Accurate and approximate analytic solutions of singularly perturbed differential equations with two-dimensional boundary layers

Authors:
Zi-Cai Li;Heng-Shuing Tsai;Song Wang;John J. H. Miller
Affiliations:
Department of Applied Mathematics, Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan;Department of Applied Mathematics, Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan;School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Hwy, Crawley, WA6009, Australia;Department of Mathematics, Trinity College, Dublin, Ireland
Venue:
Computers & Mathematics with Applications
Year:
2008

Citing 3
Cited 0

An analysis of a conforming exponentially fitted finite element method for a convection-diffusion problem

Journal of Computational and Applied Mathematics
A nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection-diffusion equation

Mathematics of Computation
Particular solutions of singularly perturbed partial differential equations with constant coefficients in rectangular domains, part I. convergence analysis

Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on boundary and interior layers - computational and asymptotic methods (BAIL 2002)

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Abstract

In this paper we construct three new test problems, called Models A, B and C, whose solutions have two-dimensional boundary layers. Approximate analytic solutions are found for these problems, which converge rapidly as the number of terms in their expansion increases. The approximations are valid for @e=10^-^8 in practical computations. Surprisingly, the algorithm for Model A can be carried out even for @e-~. Model C has a simple exact solution. These three new accurate and approximate analytic solutions with two-dimensional boundary layers may be more useful for testing numerical methods than those in [Z.C. Li, H.Y. Hu, C.H. Hsu, S. Wang, Particular solutions of singularly perturbed partial differential equations with constant coefficients in rectangular domains, I. Convergence analysis, J. Comput. Appl. Math. 166 (2004) 181-208] in the sense that the series solutions from the former converge much faster than those of the latter when @e is small.