### AN hp-FEM FOR SINGULARLY PERTURBED TRANSMISSION PROBLEMS

Serge Nicaise1, Christos Xenophontos2

1. 1. Université de Valenciennes et du Hainaut Cambrésis LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes F-59313-Valenciennes Cedex 9 France;
2. Department of Mathematics and Statistics University of Cyprus, P. O. Box 20537 Nicosia 1678, Cyprus
• Received:2015-01-13 Revised:2016-07-25 Online:2017-03-15 Published:2017-03-15
• Contact: Christos Xenophontos

Serge Nicaise, Christos Xenophontos. AN hp-FEM FOR SINGULARLY PERTURBED TRANSMISSION PROBLEMS[J]. Journal of Computational Mathematics, 2017, 35(2): 152-168.

We perform the analysis of the hp finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In[12] it was shown that this method yields robust exponential convergence, as the degree p of the approximating polynomials is increased, when the error is measured in the energy norm associated with the boundary value problem. In the present article we sharpen the result by showing that the hp-Finite Element Method (FEM) on Spectral Boundary Layer Meshes leads to robust exponential convergence in a stronger, more balanced norm. Several numerical results illustrating and extending the theory are also presented.

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 [1] Feng Kang, Differential vs. integral equations and finite vs. infinite elements, Math. Numer. Sinica, 2:1(1980), 100-105.[2] N.S. Bakhvalov, Towards optimization of methods for solving boundary value problems in the presence of boundary layers, (in Russian), Zh. Vychisl. Mat. Mat. Fiz., 9(1969), 841-859.[3] R. Lin and M. Stynes, A balanced finite element method for singularly perturbed reaction-diffusion problems, SIAM J. Numer. Anal., 50:5(2012),2729-2743.[4] A. Maghnouji and S. Nicaise, Boundary layers for transmission problems with singularities, Electronic J. Diff. Eqs, 2006:14(2006), 1-16.[5] J.M. Melenk, On the robust exponential convergence of hp finite element methods for problems with boundary layers, IMA J. Num. Anal., 17(1997), 577-601.[6] J.M. Melenk, hp-Finite Element Methods for Singular Perturbations, Vol. 1796 of Springer Lecture Notes in Mathematics, Springer Verlag, 2002.[7] M.J. Melenk, C. Xenophontos and L. Oberbroeckling, Robust exponential convergence of hp-FEM for singularly perturbed systems of reaction-diffusion equations with multiple scales, IMA J. Num. Anal., 33:2(2013), 609-628.[8] M.J. Melenk, C. Xenophontos and L. Oberbroeckling, Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales, Advances in Computational Mathematics, 39(2013), 367-394.[9] J.M. Melenk and C. Schwab, hp FEM for reaction diffusion equations I:Robust exponential convergence, SIAM J. Num. Anal., 35(1998), 1520-1557.[10] J.M. Melenk and C. Xenophontos, Robust exponential convergence of hp-FEM in balanced norms for singularly perturbed reaction-diffusion equations, Calcolo, 35(2016), 105-132.[11] J.J. Miller, E. O'Riordan and G.I. Shishkin, Fitted Numerical Methods For Singular Perturbation Problems, World Scientific, 1996.[12] S. Nicaise and C. Xenophontos, Finite element methods for a singularly perturbed transmission problem, J. Num. Math., 17, (2009), 245-275.[13] S. Nicaise and C. Xenophontos, Robust approximation of singularly perturbed delay differential equations by the hp finite element method, Comp. Meth. Appl. Math. 13(2013), 21-37.[14] S. Nicaise and C. Xenophontos, Convergence analysis of an hp Finite Element Method for singularly perturbed transmission problems in smooth domains, Num. Meth. PDEs, 39(2013), 367-394.[15] H.G. Roos and S. Franz, Error estimation in a balanced norm for a convection-diffusion problems with two different boundary layers, Calcolo, 51(2014), 423-440.[16] H.G. Roos and M. Schopf, Convergence and stability in balanced norms of finite element methods on Shishkin meshes for reaction-diffusion problems, ZAMM, 95(2016), 551-565.[17] H.G. Roos, M. Stynes and L. Tobiska, Robust numerical methods for singularly perturbed differential equations, Volume 24 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2008.[18] C. Schwab, p/hp Finite Element Methods, Oxford University Press, 1998.[19] C. Schwab and M. Suri, The p and hp versions of the finite element method for problems with boundary layers, Math. Comp. 65(1996), 1403-1429.[20] G.I. Shishkin, Grid approximation of singularly perturbed boundary value problems with a regular boundary layer, Sov. J. Numer. Anal. Math. Model., 4(1989), 397-417.
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