Increasing accuracy and efficiency in FE computations of the Leray‐Deconvolution model

Abstract

This article develops, analyzes, and tests a finite element method for approximating solutions to the Leraydeconvolution regularization of the Navier-Stokes equations. The scheme combines three ideas to create an accurate and effective algorithm: the use of an incompressible filter, a linearization that decouples the velocity-pressure system from the filtering and deconvolution operations, and a stabilization that works well with the linearization. A rigorous and complete numerical analysis of the scheme is given, and numerical experiments are presented that show clear advantages of the scheme.

A numerical study of the Navier–Stokes-αβ model

“…Leray-type models do [32,2], for example). Hence, for unconditionally stable computations, one must solve the coupled nonlinear system.…”

“…They can arise, for example, as approximations to Gaussian filters of high qualitative and quantitative accuracy [17]. The discrete filter we choose is a direct finite-element implementation of (1.12) and (1.13) and is the same filter as that used in [39,37,11,2].…”

Increasing Accuracy and Efficiency for Regularized Navier-Stokes Equations

“…An attractive alternative is to define the differential filter by a discrete Stokes problem so as to preserve incompressibility approximately [6,8,27]. In the case of internal flows under no slip boundary conditions we must use the discrete Stokes differential filter to preserve discrete incompressibility.…”

“…is a stabilization term that comes with the linearization method and it corresponds to adding ≈ − tu t to the continuous model, [6,20]. (3.20) and note that the trilinear term vanishes leaving…”

“…Our goal is also to illuminate the connections between δ, h, χ , ν, and the fully discretized algorithms used to compute the fluctuation u . We impose a more consistent discretization of (1.1)-(1.2) via an incompressible filter and add a stabilized linearization, following the work from [6]. Thus, the proposed scheme for the time relaxation model (1.1)-(1.2) incorporates three ideas: incompressible filter for better consistency outside of the periodic domains, the efficient linearization of Baker [3] that allows to solve only one linear system per time step with second order of accuracy, and the stabilization term that is natural for this linearization [20].…”