Numerische Mathematik

, Volume 109, Issue 2, pp 167–191

Stability and accuracy of adapted finite element methods for singularly perturbed problems

Article

Abstract

The stability and accuracy of a standard finite element method (FEM) and a new streamline diffusion finite element method (SDFEM) are studied in this paper for a one dimensional singularly perturbed connvection-diffusion problem discretized on arbitrary grids. Both schemes are proven to produce stable and accurate approximations provided that the underlying grid is properly adapted to capture the singularity (often in the form of boundary layers) of the solution. Surprisingly the accuracy of the standard FEM is shown to depend crucially on the uniformity of the grid away from the singularity. In other words, the accuracy of the adapted approximation is very sensitive to the perturbation of grid points in the region where the solution is smooth but, in contrast, it is robust with respect to perturbation of properly adapted grid inside the boundary layer. Motivated by this discovery, a new SDFEM is developed based on a special choice of the stabilization bubble function. The new method is shown to have an optimal maximum norm stability and approximation property in the sense that \(\|u-u_{N}\|_{\infty}\leq C\inf_{v_{N}\in V^{N}}\|u-v_{N}\|_{\infty},\) where uN is the SDFEM approximation in linear finite element space VN of the exact solution u. Finally several optimal convergence results for the standard FEM and the new SDFEM are obtained and an open question about the optimal choice of the monitor function for the moving grid method is answered.

Mathematics Subject Classification (2000)

