Numerical Algorithms

, Volume 79, Issue 1, pp 281–310 | Cite as

Boundary layer preconditioners for finite-element discretizations of singularly perturbed reaction-diffusion problems

  • Thái Anh Nhan
  • Scott MacLachlan
  • Niall MaddenEmail author
Original Paper


We consider the iterative solution of linear systems of equations arising from the discretization of singularly perturbed reaction-diffusion differential equations by finite-element methods on boundary-fitted meshes. The equations feature a perturbation parameter, which may be arbitrarily small, and correspondingly, their solutions feature layers: regions where the solution changes rapidly. Therefore, numerical solutions are computed on specially designed, highly anisotropic layer-adapted meshes. Usually, the resulting linear systems are ill-conditioned, and so, careful design of suitable preconditioners is necessary in order to solve them in a way that is robust, with respect to the perturbation parameter, and efficient. We propose a boundary layer preconditioner, in the style of that introduced by MacLachlan and Madden for a finite-difference method (MacLachlan and Madden, SIAM J. Sci. Comput. 35(5), A2225–A2254 2013). We prove the optimality of this preconditioner and establish a suitable stopping criterion for one-dimensional problems. Numerical results are presented which demonstrate that the ideas extend to problems in two dimensions.


Singularly perturbed Layer-adapted meshes Robust multigrid Preconditioning 


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  1. 1.
    Adler, J., MacLachlan, S., Madden, N.: A first-order system Petrov-Galerkin discretisation for a reaction-diffusion problem on a fitted mesh. IMA J. Numer. Anal. 36(3), 1281–1309 (2016). MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alcouffe, R.E., Brandt, A., Dendy, J.E. Jr., Painter, J.W.: The multigrid method for the diffusion equation with strongly discontinuous coefficients. SIAM J. Sci. Statist. Comput. 2(4), 430–454 (1981). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anderson, E., Bai, Z., Bischof, C., Blackford, L.S., Demmel, J., Dongarra, J.J., Du Croz, J., Hammarling, S., Greenbaum, A., McKenney, A., Sorensen, D.: LAPACK Users' Guide, 3rd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, USA (1999)Google Scholar
  4. 4.
    Bakhvalov, N.: Towards optimization of methods for solving boundary value problems in the presence of boundary layers. Zh. Vychisl. Mat. i Mat. Fiz. 9, 841–859 (1969)Google Scholar
  5. 5.
    Briggs, W.L., Henson, V.E., McCormick, S.F.: A multigrid tutorial, 2 edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2000).
  6. 6.
    Chen, Y., Davis, T.A., Hager, W.W., Rajamanickam, S.: Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate. ACM Trans. Math. Softw. 35(3), 22:1–22:14 (2008). MathSciNetCrossRefGoogle Scholar
  7. 7.
    Davis, T.A.: Direct methods for sparse linear systems, Fundamentals of Algorithms, vol. 2. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2006).
  8. 8.
    Dendy, J.E. Jr.: Black box multigrid. J. Comput. Phys. 48(3), 366–386 (1982). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. No. 16 in Applied Mathematics. Chapman & Hall/CRC, Boca Raton (2000)zbMATHGoogle Scholar
  10. 10.
    Gaspar, F., Clavero, C., Lisbona, F.: Some numerical experiments with multigrid methods on Shishkin meshes. J. Comput. Appl. Math. 138(1), 21–35 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gaspar, F., Lisbona, F., Clavero, C.: Numerical Analysis and Its Applications, Lecture Notes in Computer Science, vol. 1988, pp. 128–135. Springer Berlin / Heidelberg. In: Vulkov, L., Yalamov, P., Wasniewski, J. (eds.) (2001).
  12. 12.
    George, A.: Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 10, 345–363 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kopteva, N., Madden, N., Stynes, M.: Grid equidistribution for reaction-diffusion problems in one dimension. Numer. Algorithms 40(3), 305–322 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    LAPACK version 3.5.0: Subroutine DPTTRS.
  15. 15.
    Linß, T.: Layer-adapted meshes for reaction-convection-diffusion problems. Lecture notes in Mathematics 1985. Berlin: Springer. xi, 320 p. (2010).
  16. 16.
    Liu, F., Madden, N., Stynes, M., Zhou, A.: A two-scale sparse grid method for a singularly perturbed reaction-diffusion problem in two dimensions. IMA J. Numer. Anal. 29(4), 986–1007 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    MacLachlan, S., Madden, N.: Robust solution of singularly perturbed problems using multigrid methods. SIAM J. Sci. Comput. 35(5), A2225–A2254 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted numerical methods for singular perturbation problems, revised edn. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ. Error estimates in the maximum norm for linear problems in one and two dimensions (2012).
  19. 19.
    Nhan, T.A., Madden, N.: Cholesky factorisation of linear systems coming from finite difference approximations of singularly perturbed problems. In: BAIL 2014–Boundary and Interior Layers, Computational and Asymptotic Methods, Lect. Notes Comput. Sci. Eng., pp. 209–220. Springer International Publishing (2015).
  20. 20.
    Roos, H.G.: A note on the conditioning of upwind schemes on Shishkin meshes. IMA J. Numer. Anal. 16(4), 529–538 (1996). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer Series in Computational Mathematics, 2nd edn., vol. 24. Springer, Berlin (2008)zbMATHGoogle Scholar
  22. 22.
    Trottenberg, U., Oosterlee, C.W., Schüller, A.: Multigrid. Academic Press, Inc, San Diego (2001)zbMATHGoogle Scholar
  23. 23.
    Wathen, A.J.: Realistic eigenvalue bounds for the Galerkin mass matrix. IMA J. Numer. Anal. 7(4), 449–457 (1987). MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wesseling, P.: An introduction to multigrid methods. Pure and Applied Mathematics (New York). Wiley, Chichester (1992)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsOhlone CollegeFremontUSA
  2. 2.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt John’sCanada
  3. 3.School of Mathematics, Statistics and Applied MathematicsNational University of Ireland GalwayGalwayIreland

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