Proximal Discontinuous Galerkin Methods for Variational Inequalities
For researchers in numerical analysis and computational mechanics, this work provides a unified framework and higher-order convergence for proximal Galerkin methods, though it is incremental as it extends existing techniques.
This paper introduces a family of proximal discontinuous Galerkin methods for variational inequalities, using the obstacle problem as an example. The proximal hybrid high-order method achieves the first higher-order convergence result for any proximal Galerkin method.
We introduce a family of proximal discontinuous Galerkin methods for variational inequalities, focusing on the obstacle problem as a didactic example. Each member of this family is born from applying a different well-known nonconforming finite element discretization to the Bregman proximal point method. We explicitly treat four examples: the symmetric interior penalty discontinuous Galerkin, the enriched Galerkin, the hybridizable interior penalty and the hybrid high-order methods. We formulate a unified analysis framework for this family of methods and prove the existence and uniqueness of solutions, energy dissipation, and error estimates for both the primal and dual variables. Remarkably, the proximal hybrid high-order method with piecewise constant cell unknowns and piecewise affine cell unknowns leads to the first higher-order convergence result for any proximal Galerkin method.