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Maximum-norm error analysis of a non-monotone FEM for a singularly perturbed reaction-diffusion problem. (English) Zbl 1221.65175
This paper is concerned with the numerical solution of singularly perturbed one-dimensional boundary value problems of the type \[ - \varepsilon^2 u'' + r u = f \text{ in } (0,1), u(0)= u(1)=0, \] where \( r,f \in {\mathcal C}^2[0,1]\), \( 0 << \varepsilon < 1\), \( r(x) > \rho^2 > 0 \) for \( x \in [0,1]\), whose solution possess two layers of width \({\mathcal O} ( \varepsilon \log \varepsilon^{-1} )\) at the endpoints of the domain. The author proposes an FE discretization in an arbitrary mesh \( ( x_j )_{j=0}^N \) in \([0,1]\), where the test function \( \varphi_i\) is the standard hat function associated to the \(i\)-th mesh node. It is found that this discretization leads to a finite difference scheme that – unlike the standard central difference schemes – is not inverse monotone. In this context the author proves that it is stable in the maximum norm, and for the layer adapted meshes proposed by Shishkin and Bakhvalov some convergence results are established. The paper ends with the numerical results obtained for two test problems with the above meshes.

65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
Full Text: DOI
[1] V. B. Andreev, On the uniform convergence of a classical difference scheme on an irregular grid for the one-dimensional singularly perturbed reaction-diffusion equation, Comput. Math. Math. Phys., 44(3) (2004), pp. 449–464. · Zbl 1114.65097
[2] N. S. Bakhvalov, Towards optimization of methods for solving boundary value problems in the presence of boundary layers, Zh. Vychisl. Mat. Mat. Fiz., 9 (1969), pp. 841–859. In Russian.
[3] N. V. Kopteva, Maximum norm a posteriori error estimates for a one-dimensional singularly perturbed semilinear reaction-diffusion problem, IMA J. Numer. Anal., in press. · Zbl 1149.65066
[4] T. Linß, Sufficient conditions for uniform convergence on layer-adapted meshes for one-dimensional reaction-diffusion problems, Numer. Algorithms, 40(1) (2005), pp. 23–32. · Zbl 1082.65077
[5] T. Linß and N. Madden, A finite element analysis of a coupled system of singularly perturbed reaction-diffusion equations, Appl. Math. Comput., 148(4) (2004), pp. 869–880. · Zbl 1042.65065
[6] J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems. Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific, Singapore, 1996. · Zbl 0915.65097
[7] E. O’Riordan and M. Stynes, A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion problem, Math. Comput., 47 (1986), pp. 555–570. · Zbl 0625.65073
[8] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1967. · Zbl 0153.13602
[9] H.-G. Roos, M. Stynes, and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, vol. 24, Springer Series in Computational Mathematics. Springer, Berlin, 1996. · Zbl 0844.65075
[10] G. I. Shishkin, Grid Approximation of Singularly Perturbed Elliptic and Parabolic Equations, Second doctorial thesis, Keldysh Institute, Moscow, 1990. In Russian.
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