Surface PDE: a minimizing movement approach

我们的研究主要集中在发展曲面约束偏微分方程的极小化运动。通过将最近点方法与泛函极小化方法相结合，成功地实现了相应的逼近方法，利用我们的曲面型极小化运动，我们能够有效地模拟曲面上界面的平均曲率流和双曲平均曲率流.这些方法代表的MBO（梅里曼，本斯，和Osher）和HMBO（双曲梅里曼，本斯，和Osher）算法，专门为曲面约束setting.In此外，我们设计了一个曲面约束的有符号距离矢量场（SDVF）描述相位几何形状的表面上的多相设置。我们进一步实现了将SDVF应用于计算问题的数值算法。关于我们结合最近点方法和最小化运动的近似方法，在各种条件下对表面上的热和波动方程进行了数值误差分析。我们的表面型最小化运动的收敛性，相对于空间离散化，也进行了检查。结果表明，数值解收敛于精确解。

Our research focused on developing minimizing movments for surface-constrained partial differential equations. The corresponding approximation methods were successfully realized by incorporating the closest point method into a functional minimization scheme.Using our surface-type minimizing movements, we were able to effectively simulate mean curvature flow and hyperbolic mean curvature flow of interfaces on surfaces. These methods represent generalizations of the MBO (Merriman, Bence, and Osher) and HMBO (Hyperbolic Merriman, Bence, and Osher) algorithms, specifically tailored for the surface-constrained setting.In addition, we designed a surface-constrained signed distance vector field (SDVF) for describing phase geometries on surfaces in multiphase settings. We further implemented the numerical algorithms that enable the application of the SDVF to computational problems.Regarding our approximation method that combines the closest point method and minimizing movements, numerical error analyses were conducted for the heat and wave equations on surfaces, under various conditions. Convergence of our surface-type minimizing movement, with respect to the spatial discretization, was also examined. Our the results revealed that the numerical solution converges to the exact solution.