A Lagrangian‐Eulerian Scheme for Convection‐Diffusion Obstacle Problems

This paper presents an efficient computational framework for solving convection‐diffusion obstacle problems, designed for convection‐dominated regimes while ensuring local and global mass conservation. The method relies on an operator‐splitting strategy that decouples the problem into convection and diffusion sub‐problems, treated, respectively, in Lagrangian and Eulerian settings. The convective transport is handled by a particle‐in‐cell method, while the diffusion, formulated as a parabolic variational inequality, is discretized using mixed finite elements. This leads to symmetric saddle‐point systems with complementarity conditions, solved efficiently via a primal‐dual active set algorithm. To ensure conservative coupling between particles and mesh, a PDE‐constrained projection is employed. The effectiveness and performance of the overall approach have been established by rigorous benchmarks with analytical solutions from the literature, covering both structured and unstructured meshes.