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Journal of Scientific Computing

Erschienen in:

01.10.2020

Optimal Order of Uniform Convergence for Finite Element Method on Bakhvalov-Type Meshes

verfasst von: Jin Zhang, Xiaowei Liu

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2020

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Abstract

We propose a new analysis of convergence for a kth order (\(k\ge 1\)) finite element method, which is applied on Bakhvalov-type meshes to a singularly perturbed two-point boundary value problem. A novel interpolant is introduced, which has a simple structure and is easy to generalize. By means of this interpolant, we prove an optimal order of uniform convergence with respect to the perturbation parameter. Numerical experiments illustrate these theoretical results.

Vorheriger Artikel Strong Stability Preserving Second Derivative General Linear Methods with Runge–Kutta Stability

Nächster Artikel Admissible Concentration Factors for Edge Detection from Non-uniform Fourier Data

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Bahvalov, N.S.: On the optimization of the methods for solving boundary value problems in the presence of a boundary layer. Zh. Vychisl. Mat. Mat. Fiz. 9, 841–859 (1969)MathSciNet

Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)MathSciNetCrossRef

Brdar, M., Zarin, H.: A singularly perturbed problem with two parameters on a Bakhvalov-type mesh. J. Comput. Appl. Math. 292, 307–319 (2016). https://doi.org/10.1016/j.cam.2015.07.011MathSciNetCrossRefMATH

Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2008). https://doi.org/10.1007/978-0-387-75934-0CrossRef

The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, vol. 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). https://doi.org/10.1137/1.9780898719208

Kopteva, N.: On the convergence, uniform with respect to the small parameter, of a scheme with central difference on refined grids. Zh. Vychisl. Mat. Mat. Fiz. 39(10), 1662–1678 (1999)MathSciNet

Kopteva, N., Savescu, S.B.: Pointwise error estimates for a singularly perturbed time-dependent semilinear reaction-diffusion problem. IMA J. Numer. Anal. 31(2), 616–639 (2011). https://doi.org/10.1093/imanum/drp032MathSciNetCrossRefMATH

Linß, T.: Solution decompositions for linear convection–diffusion problems. Z. Anal. Anwendungen 21(1), 209–214 (2002). https://doi.org/10.4171/ZAA/1073MathSciNetCrossRefMATH

Linß, T.: Layer-Adapted Meshes for Reaction–Convection–Diffusion problems. Lecture Notes in Mathematics, vol. 1985. Springer, Berlin (2010)

10.

Liu, X., Stynes, M., Zhang, J.: Supercloseness of edge stabilization on Shishkin rectangular meshes for convection-diffusion problems with exponential layers. IMA J. Numer. Anal. 38(4), 2105–2122 (2018). https://doi.org/10.1093/imanum/drx055MathSciNetCrossRefMATH

11.

Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Singular Perturbation Problems. In: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, revised edn. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2012). https://doi.org/10.1142/9789814390743

12.

Nhan, T.A., Vulanovic, R.: The Bakhvalov mesh: a complete finite-difference analysis of two-dimensional singularly perturbed convection-diffusion problems. Numer. Algorithms (To appear) (2020)

13.

Roos, H., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. In: Springer Series in Computational Mathematics, vol. 24, 2nd edn. Springer, Berlin (2008)

14.

Roos, H.G.: Error estimates for linear finite elements on Bakhvalov-type meshes. Appl. Math. 51(1), 63–72 (2006). https://doi.org/10.1007/s10492-006-0005-yMathSciNetCrossRefMATH

15.

Roos, H.G., Stynes, M.: Some open questions in the numerical analysis of singularly perturbed differential equations. Comput. Methods Appl. Math. 15(4), 531–550 (2015). https://doi.org/10.1515/cmam-2015-0011MathSciNetCrossRefMATH

16.

Russell, S., Stynes, M.: Balanced-norm error estimates for sparse grid finite element methods applied to singularly perturbed reaction–diffusion problems. J. Numer. Math. 27(1), 37–55 (2019). https://doi.org/10.1515/jnma-2017-0079MathSciNetCrossRefMATH

17.

Shishkin, G.I.: Grid approximation of singularly perturbed elliptic and parabolic equations (in Russian). 2nd doctoral thesis, Keldysh Institute, Moscow (1990)

18.

Stynes, M., O’Riordan, E.: A uniformly convergent Galerkin method on a Shishkin mesh for a convection–diffusion problem. J. Math. Anal. Appl. 214(1), 36–54 (1997)MathSciNetCrossRef

19.

Teofanov, L., Brdar, M., Franz, S., Zarin, H.: SDFEM for an elliptic singularly perturbed problem with two parameters. Calcolo 55(4), 20 (2018). https://doi.org/10.1007/s10092-018-0293-0MathSciNetCrossRefMATH

20.

Zhang, J., Liu, X.: Superconvergence of finite element method for singularly perturbed convection-diffusion equations in 1D. Appl. Math. Lett. 98, 278–283 (2019). https://doi.org/10.1016/j.aml.2019.06.018MathSciNetCrossRefMATH

21.

Zhang, J., Liu, X., Yang, M.: Optimal order \(L^2\) error estimate of SDFEM on Shishkin triangular meshes for singularly perturbed convection-diffusion equations. SIAM J. Numer. Anal. 54(4), 2060–2080 (2016). https://doi.org/10.1137/15M101035XMathSciNetCrossRefMATH

22.

Zhang, J., Stynes, M.: Supercloseness of continuous interior penalty method for convection-diffusion problems with characteristic layers. Comput. Methods Appl. Mech. Eng. 319, 549–566 (2017). https://doi.org/10.1016/j.cma.2017.03.013MathSciNetCrossRefMATH

Titel: Optimal Order of Uniform Convergence for Finite Element Method on Bakhvalov-Type Meshes
verfasst von: Jin Zhang
Xiaowei Liu
Publikationsdatum: 01.10.2020
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 1/2020
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-020-01312-y

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