Error estimates for elastic flows

薄弹性结构导致了许多有趣的现象和应用。通常，两个能量稳定的弯曲状态被用来创建一个准静态开关设备或产生机械运动。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。他们的数学建模导致约束弯曲能量，其中约束捕获对薄弹性板的剪切和拉伸效应或具有小横截面的弹性杆的不可延伸性和扭曲行为的抵抗。不同的有限元离散的能量最近已经设计出来。利用调和映射的数值逼近思想，得到了一种实用的、收敛的数值方法。在平行研究的误差估计类的几何演化问题，其中包括Willmore，平均曲率，调和映射热流的推导。在这些发展的关键是确定一个等效的制定的发展问题作为一个非线性抛物系统，包括方程的几何量，如正常领域。在这个项目中，我们的目标是把发展到一起，以获得误差估计有限元离散的时间步进计划的梯度流的各种弯曲能量。这样做的动机是双重的：首先，这证明了自然的，强阻尼松弛动力学的数值模拟，第二，它导致通过某些演化过程获得的稳态近似的误差估计。此外，这也是模拟捕捉惯性效应的一般动态过程的一步，例如，振动弹性杆。其他应用包括磁性弹性薄膜和磁性弹性棒的定态测定。

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.