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 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. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.

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，它在无限维函数空间中保留了逐点边界约束的几何和代数结构。本文介绍了近端 Galerkin 方法，并将其应用于解决自由边界问题，强制执行离散最大值原则，并开发一种可扩展的、独立于网格的算法，用于具有逐点边界约束的最优设计。本文还介绍了潜在变量近端点 （LVPP） 算法，近端 Galerkin 方法就是从该算法衍生而来的。在分析经典障碍问题时，我们发现潜在的变分不等式可以被一系列二阶偏微分方程 （PDE） 所取代，这些方程很容易离散化并使用例如近端伽辽金方法求解。在这项工作中，我们得出了几个可能具有独立兴趣的贡献。这些包括 （1） 我们称为熵泊松方程的半线性偏微分方程;（2） 高阶正性保持离散化和某些无限维李群之间的代数/几何联系;（3） 基于梯度的边界保留算法，用于双场、基于密度的拓扑优化。完整的近端 Galerkin 方法结合了非线性规划、泛函分析、热带代数和微分几何的思想，并有可能在这些领域之间以及变分和数值分析中产生新的协同作用。我们方法的开源实现伴随着这项工作，以促进复制和更广泛的采用。