| Issue |
ESAIM: M2AN
Volume 53, Number 2, March-April 2019
|
|
|---|---|---|
| Page(s) | 551 - 583 | |
| DOI | https://doi.org/10.1051/m2an/2018071 | |
| Published online | 24 April 2019 | |
Stability analysis and best approximation error estimates of discontinuous time-stepping schemes for the Allen–Cahn equation
Department of Mathematics, School of Applied Mathematics and Physical Sciences, National Technical University of Athens Zografou Campus 15780 Athens Greece
* Corresponding author: chrysafinos@math.ntua.gr
Received:
8
May
2018
Accepted:
14
November
2018
Fully-discrete approximations of the Allen–Cahn equation are considered. In particular, we consider schemes of arbitrary order based on a discontinuous Galerkin (in time) approach combined with standard conforming finite elements (in space). We prove that these schemes are unconditionally stable under minimal regularity assumptions on the given data. We also prove best approximation a-priori error estimates, with constants depending polynomially upon (1/ε) by circumventing Gronwall Lemma arguments. The key feature of our approach is a carefully constructed duality argument, combined with a boot-strap technique.
Mathematics Subject Classification: Primary 65M12 / 65M60
Key words: Allen–Cahn equations / best approximation error estimates / discontinuous time-stepping schemes
© EDP Sciences, SMAI 2019
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