# Article

Full entry | PDF   (0.2 MB)
Keywords:
singular perturbation; finite element method; layer-adapted mesh; balanced norm
Summary:
Error estimates of finite element methods for reaction-diffusion problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the $H^1$ seminorm leads to a balanced norm which reflects the layer behavior correctly. We discuss the difficulties which arise for systems of reaction-diffusion problems.
References:
[1] Crouzeix, M., Thomée, V.: The stability in $L_p$ and $W_p^1$ of the $L_2$-projection onto finite element function spaces. Math. Comput. 48 (1987), 521-532. DOI 10.2307/2007825 | MR 0878688 | Zbl 0637.41034
[2] Faustmann, M., Melenk, J. M.: Robust exponential convergence of $hp$-FEM in balanced norms for singularly perturbed reaction-diffusion problems: corner domains. Comput. Math. Appl. 74 (2017), 1576-1589. DOI 10.1016/j.camwa.2017.03.015 | MR 3706618
[3] Franz, S., Roos, H.-G.: Error estimation in a balanced norm for a convection-diffusion problem with two different boundary layers. Calcolo 51 (2014), 423-440. DOI 10.1007/s10092-013-0093-5 | MR 3252075 | Zbl 1314.65141
[4] Franz, S., Roos, H.-G.: Robust error estimation in energy and balanced norms for singularly perturbed fourth order problems. Comput. Math. Appl. 72 (2016), 233-247. DOI 10.1016/j.camwa.2016.05.001 | MR 3506572
[5] Lin, R., Stynes, M.: A balanced finite element method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. 50 (2012), 2729-2743. DOI 10.1137/110837784 | MR 3022240 | Zbl 1260.65103
[6] Lin, R., Stynes, M.: A balanced finite element method for a system of singularly perturbed reaction-diffusion two-point boundary value problems. Numer. Algorithms 70 (2015), 691-707. DOI 10.1007/s11075-015-9969-6 | MR 3428676 | Zbl 1333.65084
[7] Linß, T.: Analysis of a FEM for a coupled system of singularly perturbed reaction-diffusion equations. Numer. Algorithms 50 (2009), 283-291. DOI 10.1007/s11075-008-9228-1 | MR 2487239 | Zbl 1163.65054
[8] Melenk, J. M., Xenophontos, C.: Robust exponential convergence of $hp$-FEM in balanced norms for singularly perturbed reaction-diffusion equations. Calcolo 53 (2016), 105-132. DOI 10.1007/s10092-015-0139-y | MR 3461383 | Zbl 1336.65148
[9] Oswald, P.: $L_\infty$-bounds for the $L_2$-projection onto linear spline spaces. Recent Advances in Harmonic Analysis and Applications D. Bilyk et al. Springer Proc. Math. Stat. 25, Springer, New York (2013), 303-316. DOI 10.1007/978-1-4614-4565-4_24 | MR 3066894 | Zbl 1273.65180
[10] Roos, H.-G.: Error estimates in balanced norms of finite element methods on Shishkin meshes for reaction-diffusion problems. Model. Anal. Inf. Sist. 23 (2016), 357-363. DOI 10.18255/1818-1015-2016-3-357-363 | MR 3520858
[11] Roos, H.-G.: Error estimates in balanced norms of finite element methods on layer-adapted meshes for second order reaction-diffusion problems. Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2016 Z. Huang et al. Lecture Notes in Computational Science and Engineering 120, Springer, Cham (2017), 1-18. DOI 10.1007/978-3-319-67202-1_1 | MR 3772487
[12] Roos, H.-G., Schopf, M.: Convergence and stability in balanced norms for finite element methods on Shishkin meshes for reaction-diffusion problems. ZAMM, Z. Angew. Math. Mech. 95 (2015), 551-565. DOI 10.1002/zamm.201300226 | MR 3358551 | Zbl 1326.65163
[13] Roos, H.-G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion-Reaction and Flow Problems. Springer Series in Computational Mathematics 24, Springer, Berlin (2008). DOI 10.1007/978-3-540-34467-4 | MR 2454024 | Zbl 1155.65087

Partner of