| Title:
|
Remarks on balanced norm error estimates for systems of reaction-diffusion equations (English) |
| Author:
|
Roos, Hans-Goerg |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
63 |
| Issue:
|
3 |
| Year:
|
2018 |
| Pages:
|
273-279 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
singular perturbation |
| Keyword:
|
finite element method |
| Keyword:
|
layer-adapted mesh |
| Keyword:
|
balanced norm |
| MSC:
|
65N30 |
| idZBL:
|
Zbl 06945733 |
| idMR:
|
MR3833661 |
| DOI:
|
10.21136/AM.2018.0063-18 |
| . |
| Date available:
|
2018-07-16T08:48:33Z |
| Last updated:
|
2020-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147311 |
| . |
| Reference:
|
[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. Zbl 0637.41034, MR 0878688, 10.2307/2007825 |
| Reference:
|
[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. MR 3706618, 10.1016/j.camwa.2017.03.015 |
| Reference:
|
[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. Zbl 1314.65141, MR 3252075, 10.1007/s10092-013-0093-5 |
| Reference:
|
[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. MR 3506572, 10.1016/j.camwa.2016.05.001 |
| Reference:
|
[5] Lin, R., Stynes, M.: A balanced finite element method for singularly perturbed reaction-diffusion problems.SIAM J. Numer. Anal. 50 (2012), 2729-2743. Zbl 1260.65103, MR 3022240, 10.1137/110837784 |
| Reference:
|
[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. Zbl 1333.65084, MR 3428676, 10.1007/s11075-015-9969-6 |
| Reference:
|
[7] Linß, T.: Analysis of a FEM for a coupled system of singularly perturbed reaction-diffusion equations.Numer. Algorithms 50 (2009), 283-291. Zbl 1163.65054, MR 2487239, 10.1007/s11075-008-9228-1 |
| Reference:
|
[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. Zbl 1336.65148, MR 3461383, 10.1007/s10092-015-0139-y |
| Reference:
|
[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. Zbl 1273.65180, MR 3066894, 10.1007/978-1-4614-4565-4_24 |
| Reference:
|
[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. MR 3520858, 10.18255/1818-1015-2016-3-357-363 |
| Reference:
|
[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. MR 3772487, 10.1007/978-3-319-67202-1_1 |
| Reference:
|
[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. Zbl 1326.65163, MR 3358551, 10.1002/zamm.201300226 |
| Reference:
|
[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). Zbl 1155.65087, MR 2454024, 10.1007/978-3-540-34467-4 |
| . |
