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Journals / Hacettepe Journal of Mathematics and Statistics / 2021 / Cilt: 50 - Sayı: 5

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

Ali ŞENDUR
Srinivasan NATESAN
Gautam SINGH
Pages
1306–1324
Publication Date
2021-10-15
DOI
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AI Summary & Key Takeaways

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⚡ TL;DR(Too Long; Didn't Read)

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 Key Takeaways

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Objective & Hypothesis: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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Key Findings & Data: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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Conclusion & Impact: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.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables: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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Cite This Article

ŞENDUR, A., NATESAN, S., SINGH, G. (2021). Error estimates for a fully discrete $\varepsilon-$uniform finite element method on quasi uniform meshes. Hacettepe Journal of Mathematics and Statistics, Cilt: 50 - Sayı: 5, 1306–1324. https://ajindex.com/articles/669261

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References (14)

  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. Multis