Elastic flow for closed curves can involve significant deformations. Mesh-based approximation schemes require tangentially redistributing vertices for long-time computations. We present and analyze a method that uses the Dirichlet energy for this purpose. The approach effectively also penalizes the length of the curve, and equilibrium shapes are equivalent to stationary points of the elastic energy augmented with the length functional. Our numerical method is based on linear parametric finite elements. Following the lines of K Deckelnick, and G Dziuk (Math Comp 78, 266 (2009), 645-671) we prove convergence and establish error estimates, noting that the addition of the Dirichlet energy simplifies the analysis in comparison with the length functional. We also present a simple semi-implicit time discretization and discuss some numerical result that support the theory.
翻译:封闭曲线的电磁流可能涉及巨大的变形。 以网状为基础的近似方案要求长期计算时对脊椎进行正流再分配。 我们提出和分析一种为此目的使用dirichlet能源的方法。 这种方法还有效地惩罚了曲线的长度, 平衡形状相当于弹性能量的固定点和长度功能。 我们的数字方法以线性参数有限元素为基础。 按照K Deckelnick和G Dziuk(Math Comp 78, 266(2009), 645-671)的线条, 我们证明存在趋同, 并确定了误差估计值, 指出添加dirichlet能源与长度功能相比简化了分析。 我们还提出了一个简单的半不完全的时间分解, 并讨论支持该理论的一些数字结果 。