Search

We propose new, efficient, and simple nonlinear iteration methods for the incompressible Navier-Stokes equations. The methods are constructed by applying Yosida-type algebraic splitting to the linear systems that arise from grad-div stabilized finite element implementations of incremental Picard and Newton iterations. They are efficient because at each nonlinear iteration, the same symmetric positive definite Schur complement system needs to be solved, which allows for CG to be used for inner and outer solvers, simple preconditioning, and the reusing of preconditioners. For the proposed incremental Picard-Yosida and Newton-Yosida iterations, we prove under small data conditions that the methods converge to the solution of the discrete nonlinear problem. Numerical tests are performed which illustrate the effectiveness of the method on a variety of test problems. [ABSTRACT FROM AUTHOR]

This paper continues some recent work on the numerical solution of the steady incompressible Navier–Stokes equations. We present a new method, similar to the one presented in Rebholz et al., but with superior convergence and numerical properties. The method is efficient as it allows one to solve the same symmetric positive‐definite system for the pressure at each iteration, allowing for the simple preconditioning and the reuse of preconditioners. We also demonstrate how one can replace the Schur complement system with a diagonal matrix inversion while maintaining accuracy and convergence, at a small fraction of the numerical cost. Convergence is analyzed for Newton and Picard‐type algorithms, as well as for the Schur complement approximation. [ABSTRACT FROM AUTHOR]

The unsteady incompressible Navier–Stokes equations in primitive variables are often numerically solved by segregating the computation of velocity and pressure, according to either functional analysis arguments following the pioneering work of A.J. Chorin and R. Temam (STD, Split-Then-Discretize paradigm) or linear algebra arguments based on the inexact block factorization of the discretized problem (DTS, Discretize-Then-Split paradigm). The presence of the time derivative allows for the calibration of an appropriate approximation of the pseudo-differential operator of the pressure problem and excellent results in terms of both accuracy and efficiency have been obtained as witnessed by the abundant literature. The extension of the same segregated approach to the steady Navier–Stokes equations is unclear, unless a pseudo-time advancing formulation is undertaken. In this paper we present a methodology for a segregated computation of the primitive variables in a genuinely steady formulation, so to avoid iterations to get to the steady limit. The approach is largely inspired by the algebraic factorization of the unsteady problem (DTS approach), yet we detail specific settings required by the absence of the velocity time-derivative. The basic idea relies on the introduction of some parameters in a modified Picard linearization. We discuss stability bounds and the convergence of the segregated method to the unsplit solution. Several numerical results on different test cases confirm the efficiency of the procedure. [ABSTRACT FROM AUTHOR]

Algebraic splitting methods are a common approach to solving the saddle point linear systems that arise at each time step of an incompressible flow simulation. There are two main classes of these methods, those of Yosida-type and those of Algebraic Chorin Temam (ACT)-type, with the Yosida methods being predominantly used in practice. We show herein, through new analysis and extensive numerical testing, that ACT methods that include the viscous term stiffness matrix in the modified A-block are unconditionally stable and can be superior to Yosida methods in a range of problems. Particular situations where the ACT-type solvers are advantageous include problems where numerical stability is a concern, as well as problems where strong enforcement of the divergence constraint is important. • ACT splitting can preserve the unconditional stability of BDF2-FEM and BE-FEM methods. • Analysis includes incremental formulation and pressure correction for higher accuracy. • We show ACT methods preserve the FEM divergence constraint. • Our analysis is in the function space setting with appropriate norms. • We give results of several numerical tests that compare Yosida and ACT. [ABSTRACT FROM AUTHOR]