Convergence theory for proximal Galerkin with gradient-type constraints

Establish theoretical guarantees of convergence for the proximal Galerkin algorithm applied to variational inequalities with gradient-type constraints, including the strain-constrained elasticity problem.

Background

The paper applies proximal Galerkin to three classes of variational inequalities: obstacle problems, Signorini contact problems, and strain-constrained elasticity problems. In the strain-constrained setting, the pointwise constraint bounds the norm of the symmetric strain, and the latent variable is tensor-valued rather than scalar-valued or boundary-supported.

Although the paper reports experimental evidence for proximal Galerkin on gradient-type constraints, it explicitly states that theoretical convergence guarantees are not established. Resolving this problem would provide a rigorous foundation for the observed behavior of the method beyond the numerical experiments.

References

Theoretical guarantees of convergence of PG for problems with gradient-type constraints is an open problem although experimental evidence can be found in .

— Preconditioning proximal Galerkin: mesh and degree robust solvers for variational inequalities  (2610.13591 - Papadopoulos et al., 9 Oct 2026) in Section 2, subsection “PG for strain-constrained elasticity”