We propose an energy-stable parametric finite element method (ES-PFEM) to discretize the motion of a closed curve under surface diffusion with an anisotropic surface energy $γ(θ)$ -- anisotropic surface diffusion -- in two dimensions, while $θ$ is the angle between the outward unit normal vector and the vertical axis. By introducing a positive definite surface energy (density) matrix $G(θ)$, we present a new and simple variational formulation for the anisotropic surface diffusion and prove that it satisfies area/mass conservation and energy dissipation. The variational problem is discretized in space by the parametric finite element method and area/mass conservation and energy dissipation are established for the semi-discretization. Then the problem is further discretized in time by a (semi-implicit) backward Euler method so that only a linear system is to be solved at each time step for the full-discretization and thus it is efficient. We establish well-posedness of the full-discretization and identify some simple conditions on $γ(θ)$ such that the full-discretization keeps energy dissipation and thus it is unconditionally energy-stable. Finally the ES-PFEM is applied to simulate solid-state dewetting of thin films with anisotropic surface energies, i.e. the motion of an open curve under anisotropic surface diffusion with proper boundary conditions at the two triple points moving along the horizontal substrate. Numerical results are reported to demonstrate the efficiency and accuracy as well as energy dissipation of the proposed ES-PFEM.
We propose and analyze structure-preserving parametric finite element methods (SP-PFEM) for evolution of a closed curve under different geometric flows with arbitrary anisotropic surface energy $γ(\boldsymbol{n})$ for $\boldsymbol{n}\in \mathbb{S}^1$ representing the outward unit normal vector. By introducing a novel surface energy matrix $\boldsymbol{G}_k(\boldsymbol{n})$ depending on $γ(\boldsymbol{n})$ and the Cahn-Hoffman $\boldsymbolξ$-vector as well as a nonnegative stabilizing function $k(\boldsymbol{n}):\ \mathbb{S}^1\to \mathbb{R}$, which is a sum of a symmetric positive definite matr…
For the evolution of a closed surface under anisotropic surface diffusion with a general anisotropic surface energy $γ(\boldsymbol{n})$ in three dimensions (3D), where $\boldsymbol{n}$ is the unit outward normal vector, by introducing a novel symmetric positive definite surface energy matrix $\boldsymbol{Z}_k(\boldsymbol{n})$ depending on a stabilizing function $k(\boldsymbol{n})$ and the Cahn-Hoffman $\boldsymbolξ$-vector, we present a new symmetrized variational formulation for anisotropic surface diffusion with weakly or strongly anisotropic surface energy, which preserves two important str…
High-order parametric local discontinuous Galerkin methods for anisotropic curve-shortening flows
QiuZhen College, Tsinghua University（邱成桐数学科学中心）；Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心）；Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）
We deal with a long-standing problem about how to design an energy-stable numerical scheme for solving the motion of a closed curve under {\sl anisotropic surface diffusion} with a general anisotropic surface energy $γ(\boldsymbol{n})$ in two dimensions, where $\boldsymbol{n}$ is the outward unit normal vector. By introducing a novel symmetric positive definite surface energy matrix $Z_k(\boldsymbol{n})$ depending on the Cahn-Hoffman $\boldsymbolξ$-vector and a stabilizing function $k(\boldsymbol{n})$, we first reformulate the anisotropic surface diffusion into a conservative form and then der…
A Parametric Finite Element Approach for an Anisotropic Multi-Phase Mullins-Sekerka Problem with Kinetic Undercooling
Université Claude Bernard Lyon 1, CNRS, Centrale Lyon, INSA Lyon, Université Jean Monnet, ICJ UMR5208（里昂第一大学，法国国家科学研究中心，中央里昂学院，里昂国立应用科学学院，让·莫内大学，ICJ UMR5208）；Fakultät für Mathematik, Universität Regensburg（雷根斯堡大学数学系）；Dipartimento di Mathematica, Università di Trento（特伦托大学数学系）
Motivated by nonlinear approximation results for classes of parametric partial differential equations (PDEs), we seek to better understand so-called library approximations to analytic functions of countably infinite number of variables. Rather than approximating a function of interest in a single space, a library approximation uses a collection of spaces and the best space may be chosen for any point in the domain. In the setting of this paper, we use a specific library which consists of local Taylor approximations on sufficiently small rectangular subdomains of the (rescaled) parameter domain…
Based on a recent novel formulation of parametric anisotropic curve shortening flow, we analyse a fully discrete numerical method of this geometric evolution equation. The method uses piecewise linear finite elements in space and a backward Euler approximation in time. We establish existence and uniqueness of a discrete solution, as well as an unconditional stability property. Some numerical computations confirm the theoretical results and demonstrate the practicality of our method.
