We consider formal matched asymptotics to show the convergence of a degenerate area preserving surface Allen-Cahn equation to its sharp interface limit of area preserving geodesic curvature flow. The degeneracy results from a surface de Gennes-Cahn-Hilliard energy and turns out to be essential to numerically resolve the dependency of the solution on geometric properties of the surface. We experimentally demonstrate convergence of the numerical algorithm, which considers a graph formulation, adaptive finite elements and a semi-implicit discretization in time, and uses numerical solutions of the sharp interface limit, also considered in a graph formulation, as benchmark solutions. The results provide the mathematical basis to explore the impact of curvature on cells and their collective behaviour. This is essential to understand the physical processes underlying morphogenesis.

Phase transition problems on curved surfaces can lead to a panopticon of fascinating patterns. In this paper we consider finite element approximations of phase field models with a spatially inhomogeneous and anisotropic surface energy density. The problems are either posed in $\mathbb{R}^3$ or on a two-dimensional hypersurface in $\mathbb{R}^3.$ In the latter case, a fundamental choice regarding the anisotropic energy density has to be made. One possibility is to use a density defined in the ambient space $\mathbb{R}^3.$ However, we propose and advocate for an alternative, where a density is defined on a fixed chosen tangent space, and is then moved along geodesics to the other tangent spaces. Our numerical method can be employed in all of the above situations, where for the problems on hypersurfaces the algorithm uses parametric finite elements. We prove an unconditional stability result for our schemes and present several numerical experiments, including for the modelling of ice crystal growth on a sphere.

In this paper, we develop a new linear, fully-decoupled, unconditional energy-stable BDF2-SAV-FEM scheme for solving the smectic-A liquid crystals, based on the finite element method (FEM) for spatial discretization and two-step backward differentiation formula (BDF2) for temporal discretization. To decouple the computations of the layer function and velocity field, we introduce an additional stabilization term into the constitutive equation. The nonlinear energy potential and the Navier-Stokes equations are treated by the scalar auxiliary variable (SAV) method and the rotational pressure-correction method, respectively. The unique solvability, unconditional energy stability, and error estimations of the proposed numerical scheme have been demonstrated. Several numerical experiments are carried out to validate our theoretical analysis.

The weighted and shifted seven-step BDF method is proposed by the authors [Akrivis, Chen, and Yu, IMA. Numer. Anal., DOI:10.1093/imanum/drae089] for parabolic equation on uniform meshes. In this paper, we study the weighted and shifted two-step BDF method (WSBDF2) for the Allen-Cahn equation on variable grids. In order to preserve a modified energy dissipation law at the discrete level, a novel technique is designed to deal with the nonlinear term. The stability and convergence analysis of the WSBDF2 method are rigorously proved by the energy method under the adjacent time-step ratios $r_s ≥ 4.8645.$ Finally, numerical experiments are implemented to illustrate the theoretical results. The proposed approach is applicable for the Cahn-Hilliard equation.

Diffuse domain methods (DDMs) have garnered significant attention for approximating solutions to partial differential equations on complex geometries. These methods implicitly represent the geometry by replacing the sharp boundary interface with a diffuse layer of thickness $ε,$ which scales with the minimum grid size. This approach reformulates the original equations on an extended regular domain, incorporating boundary conditions through singular source terms. In this work, we conduct a matched asymptotic analysis of a DDM for a two-sided problem with transmission-type Robin boundary conditions. Our results show that, in the one dimensional space, the solution of the diffuse domain approximation asymptotically converges to the solution of the original problem, with exactly first-order accuracy in $ε.$ Furthermore, we provide numerical simulations that validate and illustrate the analytical result.

Diffuse domain methods (DDMs) have gained significant attention for solving partial differential equations (PDEs) on complex geometries. These methods approximate the domain by replacing sharp boundaries with a diffuse layer of thickness $ε,$ which scales with the minimum grid size. This reformulation extends the problem to a regular domain, incorporating boundary conditions via singular source terms. In this work, we analyze the convergence of a DDM approximation problem with transmission-type Neumann boundary conditions. We prove that the energy functional of the diffuse domain problem $Γ$-converges to the energy functional of the original problem as $ε → 0.$ Additionally, we show that the solution of the diffuse domain problem strongly converges in $H^1 (Ω),$ up to a subsequence, to the solution of the original problem, as $ε → 0.$

The degenerate de Gennes-Cahn-Hilliard (dGCH) equation is a recent phase field model that may more accurately approximate surface diffusion. After establishing the Gamma convergence of the dGCH energy in [10], in this paper, we study the convergence of boundary force. This is done by carefully crafting a nonlinear transformation that transforms the dGCH energy into a Cahn-Hilliard-type energy with a non-smooth potential. We carry out explicit computations and analysis to this new system, which in turn enables us to establish the convergence of boundary force for the dGCH energy.