The proximal Galerkin finite element method is a high-order, low-iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of point-wise 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 with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several 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 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. This work is accompanied by open-source implementations of our methods to facilitate reproduction and broader adoption.

翻译：邻近Galerkin有限元法是一种高阶、低迭代复杂度的非线性数值方法，能够在无限维函数空间中保持点态有界约束的几何与代数结构。本文介绍了邻近Galerkin方法，并将其应用于求解自由边界问题、强化离散极值原理，以及开发具有点态有界约束的、可扩展且网格无关的最优设计算法。本文同时提出了潜变量邻近点算法，邻近Galerkin方法正是由此衍生而来。在分析经典障碍问题时，我们发现其基本变分不等式可被一系列易于离散化求解的二阶偏微分方程序列所替代，例如采用邻近Galerkin方法进行求解。本研究取得了多项具有独立价值的成果，包括：（1）我们称之为熵泊松方程的半线性偏微分方程；（2）高阶保正离散格式与特定无限维李群之间的代数/几何关联；（3）基于梯度、满足约束条件的双场密度拓扑优化算法。完整的邻近Galerkin方法体系融合了非线性规划、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。本研究附有开源代码实现，以促进方法复现与更广泛的应用。