Variational inequalities play a pivotal role in a wide array of scientific and engineering applications. This project presents two techniques for adaptive mesh refinement (AMR) in the context of variational inequalities, with a specific focus on the classical obstacle problem. We propose two distinct AMR strategies: Variable Coefficient Elliptic Smoothing (VCES) and Unstructured Dilation Operator (UDO). VCES uses a nodal active set indicator function as the initial iterate to a time-dependent heat equation problem. Solving a single step of this problem has the effect of smoothing the indicator about the free boundary. We threshold this smoothed indicator function to identify elements near the free boundary. Key parameters such as timestep and threshold values significantly influence the efficacy of this method. The second strategy, UDO, focuses on the discrete identification of elements adjacent to the free boundary, employing a graph-based approach to mark neighboring elements for refinement. This technique resembles the dilation morphological operation in image processing, but tailored for unstructured meshes. We also examine the theory of variational inequalities, the convergence behavior of finite element solutions, and implementation in the Firedrake finite element library. Convergence analysis reveals that accurate free boundary estimation is pivotal for solver performance. Numerical experiments demonstrate the effectiveness of the proposed methods in dynamically enhancing mesh resolution around free boundaries, thereby improving the convergence rates and computational efficiency of variational inequality solvers. Our approach integrates seamlessly with existing Firedrake numerical solvers, and it is promising for solving more complex free boundary problems.

翻译：变分不等式在众多科学与工程应用中发挥着关键作用。本项目针对变分不等式提出了两种自适应网格细化（AMR）技术，并特别聚焦于经典障碍问题。我们提出了两种不同的AMR策略：变系数椭圆光滑化（VCES）和非结构化膨胀算子（UDO）。VCES采用节点活动集指示函数作为瞬态热传导方程问题的初始迭代解。求解该问题的单步计算可实现自由边界附近指示函数的光滑化。通过对光滑化指示函数进行阈值处理，可识别自由边界附近的网格单元。时间步长与阈值等关键参数对该方法的效能具有显著影响。第二种策略UDO侧重于自由边界相邻单元的离散识别，采用基于图论的方法标记待细化的相邻单元。该技术类似于图像处理中的膨胀形态学操作，但专为非结构化网格设计。我们还探讨了变分不等式理论、有限元解的收敛行为及其在Firedrake有限元库中的实现。收敛分析表明，精确的自由边界估计对求解器性能至关重要。数值实验证明，所提方法能有效增强自由边界区域的网格分辨率动态适应性，从而提升变分不等式求解器的收敛速度与计算效率。本方法可与现有Firedrake数值求解器无缝集成，在解决更复杂自由边界问题方面具有良好前景。