Moving Mesh with Streamline Upwind Petrov-Galerkin (MM-SUPG) Method for Time-dependent Convection-Dominated Convection-Diffusion Problems

Abstract

Time-dependent convection-dominated convection-diffusion problems are considered. We develop a moving mesh streamline upwind Petrov-Galerkin (MM-SUPG) method by combining residual-based SUPG stabilization with a metric-based moving mesh PDE (MMPDE) approach. The key feature of the method is the interaction between mesh adaptation and stabilization: the evolving mesh modifies both the direction and magnitude of the SUPG stabilization through the local element geometry, while stabilization improves robustness in convection-dominated regimes. For isotropic diffusion, numerical results show that the proposed method reduces spurious oscillations and provides improved resolution of sharp layers compared with fixed mesh methods, while yielding accuracy comparable to moving mesh finite element methods without SUPG in some cases. For anisotropic diffusion, we introduce a weighted tensor that incorporates both the diffusion tensor and the convection field, and construct a metric tensor via intersection to guide mesh adaptation. Under suitable assumptions on the elementwise tensors and mesh geometry, we establish sufficient conditions for the discrete maximum principle (DMP) of the fully discrete scheme. The analysis is based on quantitative bounds for the convection terms, and requires structural conditions on the alignment between the velocity field and the diffusion tensor. Numerical experiments demonstrate that the proposed metric improves monotonicity properties and reduces undershoots, while maintaining overall accuracy.