A Newton--Krylov--AMG Method For An Adaptive Space--Time Finite Element Discretization Of The Allen--Cahn Equation In 3D And 4D

Dr. Huidong Yang

June 5, 2018, 3:30 p.m. S2 416-1

The aim of this work is twofold. On the one hand, in contrast to conventional time stepping methods, we provide a promising alternative to numerical simulations of the Allen--Cahn equation, which is based on a space--time finite element discretization in 3D and 4D space--time domains. In particular, we use a Galerkin--Petrov space--time finite element method, which employs piecewise linear finite elements simultaneously in space and time on admissible simplex meshes, composed of tetrahedra and pentachora in 3D and 4D, respectively. Further, an adaptive space--time finite element method is developed
using an explicit residual--type error indicator. On the other hand, combined outer--inner Newton--GMRES iterations, along with a proper algebraic multigrid preconditioner,
for solving the discrete nonlinear algebraic equations are discussed. We mainly focus on performance comparison of the solution method for the nonlinear system on both uniform and adaptive refinements. Several numerical examples in 3D and 4D are performed, which demonstrate applicability of the proposed method.