Abstract
We consider fourth order singularly perturbed boundary value problems (BVPs) in one-dimension and the approximation of their solution by the hp version of the Finite Element Method (FEM). If the given problem’s boundary conditions are suitable for writing the BVP as a second order system, then we construct an hp FEM on the so-called Spectral Boundary Layer Mesh that gives a robust approximation that converges exponentially in the energy norm, provided the data of the problem is analytic. We also consider the case when the BVP is not written as a second order system and the approximation belongs to a finite dimensional subspace of the Sobolev space H 2. For this case we construct suitable C 1 −conforming hierarchical basis functions for the approximation and we again illustrate that the hp FEM on the Spectral Boundary Layer Mesh yields a robust approximation that converges exponentially. A numerical example that validates the theory is also presented.
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References
Bakhvalov, N.S.: Towards optimization of methods for solving boundary value problems in the presence of boundary layers. Zh. Vychisl. Mat. Mat. Fiz. 9, 841–859 (1969) (in Russian)
Roos, H.-G., Stynes, M.: A uniformly convergent discretization method for a fourth order singular perturbation problem. Bonn. Math. Schr. 228, 30–40 (1991)
Melenk, M.J., Xenophontos, C., Oberbroeckling, L.: Robust exponential convergence of hp-FEM for singularly perturbed systems of reaction-diffusion equations with multiple scales. Ima. J. Num. Anal., 1–20 (2012), doi:10.1093/imanum/drs013
Pan, Z.-X.: The difference and asymptotic methods for a fourth order equation with a small parameter. In: BAIL IV Proceedings, pp. 392–397. Book Press, Dublin (1986)
Panaseti, P., Zouvani, A., Madden, N., Xenophontos, C.: A C 1–conforming hp Finite Element Method for fourth order singularly perturbed boundary value problems. Submitted to App. Numer. Math. (2013)
Roos, H.-G.: A uniformly convergent discretization method for a singularly perturbed boundary value problem of the fourth order. In: Review of Research, Faculty of Science, Mathematics Series, Univ. Novi Sad, vol. 19, pp. 51–64 (1989)
Schwab, C.: p- and hp-Finite Element Methods. Oxford Science Publications (1998)
Schwab, C., Suri, M.: The p and hp versions of the finite element method for problems with boundary layers. Math. Comp. 65, 1403–1429 (1996)
Shishkin, G.I.: A difference scheme for an ordinary differential equation of the fourth order with a small parameter at the highest derivative. Differential Equations 21, 1743–1742 (1985) (in Russian)
Shishkin, G.I.: Grid approximation of singularly perturbed boundary value problems with a regular boundary layer. Sov. J. Numer. Anal. Math. Model. 4, 397–417 (1989)
Xenophontos, C., Oberbroeckling, L.: A numerical study on the finite element solution of singularly perturbed systems of reaction-diffusion problems. Appl. Meth. Comp. 187, 1351–1367 (2007)
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Xenophontos, C., Melenk, M., Madden, N., Oberbroeckling, L., Panaseti, P., Zouvani, A. (2013). hp Finite Element Methods for Fourth Order Singularly Perturbed Boundary Value Problems. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Numerical Analysis and Its Applications. NAA 2012. Lecture Notes in Computer Science, vol 8236. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41515-9_61
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DOI: https://doi.org/10.1007/978-3-642-41515-9_61
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