65L10 65L20 65L60 76R99 

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References

  1. 1.
    Bakhalov N.S. (1969). Towards optimization of methods for solving boundary value problems in the presence of boundary layers (in Russian). Zh. Vychisl. Mater. Mater. Fiz. 9: 841–859 Google Scholar
  2. 2.
    Borouchaki, H., Castro-Diaz, M.J., George, P.L., Hecht, F., Mohammadi, B.: Anisotropic adaptive mesh generation in two dimensions for CFD. In: 5th International Conference On Numerical Grid Generation in Computational Field Simulations, vol.3, pp.197–206. Mississppi State University (1996)Google Scholar
  3. 3.
    Brezzi F., Hughes T.J.R., Marini L.D., Russo A. and Süli E. (1999). A priori error analysis of residual-free bubbles for advection-diffusion problems. SIAM J. Numer. Anal. 36(4): 1933–1948 MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brezzi F., Marini D. and Süli E. (2000). Residual-free bubbles for advection-diffusion problems: the general error analysis. Numer. Math. 85: 31–47 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Brezzi F. and Russo A. (1994). Choosing bubbles for advection-diffusion problems. Math. Models Methods Appl. Sci. 4: 571–587 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cao W., Huang W. and Russell R.D. (1999). A study of monitor functions for two dimensional adaptive mesh generation. SIAM J. Sci. Comput. 20: 1978–1994 MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Carey G.F. and Dinh H.T. (1985). Grading functions and mesh redistribution. SIAM J. Numer. Anal. 22(5): 1028–1040 MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chen, L.: Mesh smoothing schemes based on optimal Delaunay triangulations. In: 13th International Meshing Roundtable, pp.109–120. Sandia National Laboratories, Williamsburg (2004)Google Scholar
  9. 9.
    Chen L. (2005). New analysis of the sphere covering problems and optimal polytope approximation of convex bodies. J. Approx. Theory 133(1): 134–145 MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Chen, L., Sun, P., Xu, J.: Multilevel homotopic adaptive finite element methods for convection dominated problems. In: The Proceedings for 15th Conferences for Domain Decomposition Methods. Lecture Notes in Computational Science and Engineering 40, pp.459–468. Springer, Heidelberg (2004)Google Scholar
  11. 11.
    Chen L., Sun P. and Xu J. (2007). Optimal anisotropic simplicial meshes for minimizing interpolation errors in L p-norm. Math. Comput. 76: 179–204 MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chen L. and Xu J. (2004). Optimal Delaunay triangulations. J. Comput. Math. 22(2): 299–308 MATHMathSciNetGoogle Scholar
  13. 13.
    Chen, L., Xu, J.: An optimal streamline diffusion finite element method for a singularly perturbed problem. In: AMS Contemporary Mathematics Series: Recent Advances in Adaptive Computation, vol.383, pp.236–246, Hangzhou (2005)Google Scholar
  14. 14.
    Chen, Y.: Uniform pointwise convergence for a singularly perturbed problem using arc-length equidistribution. In: Proceedings of the 6th japan-china joint seminar on numerical mathematics (tsukuba, 2002). J. Comput. Appl. Math. 159(1), 25–34 (2003)Google Scholar
  15. 15.
    Chen Y. (2006). Uniform convergence analysis of finite difference approximations for singular perturbation problems on an adapted grid. Adv. Comput. Math. 24: 197–212 MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    D’Azevedo E.F. and Simpson R.B. (1989). On optimal interpolation triangle incidences. SIAM J. Sci. Statist. Comput. 6: 1063–1075 MathSciNetGoogle Scholar
  17. 17.
    de Boor C. (1973). Good approximation by splines with variable knots. Int. Seines Numer. Math, Birkhauser Verlag, Basel 21: 57–72 Google Scholar
  18. 18.
    Boor C. de(1974). Good approximation by splines with variables knots II. In: Watson, G.A. (eds) Proceedings of the Eleventh International Conference on Numerical Methods in Fluid Dynamics, vol. 363., pp 12–20. Springer, Dundee Google Scholar
  19. 19.
    Devore R.A. and Lorentz G.G. (1993). Constructive Approximation. Springer, New York MATHGoogle Scholar
  20. 20.
    Dolejšì V. and Felcman J. (2004). Anisotropic mesh adaptation for numerical solution of boundary value problems. Numer. Methods Partial Differ. Equ. 20: 576–608 CrossRefGoogle Scholar
  21. 21.
    Farrell P.A., Hegarty A.F., Miller J.J.H., O’Riordan E. and Shishkin G.I. (2004). Singularly perturbed convection-diffusion problems with boundary and weak interior layers. J. Comput. Appl. Math. 166: 131–151 CrossRefMathSciNetGoogle Scholar
  22. 22.
    Franca L.P. and Russo A. (1996). Deriving upwinding, mass lumping and selective reduced integration by residual-free bubbles. Appl. Math. Lett. 9: 83–88 MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Habashi, W.G., Fortin, M., Dompierre, J., Vallet, M.G., Ait-Ali-Yahia, D., Bourgault, Y., Robichaud, M.P., Tam, A., Boivin, S.: Anisotropic mesh optimization for structured and unstructured meshes. In: 28th Computational Fluid Dynamics Lecture Series. von Karman Institute (1997)Google Scholar
  24. 24.
    Huang W. (2001). Practical aspects of formulation and solution of moving mesh partial differential equations. J. Comput. Phys. 171: 753–775 MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Huang W. (2001). Variational mesh adaptation: isotropy and equidistribution. J. Comput. Phys. 174: 903–924 MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Huang W. and Sun W. (2003). Variational mesh adaptation. II: Error estimates and monitor functions. J. Comput. Phys. 184: 619–648 MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Hughes, T.J.R.: Multiscale phenomena: Green’s functions, the dirichlet-to-neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Eng, pp.127, no. 1–4, 387–401 (1995)Google Scholar
  28. 28.
    Hughes T.J.R. and Brooks A. (1979). A multidimensional upwind scheme with no crosswind diffusion. In: Hughes, T.J.R. (eds) Finite Element Methods for Convection Dominated Flows, AMD, vol. 34, pp 19–35. ASME, New York Google Scholar
  29. 29.
