Spectral analysis and sine transform based preconditioning for a structure preserving stabilized scheme approximating the space-fractional Allen Cahn equation with logarithmic potential
For researchers solving fractional Allen-Cahn equations, this provides preconditioners that may accelerate iterative solvers, but the work is incremental.
The paper analyzes spectral properties of matrices from a stabilized scheme for the space-fractional Allen-Cahn equation with logarithmic potential, and designs sine-transform-based preconditioners. Numerical tests show improved convergence.
We consider an initial boundary value problem of the space fractional Allen-Cahn equation with logarithmic Flory-Huggins potential. As an approximation technique, first-order weighted and shifted Grunwald difference formulae of the left and right fractional derivatives are used. The main focus of the present work is to study the spectral features of the underlying matrices and matrix sequences and to design proper preconditioners based on the spectral information. Then a computational and spectral analysis of the resulting preconditioned matrix sequences is performed. Numerical evidence and a short list of open problems complete the current study.