zbMATH — the first resource for mathematics

A posteriori error estimation for arbitrary order FEM applied to singularly perturbed one-dimensional reaction-diffusion problems. (English) Zbl 1340.65163
Extending his earlier work [BIT 47, No. 2, 379–391 (2007; Zbl 1221.65175)] on the case \(r=1\), the author establishes an a posteriori bound on the \(\infty\)-norm of the error in an \(r\)th order finite element method (FEM) approximation to the solution, \(u\), of \(-\varepsilon ^2u''+cu=f\) in \((0,1)\), satisfying \(u(0)=u(1)=0\). Here \(c\) and \(f\) are piecewise continuous real-valued functions with \(c\geq \gamma ^2\) where \(\varepsilon(\ll 1)\) and \(\gamma\) are positive constants.

65L70 Error bounds for numerical methods for ordinary differential equations
65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations
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
34B05 Linear boundary value problems for ordinary differential equations
Full Text: DOI
[1] Bakhvalov, N. S., On the optimization of the methods for solving boundary value problems in the presence of a boundary layer, Zh. Vychisl. Mat. Mat. Fiz., 9, 841-859, (1969) · Zbl 0208.19103
[2] Chadha, N. M.; Kopteva, N., A robust grid equidistribution method for a one-dimensional singularly perturbed semilinear reaction-diffusion problem, IMA J. Numer. Anal., 31, 188-211, (2011) · Zbl 1211.65099
[3] Boor, C.; Meir, A. (ed.); etal., Good approximation by splines with variable knots. spline functions and approximation theory, No. 21, 57-72, (1973), Basel
[4] Demlow, A.; Lakkis, O.; Makridakis, C., A posteriori error estimates in the maximum norm for parabolic problems, SIAM J. Numer. Anal., 47, 2157-2176, (2009) · Zbl 1196.65153
[5] Kopteva, N., Maximum norm a posteriori error estimates for a 1D singularly perturbed semilinear reaction-diffusion problem, IMA J. Numer. Anal., 27, 576-592, (2007) · Zbl 1149.65066
[6] Kopteva, N.; Linß, T., Maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions, SIAM J. Numer. Anal., 51, 1494-1524, (2013) · Zbl 1281.65121
[7] Kopteva, N.; Linß, T.; Dimov, I. (ed.); etal., Numerical study of maximum norm a posteriori error estimates for singularly perturbed parabolic problems, No. 8236, 50-61, (2013), Berlin · Zbl 1352.65354
[8] Kopteva, N.; Stynes, M., A robust adaptive method for a quasi-linear one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39, 1446-1467, (2001) · Zbl 1012.65076
[9] Kunert, G., A note on the energy norm for a singularly perturbed model problem, Computing, 69, 265-272, (2002) · Zbl 1239.65055
[10] T. Linß: Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems. Lecture Notes in Mathematics 1985, Springer, Berlin, 2010. · Zbl 1202.65120
[11] Linß, T., Maximum-norm error analysis of a non-monotone FEM for a singularly perturbed reaction-diffusion problem, BIT, 47, 379-391, (2007) · Zbl 1221.65175
[12] J. M. Melenk: hp-Finite Element Methods for Singular Perturbations. Lecture Notes in Mathematics 1796, Springer, Berlin, 2002. · Zbl 1021.65055
[13] Nochetto, R. H.; Schmidt, A.; Siebert, K.G.; Veeser, A., Pointwise a posteriori error estimates for monotone semi-linear equations, Numer. Math., 104, 515-538, (2006) · Zbl 1104.65107
[14] H.-G. Roos, M. Schopf: Convergence and stability in balanced norms of finite element methods on Shishkin meshes for reaction-diffusion problems. Z. Angew. Math. Mech., in press. · Zbl 1326.65163
[15] Roos, H.-G.; Stynes, M.; Tobiska, L., Robust numerical methods for singularly perturbed differential equations, No. 24, (2008), Berlin
[16] G. I. Shishkin: Discrete Approximation of Singularly Perturbed Elliptic and Parabolic Equations. Russian Academy of Sciences, Ural Section, Ekaterinburg, 1992. (In Russian.)
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.