    Hughes T.J.R., Feijoo G., Mazzei L. and Quincy J.B. (1998). The variational multiscale method—a paradigm for computational mechanics. Comput. Methods Appl. Mech. Eng. 166: 3–24 MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Johnson C. and Nvert U. (1981). An analysis of some finite element methods for advection-diffusion problems. In: Axelsson, O., Frank, L.S. and Vander Sluis, A. (eds) Analytical and Numerical Approaches to Asymptotic Problems in Analysis., pp 99–116. Amsterdam, NorthHolland Google Scholar
  31. 31.
    Johnson C., Schatz A.H. and Wahlbin L.B. (1987). Crosswind smear and pointwise errors in streamline diffusion finite element methods. Math. Comput. 49: 25–38 MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Kellogg R.B. and Tsan A. (1978). Analysis of some difference approximations for a singular perturbation problem without turning points. Math. Comput. 32: 1025–1039 MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Kopteva N.V. (1999). Uniform convergence with respect to a small parameter of a scheme with central difference on refining grids. Comput. Math. Math. Phys. 39(10): 1594–1610 MATHMathSciNetGoogle Scholar
  34. 34.
    Kopteva N.V. (2001). Maximum norm a posteriori error estimates for a one-dimensional convection-diffusion problem. SIAM J. Numer. Anal. 39(2): 423–441 MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Kopteva N.V. and Stynes M. (2001). A robust adaptive method for quasi-linear one-dimensional convection-diffusion problem. SIAM J. Numer. Anal. 39: 1446–1467 MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Lenferink W. (2000). Pointwise convergence of approximations to a convection-diffusion equation on a Shishkin mesh. Appl. Numer. Math. 32(1): 69–86 MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Linß T. (2001). Sufficient conditions for uniform convergence on layer-adapted grids. Appl. Numer. Math. 37: 241–255 MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Linß T. (2003). Layer-adapted meshes for convectioni-diffusion problems. Comput. Methods Appl. Mech. Eng. 192: 1061–1105 MATHCrossRefGoogle Scholar
  39. 39.
    Linß T. and Stynes M. (2001). The SDFEM on Shishkin meshes for linear convection-diffusion problems. Numer. Math. 87: 457–484 MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Miller J.J.H., O’Riordan E. and Shishkin G.I. (1995). On piecewise-uniform meshes for upwind- and central-difference operators for solving singularly perturbed problems. IMA J. Numer. Anal. 15(1): 89–99 MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Miller J.J.H., O’Riordan E. and Shishkin G.I. (1996). Fitted Numerical Methods For Singular Perturbation Problems. World Scientific, Singapore MATHGoogle Scholar
  42. 42.
    Morton K.W. (1996). Numerical Solution of Convection-Diffusion Problems, volume 12 of Applied Mathematics and Mathematical Computation. Chapman & Hall, London Google Scholar
  43. 43.
    Nadler E. (1986). Piecewise linear best L 2 approximation on triangulations. In: Chui, C.K., Schumaker, L.L. and Ward, J.D. (eds) Approximation Theory, vol. V, pp 499–502. Academic, New York Google Scholar
  44. 44.
    Niijima K. (1990). Pointwise error estimates for a streamline diffusion finite element scheme. Numer. Math. 56: 707–719 MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Qiu Y., Sloan D.M. and Tang T. (2000). Convergence analysis of an adaptive finite difference method for a singular perturbation problem. J. Comput. Appl. Math. 116: 121–143 MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Roos H.G. (1998). Layer-adapted grids for singular perturbation problems. ZAMM, Z. Angew. Math. Mech. 78(5): 291–309 MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Roos H.G., Stynes M. and Tobiska L. (1996). Numerical Methods for Singularly Perturbed Differential Equations, volume 24 of Springer series in Computational Mathematics. Springer, Heidelberg Google Scholar
  48. 48.
    Roos H.G. and Zarin H. (2003). The streamline-diffusion method for a convection-diffusion problem with a point source. J. Comput. Appl. Math. 150: 109–128 MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Sangalli G. (2003). Quasi optimality of the supg method for the one-dimensional adavection-diffusion problem. SIAM J. Numer. Anal. 41(4): 1528–1542 MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Schatz A.H. and Wahlbin L.B. (1982). On the quasi-optimality in L of the \(\overset{\circ}{H^1}\) -projection into finite element spacesMath. Comput. 38(157): 1–22 MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Schatz A.H. and Wahlbin L.B. (1983). On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions. Math. Comput. 40(161): 47–89 MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Shishkin, G.I.: Grid approximation of singularly perturbed elliptic and parabolic equations (in Russian). PhD thesis, Second doctorial thesis, Keldysh Institute, Moscow (1990)Google Scholar
  53. 53.
    Stynes M. and Tobiska L. (1998). A finite difference analysis of a streamline diffusion method on a Shishkin mesh. Numer. Algorithms 18: 337–360 MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Stynes M. and Tobiska L. (2003). The SDFEM for a convection-diffusion problem with a boundary layer: optimal error analysis, enhancement of accuracy. SIAM J. Numer. Anal. 41(5): 1620–1642 MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    White A.B. (1979). On selection of equidistributing meshes for two-point boundary-value problems. SIAM J. Numer. Anal. 16: 472–502 MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Zhang Z.M. (2002). Finite element superconvergence approximation for one-dimensional singularly perturbed problems. Numer. Meth. PDEs 18: 374–395 MATHGoogle Scholar
  57. 57.
    Zhang Z.M. (2003). Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems. Math. Comput. 72(243): 1147–1177 MATHCrossRefGoogle Scholar
  58. 58.
    Zhou G. and Rannacher R. (1996). Pointwise superconvergence of the streamline diffusion finite element method. Numer. Meth. PDEs 12, CMP 96(05): 123–145MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at IrvineIrvineUSA
  2. 2.The School of Mathematical SciencePeking UniversityBeijingChina

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