The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design problems with pointwise bound constraints. This paper also provides a derivation of the latent variable proximal point (LVPP) algorithm, an unconditionally stable alternative to the interior point method. LVPP is an infinite-dimensional optimization algorithm that may be viewed as having an adaptive barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of its main benefits is witnessed when analyzing the classical obstacle problem. Therein, we find that the original variational inequality can be replaced by a sequence of partial differential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout this work, we arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field density-based topology optimization. The complete latent variable proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis.

翻译：近端Galerkin有限元方法是一种高阶、低迭代复杂度的非线性数值方法，能够保持无限维函数空间中逐点有界约束的几何与代数结构。本文介绍了近端Galerkin方法，并将其应用于求解自由边界问题、强制离散最大原理，以及为具有逐点有界约束的最优设计问题开发可扩展、网格无关的算法。本文还推导了潜变量近端点（LVPP）算法，这是一种无条件稳定的内点法替代方案。LVPP是一种无限维优化算法，可视为具有自适应障碍函数，该函数在每次（外循环）优化迭代中通过新的先验信息进行更新。其主要优势之一在分析经典障碍问题时得以体现：我们发现在该问题中，原始变分不等式可被一系列易于离散化并利用高阶有限元等方法求解的偏微分方程（PDE）所取代。本文在研究过程中取得了若干可能具有独立意义的意外贡献，包括：(1) 一种称为熵泊松方程的半线性PDE；(2) 高阶正值保持离散化与某些无限维李群之间的代数/几何联系；(3) 一种基于梯度、保持有界性的两场密度拓扑优化算法。完整的潜变量近端Galerkin方法整合了非线性规划、泛函分析、热带代数与微分几何的思想，有望在这些领域之间以及变分分析与数值分析中建立新的协同关系。