We leverage the proximal Galerkin algorithm (Keith and Surowiec, Foundations of Computational Mathematics, 2024, DOI: 10.1007/s10208-024-09681-8), a recently introduced mesh-independent algorithm, to obtain a high-order finite element solver for variational problems with pointwise inequality constraints. This is achieved by discretizing the saddle point systems, arising from the latent variable proximal point method, with the hierarchical $p$-finite element basis. This results in discretized sparse Newton systems that admit a simple and effective block preconditioner. The solver can handle both obstacle-type, $u \leq \varphi$, and gradient-type, $|\nabla u| \leq \varphi$, constraints. We apply the resulting algorithm to solve obstacle problems with $hp$-adaptivity, a gradient-type constrained problem, and the thermoforming problem, an example of an obstacle-type quasi-variational inequality. We observe $hp$-robustness in the number of Newton iterations and only mild growth in the number of inner Krylov iterations to solve the Newton systems. Crucially we also provide wall-clock timings that are faster than low-order discretization counterparts.

翻译：我们利用近期提出的网格无关算法——近端伽辽金算法（Keith and Surowiec, Foundations of Computational Mathematics, 2024, DOI: 10.1007/s10208-024-09681-8），构建了一种用于处理带点式不等式约束变分问题的高阶有限元求解器。该方法通过采用分层$p$-有限元基函数对隐变量近端点法产生的鞍点系统进行离散化实现。由此得到的离散化稀疏牛顿系统可采用一种简单高效的块预条件子进行处理。该求解器能够同时处理障碍型约束（$u \leq \varphi$）与梯度型约束（$|\nabla u| \leq \varphi$）。我们将所提算法应用于三类问题：采用$hp$自适应技术的障碍问题、梯度型约束问题以及热成型问题（后者属于障碍型拟变分不等式范畴）。数值实验表明，牛顿迭代次数具有$hp$-鲁棒性，且求解牛顿系统所需的内层Krylov迭代次数仅呈现温和增长。尤为关键的是，我们提供的实际计算耗时数据表明，该方法相较于低阶离散化方案具有更快的求解速度。