Choice Of Stabilization Parameters Research Articles

Fully Computable a Posteriori Error Bounds for Hybridizable Discontinuous Galerkin Finite Element Approximations

We derive a posteriori error estimates for the hybridizable discontinuous Galerkin (HDG) methods, including both the primal and mixed formulations, for the approximation of a linear second-order elliptic problem on conforming simplicial meshes in two and three dimensions. We obtain fully computable, constant free, a posteriori error bounds on the broken energy seminorm and the HDG energy (semi)norm of the error. The estimators are also shown to provide local lower bounds for the HDG energy (semi)norm of the error up to a constant and a higher-order data oscillation term. For the primal HDG methods and mixed HDG methods with an appropriate choice of stabilization parameter, the estimators are also shown to provide a lower bound for the broken energy seminorm of the error up to a constant and a higher-order data oscillation term. Numerical examples are given illustrating the theoretical results.

LU factorizations and ILU preconditioning for stabilized discretizations of incompressible Navier–Stokes equations

SummaryThe paper studies numerical properties of LU and incomplete LU factorizations applied to the discrete linearized incompressible Navier–Stokes problem also known as the Oseen problem. A commonly used stabilized Petrov–Galerkin finite element method for the Oseen problem leads to the system of algebraic equations having a 2 × 2‐block structure. While enforcing better stability of the finite element solution, the Petrov–Galerkin method perturbs the saddle‐point structure of the matrix and may lead to less favorable algebraic properties of the system. The paper analyzes the stability of the LU factorization. This analysis quantifies the effect of the streamline upwind Petrov–Galerkin stabilization in terms of the perturbation made to a nonstabilized system. The further analysis shows how the perturbation depends on the particular finite element method, the choice of stabilization parameters, and flow problem parameters. The analysis of LU factorization and its stability helps to understand the properties of threshold ILU factorization preconditioners for the system. Numerical experiments for a model problem of blood flow in a coronary artery illustrate the performance of the threshold ILU factorization as a preconditioner. The dependence of the preconditioner properties on the stabilization parameters of the finite element method is also studied numerically.

Assessment of a vorticity based solver for the Navier–Stokes equations

We investigate numerically a recently proposed vorticity based formulation of the incompressible Navier–Stokes equations. The formulation couples a velocity–pressure system with a vorticity–helicity system, and is intended to provide a numerical scheme with enhanced accuracy and superior conservation properties. For a few benchmark problems, we study the performance of a finite element method for this formulation and compare it with the commonly used velocity–pressure based finite element method. It is shown that both steady and unsteady discrete problems in the new formulation admit simple decoupling strategies followed by the application of iterative solves to auxiliary subproblems. Further, we compare several iterative strategies to solve the discrete problems and study the interplay between the choice of stabilization parameters in the finite element method and the efficiency of linear algebra solvers.