Stability Analysis and Best Approximation Error Estimates of Discontinuous Time-Stepping Schemes for the Allen-Cahn Equation
Provides rigorous theoretical guarantees for high-order time-stepping methods for phase-field models, benefiting numerical analysts and computational scientists.
This paper proves unconditional stability and best approximation error estimates for arbitrary-order discontinuous time-stepping schemes for the Allen-Cahn equation, with constants depending polynomially on 1/ε, avoiding Grönwall arguments.
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 best approximation a-priori error estimates, with constants depending polynomially ypon $(1/ε)$ by circumventing Grönwall Lemma arguments. We also prove that these schemes are unconditionally stable under minimal regularity assumptions on the given data. The key feature of our approach is an appropriate duality argument, combined with a boot-strap technique.