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In this paper, we provide a theoretical analysis for a preconditioned steepest descent (PSD) iterative solver that improves the computational time of a finite difference numerical scheme for the Cahn-Hilliard equation with Flory-Huggins energy potential. In the numerical design, a convex splitting approach is applied to the chemical potential such that the logarithmic and the surface diffusion terms are treated implicitly while the expansive concave term is treated with an explicit update. The nonlinear and singular nature of the logarithmic energy potential makes the numerical implementation very challenging. However, the positivity-preserving property for the logarithmic arguments, unconditional energy stability, and optimal rate error estimates have been established in a recent work and it has been shown that successful solvers ensure a similar positivity-preserving property at each iteration stage. Therefore, in this work, we will show that the PSD solver ensures a positivity-preserving property at each iteration stage. The PSD solver consists of first computing a search direction (which requires solving a constant-coefficient Poisson-like equation) and then takes a one-parameter optimization step over the search direction in which the Newton iteration becomes very powerful. A theoretical analysis is applied to the PSD iteration solver and a geometric convergence rate is proved for the iteration. In particular, the strict separation property of the numerical solution, which indicates a uniform distance between the numerical solution and the singular limit values of ±1 for the phase variable, plays an essential role in the iteration convergence analysis. A few numerical results are presented to demonstrate the robustness and efficiency of the PSD solver. 

We present a C^0 interior penalty finite element method for the sixth-order modified phase field crystal equation. We demonstrate that the numerical scheme is uniquely solvable, unconditionally energy stable, and convergent. Additionally, the error analysis presented develops a detailed methodology for analyzing time dependent problems utilizing the Cinterior penalty method. We furthermore support the theory with several numerical experiments. 

Abstract Mother’s Curse alleles represent a significant source of potential male fitness defects. The maternal inheritance of mutations with the pattern of sex-specific fitness effects, s♀>0>s♂, allows Mother’s Curse alleles to spread through a population even though they reduce male fitness. Although the mitochondrial genomes of animals contain only a handful of protein-coding genes, mutations in many of these genes have been shown to have a direct effect on male fertility. The evolutionary process of nuclear compensation is hypothesized to counteract the male-limited mitochondrial defects that spread via Mother’s Curse. Here we use population genetic models to investigate the evolution of compensatory autosomal nuclear mutations that act to restore the loss of fitness caused by mitochondrial mutation pressures. We derive the rate of male fitness deterioration by Mother’s Curse and the rate of restoration by nuclear compensatory evolution. We find that the rate of nuclear gene compensation is many times slower than that of its deterioration by cytoplasmic mutation pressure, resulting in a significant lag in the recovery of male fitness. Thus, the numbers of nuclear genes capable of restoring male mitochondrial fitness defects must be large in order to sustain male fitness in the face of mutation pressures.