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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 50 Issue: 5

Error estimates for a fully discrete $\varepsilon-$uniform finite element method on quasi uniform meshes

Ali Şendur(ALANYA ALAADDIN KEYKUBAT UNIVERSITY)0000-0001-8628-5497
Srinivasan Natesan(Indian Institute of Technology)0000-0001-7527-1989
Gautam Sıngh(Indian Institute of Technology)0000-0002-0570-2170
Sayfa
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Yayın Tarihi
2021-10-15
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

Further, we determine the amount of error introduced by the discrete $L-$spline basis functions in the overall numerical method, and explore the possibility of recovering the order of convergence that are consistent with the classical order of convergence for the numerical methods using the exact $L-$splines.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:In this article, we analyze a fully discrete $\varepsilon-$uniformly convergent finite element method for singularly perturbed convection-diffusion-reaction boundary-value problems, on piecewise-uniform meshes.
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Temel Bulgular & Veri:We will concentrate on the convergence analysis of the finite element method which employ the discrete $L-$spline basis functions instead of their continuous counterparts.
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Sonuç & Katkı:Further, we determine the amount of error introduced by the discrete $L-$spline basis functions in the overall numerical method, and explore the possibility of recovering the order of convergence that are consistent with the classical order of convergence for the numerical methods using the exact $L-$splines.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:singularly perturbed convection-diffusion-reaction boundary-value problem · finite element method · boundary layers

Abstract

In this article, we analyze a fully discrete $\varepsilon-$uniformly convergent finite element method for singularly perturbed convection-diffusion-reaction boundary-value problems, on piecewise-uniform meshes. Here, we choose $L-$splines as basis functions. We will concentrate on the convergence analysis of the finite element method which employ the discrete $L-$spline basis functions instead of their continuous counterparts. The $L-$splines are approximated on the piecewise-uniform Shishkin mesh inside each element. These approximations are used as basis functions in the frame of Galerkin FEM on a coarse piecewise-uniform mesh to discretize the domain. Further, we determine the amount of error introduced by the discrete $L-$spline basis functions in the overall numerical method, and explore the possibility of recovering the order of convergence that are consistent with the classical order of convergence for the numerical methods using the exact $L-$splines.

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Bu Makaleye Atıf Yap

Şendur, A., Natesan, S., Sıngh, G. (2021). Error estimates for a fully discrete $\varepsilon-$uniform finite element method on quasi uniform meshes. Hacettepe Journal of Mathematics and Statistics, Volume: 50 Issue: 5. https://doi.org/10.15672/hujms.691017

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Kaynakça (31)

