Error estimates for elastic flows

薄层弹性结构导致了各种有趣的现象和应用。通常，两个能量稳定的弯曲状态被用来创建准静态开关装置或产生机械运动。对于所有这些技术发展来说，薄弹性物体的大变形是重要的。它们的数学模型导致了约束弯曲能，其中约束反映了弹性薄板的抗剪切和拉伸效应，或小截面弹性杆的不可伸长和扭转行为。最近已经设计出了这种能量的不同有限元离散形式。通过遵循调和映射的数值逼近思想，得到了实用且收敛的数值方法。在并行研究中，给出了一类几何演化问题的误差估计的推导，这些问题包括Willmore、平均曲率和调和映射热流。在这些发展中至关重要的是，将发展问题的等价公式确定为一个非线性抛物型系统，其中包括几何量的方程，如法场。在这个项目中，我们的目标是将这些发展结合在一起，以获得不同弯曲能梯度流的时间步长格式有限元离散的误差估计。这样做的动机有两个：第一，这证明了对自然的、强阻尼弛豫动力学的数值模拟是正确的，第二，它导致了通过某些演化过程获得的定态近似的误差估计。此外，这是向模拟捕捉惯性效应的一般动力学过程迈出的一步，例如，振动弹性杆。其他应用包括测定磁性弹性膜和棒的静止状态。

Thin elastic structures lead to a variety of interesting phenomenaand applications. Often, two energy stable bending states are used to create a quasistationary switching device or to generate mechanical locomotion. For all of these technical developments large deformations of thin elastic objects are important. Their mathematical modeling leads to constrained bending energies in which the constraint captures the resistance to shearing and stretching effects of thin elastic plates or the inextensibility and twist behavior of an elastic rod with small cross section. Different finite element discretizations of such energies have recently been devised. By following ideas for the numerical approximation of harmonic maps, which describe director fields with values in given manifolds, practical and convergent numerical methods were obtained. In parallel research the derivation of error estimates for classes of geometric evolution problems was addressed, which include the Willmore, the mean curvature, and the harmonic map heat flow. Crucial in those developments was the identification of an equivalent formulation of the evolution problems as a nonlinear parabolic system that includes equations for geometric quantities such as the normal field. In this project we aim at bringing the developments together to obtain error estimates for finite element discretizations of time stepping schemes for gradient flows of various bending energies. The motivation for this is twofold: first, this justifies the numerical simulation of natural, strongly damped relaxation dynamics, and second, it leads to error estimates for the approximation of stationary states obtained via certain evolution processes. In addition, this serves as a step towards simulating general dynamical processes that capture inertial effects, e.g., vibrating elastic rods. Other applications include the determination of stationary states of magnetic elastic films and rods.