A streamline upwind/Petrov-Galerkin method for the magnetic advection-diffusion problem

Abstract: This paper presents the development and analysis of a streamline upwind/Petrov-Galerkin (SUPG) method for the magnetic advection-diffusion problem. A key feature of the method is an SUPG-type stabilization term based on the residuals and weighted advection terms of the test function. By introducing a lifting operator to characterize the jumps of finite element functions across element interfaces, we define a discrete magnetic advection operator, which subsequently enables the formulation of the desired SUPG method. Under mild assumptions, we establish the stability of the scheme and derive optimal error estimates. Furthermore, by introducing a stabilization term that depends on the residual, we propose a nonlinear extension aimed at more effectively reducing numerical oscillations in sharp layers. Numerical examples in both two and three dimensions are provided to demonstrate the theoretical convergence and stabilization properties of the proposed method.