Core Concepts
Developing robust a posteriori error control for the Allen-Cahn equation with variable mobility.
Abstract
The study focuses on deriving a γ-robust a posteriori error estimator for finite element approximations of the Allen-Cahn equation with variable non-degenerate mobility. The estimator utilizes spectral estimates and conditional stability estimates based on Bregman distances. The paper extends results to the case of non-degenerate non-constant mobility, emphasizing the importance of robustness with respect to small values of γ. Standard error estimates exponentially depend on γ⁻¹, while this study aims for low-order polynomial behavior in γ⁻¹. Stability estimates are derived, and numerical methods are described along with suitable reconstructions of the numerical solution. The analysis is structured into sections covering basic definitions, stability, numerical methods, weak solutions, and more.
Stats
"The physical free energy is given by E(ϕ) = Z Ω 1/2|∇ϕ|² + 1/γ f(ϕ) dx."
"Thin phase transition layers make adaptive numerical schemes attractive."
"For continuous finite elements in the L∞(0, T, L2(Ω))-norm have been extended to estimates in the L∞(0, T, Lr(Ω)) norm with r ∈ [2, ∞)."
Quotes
"It is important that they are robust with respect to small values of γ."
"Our estimates are analogous to the results for constant mobility but with a slightly different eigenvalue."