.H., O’Riordan E., Shishkin G.I.: Singularly perturbed differential equations with discontinuous source terms. In: Vulkov, L.G., Miller, J.J.H., Shishkin, G.I.(eds) Analytical and Numerical Methods for Convection-Dominated and Singularly Perturbed Problems, pp. 23–31. Nova Science Publishers, Inc., New York (2000)

    Google Scholar 

  28. Gerritsma M.I., Proot M.M.J.: Analysis of a discontinuous least squares spectal element method. J. Sci. Comput. 17, 297–306 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  29. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224. Springer, Berlin (1983)

  30. Houston P., Jensen M., Süli E.: hp-discontinuous Galerkin finite element methods with least-squares stabilization. J. Sci. Comput. 17, 3–25 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  31. Huang J., Zou J.: A mortar element method for elliptic problems with discontinuous coefficients. IMA J. Numer. Anal. 22, 549–576 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  32. Jiang B.N.: The Least-squares finite element method. Theory and Applications in Computational Fluid Dynamics and Electromagnetics. Springer, Berlin (1998)

    Google Scholar 

  33. Lee E., Manteuffel T.A., Westphal C.R.: Weighted-norm first-order system least squares (FOSLS) for problems with corner singularities. SIAM J. Numer. Anal. 44, 1974–1996 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. Lee E., Manteuffel T.A., Westphal C.R.: Weighted-norm first-order system least-squares (FOSLS) for div/curl systems with three dimensional edge singularities. SIAM J. Numer. Anal. 46, 1619–1639 (2008)

    Article  MathSciNet  Google Scholar 

  35. Lin R.: Discontinuous discretization for least-squares formulation of singularly perturbed reaction-diffusion problems in one and two dimensions. SIAM J. Numer. Anal. 47, 89–108 (2008)

    Article  MathSciNet  Google Scholar 

  36. Manteuffel T.A., McCormick S.F., Ruge J., Schmidt J.G.: First-order system \({\mathcal{LL}^*}\) (FOSLL*) for general scalar elliptic problems in the plane. SIAM J. Numer. Anal. 43, 2098–2120 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  37. Marcinkowski L.: Additive Schwarz method for mortar discretization of elliptic problems with P 1 nonconforming finite elements. BIT 45, 375–394 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  38. Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted numerical methods for singular perturbation problems. Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. World Scientific Publishing Co., Inc., River Edge (1996)

  39. Miller J.J.H., O’Riordan E., Shishkin G.I., Wang S.: A parameter-uniform Schwarz method for a singularly perturbaed reaction-diffusion problem with an interior-layer. Appl. Numer. Math. 35, 323–337 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  40. Morton, K.W.: Numerical solution of convection-diffusion problems. Applied Mathematics and Mathematical Computation, vol. 12. Chapman & Hall, London (1996)

  41. Oden J.T., Babuška I., Baumann C.E.: A discontinuous hp finite element method for diffusion problems. J. Comput. Phys. 146, 491–519 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  42. Oden, J.T., Carey, G.F.: Finite elements, mathematical aspects. The Texas Finite Element Series, vol IV. Prentice-Hall, New Jersey (1983)

  43. O’Riordan E., Stynes M.: A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions. Math. Comp. 57, 47–62 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  44. Pontaza J.P., Reddy J.N.: Least-squares finite element formulations for viscous incompressible and compressible fluid flows. Comput. Methods Appl. Mech. Eng. 195, 2454–2494 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  45. Rahman T., Xu X., Hoppe R.: Additive Schwarz methods for the Crouzeix-Raviart mortar finite element for elliptic problems with discontinuous coefficients. Numer. Math. 101, 551–572 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  46. Roos H.-G., Stynes M., Tobiska L.: Numerical methods for singularly perturbed differential equations. Convection-Diffusion and Flow Problems. Springer, Berlin (1996)

    MATH  Google Scholar 

  47. Roos H.-G., Zarin H.: A second-order scheme for singularly perturbed differential equations with discontinuous source term. J. Numer. Math. 10, 275–289 (2002)

    MATH  MathSciNet  Google Scholar 

  48. Sauter S.A., Warnke R.: Composite finite elements for elliptic boundary value problems with discontinuous coefficients. Computing 77, 29–55 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  49. Stynes M., O’Riordan E.: An analysis of a singularly perturbed two-point boundary value problem using only finite element techniques. Math. Comp. 56, 663–675 (1991)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  51. Zarin H., Roos H.-G.: Interior penalty discontinuous approximations of convection-diffusion problems with parabolic layers. Numer. Math. 100, 735–759 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematical and Physical Sciences, Texas A&M International University, Laredo, TX, 78041-1900, USA

    Runchang Lin

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Lin, R. Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities. Numer. Math. 112, 295–318 (2009). https://doi.org/10.1007/s00211-008-0208-0

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  • DOI: https://doi.org/10.1007/s00211-008-0208-0

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