Fast Solver for Advection Dominated Diffusion Using Residual Minimization and Neural Networks

Abstract

Advection-dominated diffusion is a challenging computational problem that requires special stabilization efforts. Unfortunately, the numerical solution obtained with the commonly used Galerkin method delivers unexpected oscillation resulting in an inaccurate numerical solution. The theoretical background resulting from the famous inf-sup condition tells us that the finite-dimensional test space employed by the Galerkin method does not allow us to reach the supremum necessary for problem stability. We enlarge the test space to overcome this problem. We do it for a fixed trial space. The method that allows us to do so is the residual minimization method. This method, however, requires the solution to a much larger system of linear equations than the standard Galerkin method. We represent the larger test space by its set of optimal test functions, forming a basis of the same dimension as the trial space in the Galerkin method. The resulting Petrov-Galerkin method stabilizes our challenging advection-dominated problem. We train the optimal test functions offline with the neural network to speed up the computations. We also observe that the optimal test functions, usually global, can be approximated with local support functions, resulting in a low computational cost for the solver and a stable numerical solution.