Computational Mathematics and Scientific Computing Seminar

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This talk presents the Proximal Galerkin (PG) method, a high-order numerical method for solving variational problems with inequality constraints. PG combines two foundational ideas from applied mathematics: Galerkin discretizations of partial differential equations (PDEs), and (Bregman) proximal point algorithms for nonsmooth or constrained optimization. Conceptually, PG is a discretized Riemannian gradient flow within a finite-dimensional function space, such as a finite element subspace. Each iteration of the method solves a regularized subproblem equivalent to a PDE formulated as a nonlinear saddle-point system. This unified framework systematically handles a broad class of variational inequalities, yielding high-order, constraint-preserving solution approximations without specialized basis functions. The talk will outline the theoretical foundations of PG, highlight its connections to convex analysis, and showcase recent applications in contact mechanics, fracture, and multi-phase flows, among others.