On the stability of bubble functions and a stabilized mixed finite element formulation for the Stokes problem
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
D. Z. Turner, K. B. Nakshatrala, K. D. Hjelmstad
备注：
25 pages, 13 figures (The previous version was compiled by mistake with the wrong style file, the current one uses amsart, and there is no difference in the text or the figures)
In this paper we investigate the relationship between stabilized and enriched finite element formulations for the Stokes problem. We also present a new stabilized mixed formulation for which the stability parameter is derived purely by the method of weighted residuals. This new formulation allows equal order interpolation for the velocity and pressure fields. Finally, we show by counterexample that a direct equivalence between subgrid-based stabilized finite element methods and Galerkin methods enriched by bubble functions cannot be constructed for quadrilateral and hexahedral elements using standard bubble functions.
Postprocessing and Higher Order Convergence of Stabilized Finite Element Discretizations of the Stokes Eigenvalue Problem
LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算实验室）；Institute for Analysis and Computational Mathematics, Otto-von-Guericke-University Magdeburg（马格德堡奥托·冯·格里克大学分析与计算数学研究所）
In this master thesis we have compared different second order stabilized explicit Runge-Kutta methods when applied to the incompressible Navier-Stokes equations by means of a projection method and a differential algebraic approach. We explored the stability and accuracy properties of the RKC, ROCK2 and PIROCK schemes when coupled with the projection and the differential algebraic approach. PIROCK has shown unexpected instabilities, ROCK2 resulted to be the most efficient and versatile Runge-Kutta method taken into account. The differential algebraic approach sounds computationally costly but i…
In this paper we develop an a priori error analysis of a new unified mixed finite element method for the coupling of fluid flow with porous media flow in $\mathbb{R}^N$, $N\in\{2,3\}$ on isotropic meshes. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The approach utilizes a modification of the Darcy problem which allows us to apply a variant nonconforming Crouzeix-Raviart finite element to the whole coupled Stokes-Darcy problem. The w…
In this study, we present a stabilized finite element analysis for completely unified Stokes-Brinkman problems fully coupled with variable coefficient transient Advection-Diffusion-Reaction equation(VADR). As well we have carried out the stabilized finite element analysis for Stokes-Brinkman model with interface conditions fully coupled with VADR. The viscosity of the fluid, involved in flow problem, depends on the concentration of the solute, whose transport is described by VADR equation. The algebraic subgrid multiscale approach has been employed to arrive at the stabilized coupled variation…
In this paper, a stabilized extended finite element method is proposed for Stokes interface problems on unfitted triangulation elements which do not require the interface align with the triangulation. The velocity solution and pressure solution on each side of the interface are separately expanded in the standard nonconforming piecewise linear polynomials and the piecewise constant polynomials, respectively. Harmonic weighted fluxes and arithmetic fluxes are used across the interface and cut edges (segment of the edges cut by the interface), respectively. Extra stabilization terms involving ve…
In this paper a fully coupled system of transient $Navier$-$Stokes$ ($NS$) fluid flow model and variable coefficient unsteady Advection-Diffusion-Reaction ($VADR$) transport model has been studied through subgrid multiscale stabilized finite element method. In particular algebraic approach of approximating the subscales has been considered to arrive at stabilized variational formulation of the coupled system. This system is strongly coupled since viscosity of the fluid depends upon the concentration, whose transportation is modelled by $VADR$ equation. Fully implicit schemes have been consider…
A stabilized finite element method is introduced for the simulation of time-periodic creeping flows, such as those found in the cardiorespiratory systems. The new technique, which is formulated in the frequency rather than time domain, strictly uses real arithmetics and permits the use of similar shape functions for pressure and velocity for ease of implementation. It involves the addition of the Laplacian of pressure to the continuity equation with a complex-valued stabilization parameter that is derived systematically from the momentum equation. The numerical experiments show the excellent a…
Robust error estimates for stabilized finite element approximations of the two dimensional Navier-Stokes equations with application to implicit large eddy simulation
Equal order stabilized finite elements with Nitsche for stationary Navier-Stokes problem with slip boundary conditions : a priori and a posteriori error analysis
This paper considers the discretization of the time-dependent Navier-Stokes equations with the family of inf-sup stabilized Scott-Vogelius pairs recently introduced in [John/Li/Merdon/Rui, arXiv:2206.01242, 2022] for the Stokes problem. Therein, the velocity space is obtained by enriching the H^1-conforming Lagrange element space with some H(div)-conforming Raviart-Thomas functions, such that the divergence constraint is satisfied exactly. In these methods arbitrary shape-regular simplicial grids can be used. In the present paper two alternatives for discretizing the convective terms are consi…
A priori analysis for a generalized local projection stabilized conforming finite element approximation of Darcy flow and Stokes problems is presented in this paper. A first-order conforming P1 finite element space is used to approximate both the velocity and the pressure. It is shown that the stabilized discrete bilinear form satisfy the inf-sup condition with respect to a generalized local projection norm. Moreover, a priori error estimates are derived for both problems. Finally, the validation of the proposed stabilization scheme is demonstrated with appropriate numerical examples.
