no. 2
Stability analysis and best approximation error estimates of discontinuous time-stepping schemes for the Allen–Cahn equation
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 2, pp. 551-583

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.

DOI : 10.1051/m2an/2018071
Classification : Primary 65M12, 65M60
Keywords: Allen–Cahn equations, best approximation error estimates, discontinuous time-stepping schemes

Chrysafinos, Konstantinos 1

1 
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Chrysafinos, Konstantinos. Stability analysis and best approximation error estimates of discontinuous time-stepping schemes for the Allen–Cahn equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 2, pp. 551-583. doi: 10.1051/m2an/2018071

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