A fast Galerkin finite element method for a space-time fractional Allen-Cahn equation

Abstract In this paper, a space–time fractional Allen–Cahn equation is investigated to account for the memory effect of certain materials or the anomalous diffusion processes in heterogeneous distributed media for the mixture of two immiscible phases. Due to the nonlocal features of fractional derivatives, conventional solvers such as the Gauss elimination method require O ( M 2 + M N ) memory units and O ( M N 2 + M 3 N ) operations to solve the space–time fractional Allen–Cahn equation, where M is the number of spatial unknowns and N is the number of time steps. By exploring the special structure of the stiffness matrix and utilizing the “Invariant Energy Quadratization” approach, we developed an efficient linearized divide-and-conquer Galerkin finite element method without resorting to any lossy compression. This new method is efficient as it significantly reduces the memory requirement to O ( M N ) and the operation count to O ( M N ( log 2 N + log M ) ) . We have developed a fast Galerkin finite element method with local refinement that allows accurate capture of the coarsening dynamics at the interface. Furthermore, we have investigated the modeling capacities and energy dissipation properties of the space–time fractional Allen–Cahn equation. Numerical experiments are presented to demonstrate the efficiency of the new methods and the flexibility of turnable sharpness and decay behavior of the space–time fractional Allen–Cahn equation.

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