Kronecker product-based solvers for higher-order finite element method Navier-Stokes simulations

Transient time-dependent problems solved with higher-order finite element methods and time integration schemes sometimes encounter instabilities in time steps due to varying model parameters. This problem is commonly illustrated on a transient cavity flow modeled with Navier-Stokes equations, where for large Reynolds numbers, finite element discretizations Bu = f become unstable. The instability comes from the discrete inf-sup condition not fulfilled by the Galerkin method. To stabilize time steps, we employ a Petrov-Galerkin method BTWx = WT f with optimal test functions. However, this method commonly has two disadvantages. First, having a larger test space fixed, we must compute the matrix of coefficients of the optimal test functions W on the fly, which requires solving a system of linear equations GW = B with proper Gram matrix G each time step for varying model parameters. Second, the matrix of coefficients of optimal test functions is dense, and thus, the cost of multiplying it by other matrices BTW (which is needed) is high. To overcome these problems, we explore the Kronecker product structure of the matrix of coefficients of the optimal test functions G as well as of the matrices B resulting from the variational splitting of the time-integration scheme. Our solver can be successfully applied to the high Reynolds number Navier-Stokes equations.