In this work, we consider a hybrid mixed finite element method for Biot's model. The hybrid P1-RT0-P0 discretization of the displacement-pressure-Darcy's velocity system of Biot's model presented in \cite{C. Niu} is not uniformly stable with respect to the physical parameters, resulting in some issues in numerical simulations. To alleviate such problems, following \cite{V. Girault}, we stabilize the hybrid scheme with face bubble functions and show the well-posedness with respect to physical and discretization parameters, which provide optimal error estimates of the stabilized method. We intro…
Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. II: The incompressible Navier-Stokes equations
The Volume-Averaged Navier-Stokes equations are used to study fluid flow in the presence of fixed or moving solids such as packed or fluidized beds. We develop a high-order finite element solver using both forms A and B of these equations. We introduce tailored stabilization techniques to prevent oscillations in regions of sharp gradients, to relax the Ladyzhenskaya-Babuska-Brezzi inf-sup condition, and to enhance the local mass conservation and the robustness of the formulation. We calculate the void fraction using the Particle Centroid Method. Using different drag models, we calculate the dr…
Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(Δt)^2$ leapfrog finite difference scheme run backward in time
A higher order pressure-stabilized virtual element formulation for the Stokes-Poisson-Boltzmann equations
Indian Institute of Technology Madras（印度马德拉斯理工学院）；Indian Institute of Space Science and Technology（印度空间科学与技术学院）；Istituto di Matematica Applicata e Tecnologie Informatiche Consiglio Nazionale delle Ricerche（意大利国家研究委员会应用数学与信息技术研究所）
Fully discrete parameter-robust error analysis of a grad-div stabilized Crank-Nicolson artificial compressibility method for the Navier-Stokes equations
Ocean University of China（中国海洋大学）；University of South Carolina（南卡罗来纳大学）
We model incompressible flows with an adaptive stabilized finite element method Stokes flows, which solves a discretely stable saddle-point problem to approximate the velocity-pressure pair. Additionally, this saddle-point problem delivers a robust error estimator to guide mesh adaptivity. We analyze the accuracy of different discrete velocity-pressure pairs of continuous finite element spaces, which do not necessarily satisfy the discrete inf-sup condition. We validate the framework's performance with numerical examples.
An adapted bubble approach which is a modifiation of the residual-free bubbles (RFB) method, is proposed for the Helmhotz problem in 2D. A new two-level finite element method is introduced for the approximations of the bubble functions. Unlike the other equations such as the advection-diffusion equation, RFB method when applied to the Helmholtz equation, does not depend on another stabilized method to obtain approximations to the solutions of the sub-problems. Adapted bubbles (AB) are obtained by a simple modification of the sub-problems. This modification increases the accuracy of the numeric…
In this paper we develop an a posteriori error analysis for the stationary Stokes-Darcy coupled problem approximated by conforming finite element method on isotropic meshes in $\mathbb{R}^d$, $d\in\{2,3\}$. The approach utilizes a new robust stabilized fully mixed discretization developed by Jiaping Yu et al. (Advances in Difference Equations, SpringerOpen Journal, 2018). The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution plus the stabilization terms. It is proven that the a posteriori error estimate provided in this paper is both r…
