Analysis of symmetric interior penalty discontinuous Galerkin methods for the Allen-Cahn equation and the mean curvature flow. (English) Zbl 1328.65205
The authors propose two new fully discrete interior penalty discontinuous Galerkin methods for the 2D and 3D Allen-Cahn equation
\[
u_t-\Delta u+\frac{1}{\epsilon^2}f(u)=0,
\]
where \(f(u)=u(u^2-1)\). In this setting, the authors discuss a fully implicit and energy-splitting time stepping scheme. Sharp error bounds depending on \(\epsilon\) are deduced. Further, convergence of the zero level sets for the fully discrete interior penalty discontinuous Galerkin solutions for the classical and generalized mean curvature flow is investigated. Numerical experiments are also presented to support the theoretical findings.
Reviewer: Marius Ghergu (Dublin)
MSC:
65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
35Q35 | PDEs in connection with fluid mechanics |
76T99 | Multiphase and multicomponent flows |
65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |