A local discontinuous Galerkin (LDG) method for approximating large deformations of prestrained plates is introduced and tested on several insightful numerical examples in Bonito et al. (2022, LDG approximation of large deformations of prestrained plates. J. Comput. Phys., 448, 110719). This paper presents a numerical analysis of this LDG method, focusing on the free boundary case. The problem consists of minimizing a fourth-order bending energy subject to a nonlinear and nonconvex metric constraint. The energy is discretized using LDG and a discrete gradient flow is used for computing discrete minimizers. We first show $\varGamma $-convergence of the discrete energy to the continuous one. Then we prove that the discrete gradient flow decreases the energy at each step and computes discrete minimizers with control of the metric constraint defect. We also present a numerical scheme for initialization of the gradient flow and discuss the conditional stability of it.

中文翻译：

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预应力板大变形LDG法的数值分析

在 Bonito 等人的几个有见地的数值例子中，介绍了一种用于近似预应变板大变形的局部不连续 Galerkin (LDG) 方法。（2022 年，预应力板大变形的 LDG 近似。J. Comput. Phys., 448, 110719）。本文介绍了这种 LDG 方法的数值分析，重点是自由边界情况。该问题包括最小化受非线性和非凸度量约束的四阶弯曲能量。能量使用 LDG 离散化，离散梯度流用于计算离散极小值。我们首先展示$\varGamma $-离散能量到连续能量的收敛性。然后我们证明了离散梯度流在每一步都会减少能量，并在控制度量约束缺陷的情况下计算离散极小值。