  1. F. Brezzi, P. Houston, D. Marini and E. Süli, Modeling subgrid viscosity for advectiondiffusion problems, Comput. Meth. Appl. Mech. Eng. 190, 1601–1610, 2000.
  2. F. Brezzi, T.J.R. Hughes, L.D. Marini, A. Russo and E. Süli, A priori error analysis of residual-free bubbles for advection-diffusion problems, SIAM J. Numer. Anal. 36 (6), 1933-1948, 1999.
  3. F. Brezzi, D. Marini and A. Russo, Applications of pseudo residual-free bubbles to the stabilization of convection-diffusion problems, Comput. Meth. Appl. Mech. Eng. 166, 51–63, 1998.
  4. F. Brezzi and A. Russo, Choosing bubbles for advection-diffusion problems, Math. Models Meth. Appl. Sci. 4, 571–587, 1994.
  5. A.N. Brooks and T.J.R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection-dominated flows with particular emphasis on the incompressible Navier- Stokes equations, Comput. Meth. Appl. Mech. Eng. 32, 199–259, 1982.
  6. A. Buffa, T.J.R. Hughes and G. Sangalli, Analysis of a Multiscale Discontinuous Galerkin Method for Convection Diffusion Problems, SIAM J. Numer. Anal. 44, 1420- 1440, 2006.
  7. A. Cangiani and E. Süli, Enhanced residual-free bubble method for convection-diffusion problems, Inter. J. Numer. Methods Fluids 47, 1307-1313, 2005.
  8. B. Cockburn, Discontinuous Galerkin methods for convection-dominated problems, Springer Berlin Heidelberg, 1999.
  9. L.P. Franca, S.L. Frey and T.J.R. Hughes, Stabilized finite element methods: I. Application to the advective-diffusive model, Comput. Meth. Appl. Mech. Eng. 95, 253–276, 1992.
  10. L.P. Franca and F.N. Hwang, Refining the submesh strategy in the two-level finite element method: application to the advection diffusion equation, Int. J. Numer. Methods Fluids 39, 161–187, 2002.
  11. L.P. Franca, A.L. Madureira and F. Valentin, Towards multiscale functions: enriching finite element spaces with local but not bubble-like functions, Comput. Meth. Appl. Mech. Eng. 194, 3006–3021, 2005.
  12. L.P. Franca and A. Nesliturk, On a two-level finite element method for the incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids 52, 433–453, 2001.
  13. L.P. Franca, A. Nesliturk and M. Stynes, On the stability of residual-free bubbles for convection-diffusion problems and their approximation by a two-level finite element method, Comput. Meth. Appl. Mech. Eng. 166, 35–49, 1998.
  14. L.P. Franca, J.V.A. Ramalho and F. Valentin, Multiscale and residual-free bubble functions for reaction-advection-diffusion problems, Int. J. Multiscale Com. 3, 297– 312, 2005.
  15. L.P. Franca and L. Tobiska, Stability of the residual free bubble method for bilinear finite elements on rectangular grids, IMA J. Numer. Anal. 22, 73–87, 2002.
  16. E. C. Gartland, Graded-mesh difference schemes for singularly perturbed two-point boundary-value problems, Math. Comp. 51, 631–657, 1988.
  17. T.J.R. Hughes, Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origin of stabilized methods, Comput. Meth. Appl. Mech. Eng. 127, 387–401, 1995.
  18. T.J.R. Hughes, G.R. Feijoo, L. Mazzei and J.B. Quincy, The variational multiscale method -a paradigm for computational mechanics, Comput. Meth. Appl. Mech. Eng. 166 (1-2), 3–24, 1998.
  19. R.B. Kellog and A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comp. 32, 1025–1039, 1978.
  20. N. Kopteva and E. O’Riordan, Shishkin meshes in the numerical solution of singularly perturbed differential equations, Int. J. Numer. Anal. Model. 7, 393–415, 2010.
  21. T. Linss, The necessity of Shishkin-decompositions, Appl. Math. Lett. 14, 891–896, 2001.
  22. J.J.H. Miller, E. O’Riordan and G.I. Shishkin, Fitted Numerical Methods For Singular Perturbation Problems, Singapore, New Jersey, London, Hong Kong: World Scientific, 1996.
  23. E. O’Riordan and M. Stynes, A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions, Math. Comp. 57, 47–62, 1991.
  24. H.G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods For Singularly Perturbed Differential Equations, Verlag, Berlin: Springer, 2008.
  25. A. Sendur and A. Nesliturk, Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion-reaction problems, Calcolo 49, 1–19, 2012.
  26. G.I. Shishkin, Grid approximation of singularly perturbed parabolic equations with internal layers, Soviet J. Numer. Anal. Math. Appl. Modelling. 3, 393–407, 1988.
  27. M. Stynes, Steady-state convection-diffusion problems, Acta Numer. 14, 445–508, 2005.
  28. M. Stynes and E. O’Riordan, An analysis of a singularly perturbed two-point boundary value problem using only finite element techniques, Math. Comp. 56, 663–675, 1991.
  29. M. Stynes and E. O’Riordan, A uniformly convergent Galerkin method on a Shishkin mesh for a convection-diffusion problem, J. Math. Anal. Appl. 214, 36–54, 1997.
  30. G. Sun and M. Stynes, Finite element methods for singularly perturbed high order elliptic two-point boundary-value problems. II: Convection-diffusion type problems, IMA. J. of Numer. Anal. 15, 197–219, 1995.
  31. H.G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods For Singularly Perturbed Differential Equations, Verlag, Berlin: Springer, 2008.