We introduce the proximal Galerkin (PG) method for non-symmetric variational inequalities. The proposed approach is asymptotically mesh-independent and yields constraint-preserving approximations. We present both a conforming PG formulation and a hybrid mixed first-order system variant (FOSPG). We establish optimal a priori error estimates for each variant, which are verified numerically. We conclude by applying the method to American option pricing, free boundary problems in porous media, advection-diffusion with a semipermeable boundary, and the enforcement of discrete maximum principles.
Proximal Discontinuous Galerkin Methods for Variational Inequalities
CERMICS, CNRS, ENPC, Institut Polytechnique de Paris（CERMICS，法国国家科学研究中心，巴黎国立高等路桥学校，巴黎理工学院）；Centre Inria de Paris（巴黎Inria中心）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）；Department of Computational Applied Mathematics and Operations Research, Rice University（莱斯大学计算应用数学与运筹学系）
Universidad Politécnica de Madrid（马德里理工大学）；IMDEA Materials Institute（IMDEA材料研究所）；Brown University（布朗大学）；Simula Research Laboratory（西穆拉研究实验室）；Vanderbilt University（范德堡大学）
A Discontinuous Galerkin Method for H(curl)-Elliptic Hemivariational Inequalities
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Shanghai Artificial Intelligence Laboratory（上海人工智能实验室）；Department of Mathematics, University of Iowa（爱荷华大学数学系）；School of Mathematical Sciences, Inner Mongolia University（内蒙古大学数学科学学院）
Recovering the uniform boundary observability with spectral Legendre-Galerkin formulations of the 1-D wave equation
Laboratoire J.A. Dieudonné, Université Côte d’Azur（J.A.迪厄多内实验室，蔚蓝海岸大学）；GIREF, Département de mathématiques et de statistique, Université Laval（GIREF，数学与统计系，拉瓦尔大学）
