Submitted

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

Submitted

arXiv   Bibtex

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.