  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.
