Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints

Keith, Brendan; Surowiec, Thomas M.

doi:10.1007/s10208-024-09681-8

Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints

Journal Article · Tue Nov 19 23:00:00 EST 2024 · Foundations of Computational Mathematics

DOI:https://doi.org/10.1007/s10208-024-09681-8· OSTI ID:2534643

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Brown University, Providence, RI (United States); Brown University
Simula Research Laboratory, Olso (Norway)

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.

View Accepted Manuscript (DOE)

Research Organization:: Brown University, Providence, RI (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: SC0024335

OSTI ID:: 2534643

Journal Information:: Foundations of Computational Mathematics, Journal Name: Foundations of Computational Mathematics; ISSN 1615-3375

Country of Publication:: United States

Language:: English

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