Summation-by-parts (SBP) finite difference methods have several desirable properties for second-order wave equations. They combine the computational efficiency of narrow-stencil finite difference operators with provable stability on curvilinear multiblock grids. While several techniques for boundary and interface conditions exist, weak imposition via simultaneous approximation terms (SATs) is perhaps the most flexible one. Although SBP methods have been applied to elastic wave equations many times, an SBP-SAT method for general anisotropic elastic wave equations has not yet been presented in t…
Most classical finite element schemes for the (Navier-)Stokes equations are neither pressure-robust, nor are they inf-sup stable on general anisotropic triangulations. A lack of pressure-robustness may lead to large velocity errors, whenever the Stokes momentum balance is dominated by a strong and complicated pressure gradient. It is a consequence of a method, which does not exactly satisfy the divergence constraint. However, inf-sup stable schemes can often be made pressure-robust just by a recent, modified discretization of the exterior forcing term, using $\mathbf{H}(\operatorname{div})$-co…
Parametric finite elements lead to very efficient numerical methods for surface evolution equations. We introduce several computational techniques for curvature driven evolution equations based on a weak formulation for the mean curvature. The approaches discussed, in contrast to many other methods, have good mesh properties that avoid mesh coalescence and very non-uniform meshes. Mean curvature flow, surface diffusion, anisotropic geometric flows, solidification, two-phase flow, Willmore and Helfrich flow as well as biomembranes are treated. We show stability results as well as results explai…
Decoupled energy-stable Runge-Kutta schemes of arbitrary order for the anisotropic phase-field dendritic crystal growth model
School of Mathematics and Statistics, Guangdong University of Technology（广东工业大学数学与统计学院）；Institute of Natural Sciences, Shanghai Jiao Tong University（上海交通大学自然科学研究院）
A provably stable numerical method for the anisotropic diffusion equation in confined magnetic fields: Curvilinear coordinates and multi-block domains
Max Plank Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）；The University of Texas at El Paso（得克萨斯大学埃尔帕索分校）；Proxima Fusion GmbH（Proxima Fusion有限公司）；Mathematical Sciences Institute, Australian National University（澳大利亚国立大学数学科学研究所）
We propose and analyze a semi-discrete parametric finite element scheme for solving the area-preserving curve shortening flow. The scheme is based on Dziuk's approach (SIAM J. Numer. Anal. 36(6): 1808-1830, 1999) for the anisotropic curve shortening flow. We prove that the scheme preserves two fundamental geometric structures of the flow with an initially convex curve: (i) the convexity-preserving property, and (ii) the perimeter-decreasing property. To the best of our knowledge, the convexity-preserving property of numerical schemes which approximate the flow is rigorously proved for the firs…
In this paper, we describe a stable finite element formulation for advection-diffusion-reaction problems that allows for robust automatic adaptive strategies to be easily implemented. We consider locally vanishing, heterogeneous, and anisotropic diffusivities, as well as advection-dominated diffusion problems. The general stabilized finite element framework was described and analyzed in arXiv:1907.12605v3 for linear problems in general, and tested for pure advection problems. The method seeks for the discrete solution through a residual minimization process on a proper stable discontinuous Gal…
We propose an energy-stable parametric finite element method (ES-PFEM) for simulating solid-state dewetting of thin films in two dimensions via a sharp-interface model, which is governed by surface diffusion and contact line (point) migration together with proper boundary conditions. By reformulating the relaxed contact angle condition into a Robin-type boundary condition and then treating it as a natural boundary condition, we obtain a new variational formulation for the problem, in which the interface curve and its contact points are evolved simultaneously. Then, the variational problem is d…
