Improving Convergence of Time-Simultaneous Multigrid Methods for Convection-Dominated Problems using VMS Stabilization Techniques

Abstract

We present the application of a time-simultaneous multigrid algorithm closely related to multigrid waveform relaxation for stabilized convection-diffusion equations in the regime of small diffusion coefficients. We use Galerkin finite elements and the Crank-Nicolson scheme for discretization in space and time. The multigrid method blocks all time steps for each spatial unknown, enhancing parallelization in space. While the number of iterations of the solver is bounded above for the 1D heat equation, convergence issues arise in convection-dominated cases. In singularly perturbed advection-diffusion scenarios, Galerkin FE discretizations are known to show instabilities in the numerical solution.We explore a higher-order variational multiscale stabilization, aiming to enhance solution smoothness and improve convergence without compromising accuracy.