The proximal Galerkin finite element method is a high-order, nonlinear numerical method that preserves the geometric and algebraic structure of 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 scalable, mesh-independent algorithms for optimal design. The paper begins with a derivation of the latent variable proximal point (LVPP) method: an unconditionally stable alternative to the interior point method. LVPP is an infinite-dimensional optimization algorithm that may be viewed as having an adaptive (Bayesian) barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of the main benefits of this algorithm is witnessed when analyzing the classical obstacle problem. Therein, we find that the original variational inequality can be replaced by a sequence of semilinear 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 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：近端伽辽金有限元方法是一种高阶非线性数值方法，能够保持无穷维函数空间中界约束的几何与代数结构。本文介绍了近端伽辽金方法，并将其应用于求解自由边界问题、强制执行离散极大值原理，以及开发可扩展、网格无关的最优设计算法。我们首先推导了隐变量近端点（LVPP）方法：这是一种无条件稳定的内点法替代方案。LVPP是一种无穷维优化算法，可视为具有自适应（贝叶斯）障碍函数，该函数在每次（外循环）优化迭代中通过新的信息性先验进行更新。该算法的主要优势之一体现在经典障碍问题的分析中。在此我们发现，原始变分不等式可被一系列半线性偏微分方程（PDE）替代，这些方程易于采用高阶有限元等方法进行离散与求解。本研究得出若干可能具有独立意义的意外贡献，包括：（1）我们称之为熵泊松方程的半线性偏微分方程；（2）高阶保正离散与无穷维李群之间的代数/几何联系；（3）用于双场密度型拓扑优化的基于梯度且保界的算法。完整的隐变量近端伽辽金方法论融合了非线性规划、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内部催生新的协同效应。