A Novel Preconditioning Framework for Solving Nonlinear PDEs based on Fenchel-Rockafellar Duality and Transformed Primal-Dual Techniques
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.
Keywords
Cite
@article{arxiv.2510.15351,
title = {A Novel Preconditioning Framework for Solving Nonlinear PDEs based on Fenchel-Rockafellar Duality and Transformed Primal-Dual Techniques},
author = {Long Chen and Ruchi Guo and Jingrong Wei and Jun Zou},
journal= {arXiv preprint arXiv:2510.15351},
year = {2025}
}