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A Novel Preconditioning Framework for Solving Nonlinear PDEs based on Fenchel-Rockafellar Duality and Transformed Primal-Dual Techniques

Long Chen, Ruchi Guo, Jingrong Wei, Jun Zou

TL;DR

This work addresses the efficient solution of nonlinear PDEs by decoupling nonlinear terms from differential operators through Fenchel--Rockafellar duality, yielding a dual saddle-point formulation with block-diagonal nonlinear updates. It advances the transformed primal--dual (TPD) methodology into a DualTPD algorithm that leverages carefully designed preconditioners and multigrid solvers to achieve mesh-independent convergence for problems such as the $p$-Laplacian and nonlinear Maxwell equations, including $p$-curl variants. The paper provides detailed discretization strategies, per-element and Schur-complement preconditioners, and extensive numerical demonstrations showing robust performance and favorable scaling, often outperforming primal-based solvers. The framework is flexible and broadly applicable to nonlinear saddle-point systems, with clear paths for extensions to adaptive strategies, stronger nonlinearities, and coupled multiphysics problems.

Abstract

A DualTPD method is proposed for solving nonlinear partial differential equations. The method is characterized by three main features. First, decoupling via Fenchel--Rockafellar duality is achieved, so that nonlinear terms are discretized by discontinuous finite element spaces, yielding block-diagonal mass matrices and closed-form updates. Second, improved convergence is obtained by applying transformed primal--dual (TPD) dynamics to the nonlinear saddle-point system, which yields strongly monotone behavior. Third, efficient preconditioners are designed for the elliptic-type Schur complement arising from the separated differential operators, and multigrid solvers are applied effectively. Extensive numerical experiments on elliptic $p$-Laplacian and nonlinear $H(\curl)$ problems are presented, showing significant efficiency gains with global, mesh-independent convergence.

A Novel Preconditioning Framework for Solving Nonlinear PDEs based on Fenchel-Rockafellar Duality and Transformed Primal-Dual Techniques

TL;DR

This work addresses the efficient solution of nonlinear PDEs by decoupling nonlinear terms from differential operators through Fenchel--Rockafellar duality, yielding a dual saddle-point formulation with block-diagonal nonlinear updates. It advances the transformed primal--dual (TPD) methodology into a DualTPD algorithm that leverages carefully designed preconditioners and multigrid solvers to achieve mesh-independent convergence for problems such as the -Laplacian and nonlinear Maxwell equations, including -curl variants. The paper provides detailed discretization strategies, per-element and Schur-complement preconditioners, and extensive numerical demonstrations showing robust performance and favorable scaling, often outperforming primal-based solvers. The framework is flexible and broadly applicable to nonlinear saddle-point systems, with clear paths for extensions to adaptive strategies, stronger nonlinearities, and coupled multiphysics problems.

Abstract

A DualTPD method is proposed for solving nonlinear partial differential equations. The method is characterized by three main features. First, decoupling via Fenchel--Rockafellar duality is achieved, so that nonlinear terms are discretized by discontinuous finite element spaces, yielding block-diagonal mass matrices and closed-form updates. Second, improved convergence is obtained by applying transformed primal--dual (TPD) dynamics to the nonlinear saddle-point system, which yields strongly monotone behavior. Third, efficient preconditioners are designed for the elliptic-type Schur complement arising from the separated differential operators, and multigrid solvers are applied effectively. Extensive numerical experiments on elliptic -Laplacian and nonlinear problems are presented, showing significant efficiency gains with global, mesh-independent convergence.
Paper Structure (19 sections, 1 theorem, 72 equations, 2 figures, 8 tables)

This paper contains 19 sections, 1 theorem, 72 equations, 2 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Primal and Dual Formulations
  3. The Fenchel--Rockafellar Duality
  4. Examples
  5. Discretization with Block-diagonal Mass Matrices
  6. The DualTPD Algorithm
  7. Transformation and Preconditioning
  8. Transformed Primal--Dual Algorithm for Dual System
  9. Connection to Other Algorithms
  10. Application I: $p$-Laplacian Equations
  11. Discretization and Matrix Form
  12. Preconditioning
  13. Numerical Results
  14. Application II: Nonlinear Maxwell Equations
  15. Augmented Lagrangian formulation
  16. ...and 4 more sections

Key Result

Theorem 2.4

Let $\mathcal{F}^*$ be proper, differentiable, and convex. Under Assumptions assump: F^* and assump: D inf-sup, the KKT system eq_saddle_intro, associated with the dual problem, admits a unique solution.

Figures (2)

  • Figure 1: The relation of various problems derived via Fenchel–Rockafellar duality.
  • Figure 2: Computation performance for the proposed DualTPD algorithm on nonlinear ferromagnetism model.

Theorems & Definitions (5)

  • Remark 2.2
  • Theorem 2.4: 1977Scheurer
  • Remark 2.5
  • Example 2.6: $p$-Laplacian
  • Example 2.7: Nonlinear ferromagnetism