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

Abstract

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 NavierStokes equations, where for large Reynolds numbers, finite element discretizations B�� = �� 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 B ^��W�� = W^�� �� 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 B ^��W (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.