Preconditioning proximal Galerkin: mesh and degree robust solvers for variational inequalities
Abstract
The proximal Galerkin algorithm is a structure-preserving solver for variational inequalities that results in discretization-independent nonlinear iteration counts. However, in large three-dimensional simulations, the arising Newton saddle point systems cannot be feasibly solved by direct methods. To leverage an iterative linear solver, we propose a robust and scalable block preconditioner. Instead of using the usual dual Schur complement that acts on the latent variable, we design a preconditioner for the primal Schur complement. This leads to a block preconditioner that requires two approximate inverses of the latent block and only one of the primal Schur complement. The latent block represents an algebraic equation which, after a small perturbation, can be cheaply inverted. We then introduce two choices of approximations for the primal Schur complement: a PDE-operator-based and an algebraic approach. Either approximation can be efficiently inverted by geometric multigrid. We investigate the effectiveness of these strategies when proximal Galerkin is applied to obstacle, Signorini, and strain-constrained elasticity problems and also compare with a monolithic multigrid approach. Choices of standard relaxation methods, in the block preconditioner route, lead to robust iteration counts with respect to the mesh size and discretization degree. Moreover, in the context of the Signorini, specialized relaxation and prolongation also result in effective solvers for nearly-incompressible elastic materials. The linear solves are parameter-robust in 2D, with only mild growth in 3D. This allows for the solution of a three-dimensional contact problem, with a Poisson's ratio of 0.499, discretized with over ten million degrees of freedom.
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BibTeX
@misc{papadopoulos2026preconditioning,
title = {{Preconditioning proximal Galerkin: mesh and degree robust solvers for variational inequalities}},
author = {Ioannis P. A. Papadopoulos and Pablo D. Brubeck},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13591},
eprint = {2610.13591},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13591},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Preconditioning proximal Galerkin: mesh and degree robust solvers for variational inequalities
AU - Ioannis P. A. Papadopoulos
AU - Pablo D. Brubeck
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13591
UR - https://arxiv.org/abs/2610.13591
AB - The proximal Galerkin algorithm is a structure-preserving solver for variational inequalities that results in discretization-independent nonlinear iteration counts. However, in large three-dimensional simulations, the arising Newton saddle point systems cannot be feasibly solved by direct methods. To leverage an iterative linear solver, we propose a robust and scalable block preconditioner. Instead of using the usual dual Schur complement that acts on the latent variable, we design a preconditioner for the primal Schur complement. This leads to a block preconditioner that requires two approximate inverses of the latent block and only one of the primal Schur complement. The latent block represents an algebraic equation which, after a small perturbation, can be cheaply inverted. We then introduce two choices of approximations for the primal Schur complement: a PDE-operator-based and an algebraic approach. Either approximation can be efficiently inverted by geometric multigrid. We investigate the effectiveness of these strategies when proximal Galerkin is applied to obstacle, Signorini, and strain-constrained elasticity problems and also compare with a monolithic multigrid approach. Choices of standard relaxation methods, in the block preconditioner route, lead to robust iteration counts with respect to the mesh size and discretization degree. Moreover, in the context of the Signorini, specialized relaxation and prolongation also result in effective solvers for nearly-incompressible elastic materials. The linear solves are parameter-robust in 2D, with only mild growth in 3D. This allows for the solution of a three-dimensional contact problem, with a Poisson's ratio of 0.499, discretized with over ten million degrees of freedom.
ER -