zbMATH — the first resource for mathematics

A finite element analysis of a coupled system of singularly perturbed reaction-diffusion equations. (English) Zbl 1042.65065
This paper deals with the finite element method for two coupled singularly perturbed reaction-diffusion problems on general meshes. A general criterion that guarantees parameter uniform convergence in the energy norm is presented. The authors give interpolation and approximation error bounds for arbitrary meshes. The paper concludes with supporting numerical results.

65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34E15 Singular perturbations, general theory for ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
Full Text: DOI
[1] Bakhvalov, N.S, Towards optimization of methods for solving boundary value problems in the presence of boundary layers, Zh. vychisl. mat. i mat. fiz., 9, 841-859, (1969), (in Russian)
[2] C. de Boor, Good approximation by splines with variable knots, in: A. Meir, A. Sharma (Eds.), Spline Functions Approx. Theory, Proc. Sympos. Univ. Alberta, Edmonton 1972, Birkhäuser, Basel and Stuttgart, 1973, pp. 57-72
[3] Linß, T, The necessity of shishkin-decompositions, Appl. math. lett., 14, 891-896, (2001) · Zbl 0986.65071
[4] Linß, T, Sufficient conditions for uniform convergence on layer-adapted grids, Appl. numer. anal., 37, 1-2, 241-255, (2001) · Zbl 0976.65084
[5] N. Madden, M. Stynes, A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction – diffusion problems, Preprint Nr. 1 (2002), School of Mathematics, Applied Mathematics and Statistics, National University of Ireland, Cork, submitted for publication · Zbl 1048.65076
[6] Matthews, S; Miller, J.J.H; O’Riordan, E; Shishkin, G.I, A parameter robust numerical method for a system of singularly perturbed ordinary differential equations, (), 219-224
[7] S. Matthews, E. O’Riordan, G.I. Shishkin, Numerical methods for a system of singularly perturbed reaction – diffusion equations, Preprint MS-00-06, Dublin City University, 2000
[8] Matthews, S; O’Riordan, E; Shishkin, G.I, A numerical method for a system of singularly perturbed reaction – diffusion equations, J. comput. appl. math., 145, 151-166, (2002) · Zbl 1004.65079
[9] Miller, J.J.H; O’Riordan, E; Shishkin, G.I, Fitted numerical methods for singular perturbation problems. error estimates in the maximum norm for linear problems in one and two dimensions, (1996), World Scientific Singapore · Zbl 0915.65097
[10] G.I. Shishkin, Grid Approximation of Singularly Perturbed Elliptic and Parabolic Equations, Second doctorial thesis, Keldysh Institute, Moscow, 1990 (in Russian)
[11] Shishkin, G.I, Mesh approximation of singularly perturbed boundary-value problems for systems of elliptic and parabolic equations, Comput. math. math. phys., 35, 4, 429-446, (1995) · Zbl 0852.65071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.