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The proximal Galerkin finite element method is a high-order, low iterationcomplexity, nonlinear numerical method that preserves the geometric andalgebraic structure of pointwise bound constraints in infinite-dimensionalfunction spaces. This paper introduces the proximal Galerkin method and appliesit to solve free boundary problems, enforce discrete maximum principles, anddevelop a scalable, mesh-independent algorithm for optimal design problems withpointwise bound constraints. This paper also provides a derivation of thelatent variable proximal point (LVPP) algorithm, an unconditionally stablealternative to the interior point method. LVPP is an infinite-dimensionaloptimization algorithm that may be viewed as having an adaptive barrierfunction that is updated with a new informative prior at each (outer loop)optimization iteration. One of its main benefits is witnessed when analyzingthe classical obstacle problem. Therein, we find that the original variationalinequality can be replaced by a sequence of partial differential equations(PDEs) that are readily discretized and solved with, e.g., high-order finiteelements. Throughout this work, we arrive at several unexpected contributionsthat may be of independent interest. These include (1) a semilinear PDE werefer to as the entropic Poisson equation; (2) an algebraic/geometricconnection between high-order positivity-preserving discretizations and certaininfinite-dimensional Lie groups; and (3) a gradient-based, bound-preservingalgorithm for two-field density-based topology optimization. The completelatent variable proximal Galerkin methodology combines ideas from nonlinearprogramming, functional analysis, tropical algebra, and differential geometryand can potentially lead to new synergies among these areas as well as withinvariational and numerical analysis.

Proximal Galerkin: A structure-preserving finite element method for pointwise bound constraints