Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints.
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| Title: | Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints. |
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| Authors: | Keith, Brendan1 (AUTHOR) brendan_keith@brown.edu, Surowiec, Thomas M.2 (AUTHOR) thomasms@simula.no |
| Source: | Foundations of Computational Mathematics. Feb2026, Vol. 26 Issue 1, p385-481. 97p. |
| Subjects: | Finite element method, Constraint satisfaction, Numerical solutions to nonlinear differential equations, Variational inequalities (Mathematics), Galerkin methods, Partial differential equations, Structural optimization |
| Abstract: | 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. [ABSTRACT FROM AUTHOR] |
| Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Items | – Name: Title Label: Title Group: Ti Data: Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Keith%2C+Brendan%22">Keith, Brendan</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> brendan_keith@brown.edu</i><br /><searchLink fieldCode="AR" term="%22Surowiec%2C+Thomas+M%2E%22">Surowiec, Thomas M.</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> thomasms@simula.no</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Foundations+of+Computational+Mathematics%22">Foundations of Computational Mathematics</searchLink>. Feb2026, Vol. 26 Issue 1, p385-481. 97p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Finite+element+method%22">Finite element method</searchLink><br /><searchLink fieldCode="DE" term="%22Constraint+satisfaction%22">Constraint satisfaction</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+solutions+to+nonlinear+differential+equations%22">Numerical solutions to nonlinear differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Variational+inequalities+%28Mathematics%29%22">Variational inequalities (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Galerkin+methods%22">Galerkin methods</searchLink><br /><searchLink fieldCode="DE" term="%22Partial+differential+equations%22">Partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Structural+optimization%22">Structural optimization</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: 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. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s10208-024-09681-8 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 97 StartPage: 385 Subjects: – SubjectFull: Finite element method Type: general – SubjectFull: Constraint satisfaction Type: general – SubjectFull: Numerical solutions to nonlinear differential equations Type: general – SubjectFull: Variational inequalities (Mathematics) Type: general – SubjectFull: Galerkin methods Type: general – SubjectFull: Partial differential equations Type: general – SubjectFull: Structural optimization Type: general Titles: – TitleFull: Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Keith, Brendan – PersonEntity: Name: NameFull: Surowiec, Thomas M. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 02 Text: Feb2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 16153375 Numbering: – Type: volume Value: 26 – Type: issue Value: 1 Titles: – TitleFull: Foundations of Computational Mathematics Type: main |
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