The numerical implementation of finite element discretization method for the stream function formulation of a linearized Navier-Stokes equations is considered. Algorithm 1 is applied using Argyris element. Three global orderings of nodes are selected and registered in order to conclude the best banded structure of matrix and a fluid flow calculation is considered to test a problem which has a known solution. Visualization of global node orderings, matrix sparsity patterns and stream function contours are displayed showing the main features of the flow.
In this paper, we present linearized learning methods to accelerate the convergence of training for stationary nonlinear Navier-Stokes equations. To solve the stationary nonlinear Navier-Stokes (NS) equation, we integrate the procedure of linearization of the nonlinear convection term in the NS equation into the training process of multi-scale deep neural network approximation of the NS solution. Four forms of linearizations are considered. After a benchmark problem, we solve the highly oscillating stationary flows utilizing the proposed linearized learning with multi-scale neural network for …
In this article, we design and analyze an arbitrary-order stabilized finite element method to approximate the unique continuation problem for laminar steady flow described by the linearized incompressible Navier--Stokes equation. We derive quantitative local error estimates for the velocity, which account for noise level and polynomial degree, using the stability of the continuous problem in the form of a conditional stability estimate. Numerical examples illustrate the performances of the method with respect to the polynomial order and perturbations in the data. We observe that the higher ord…
We propose, analyze, and investigate numerically a novel two-level Galerkin reduced order model (2L-ROM) for the efficient and accurate numerical simulation of the steady Navier-Stokes equations. In the first step of the 2L-ROM, a relatively low-dimensional nonlinear system is solved. In the second step, the Navier-Stokes equations are linearized around the solution found in the first step, and a higher-dimensional system for the linearized problem is solved. We prove an error bound for the new 2L-ROM and compare it to the standard one level ROM (1L-ROM) in the numerical simulation of the stea…
We study a low-rank iterative solver for the unsteady Navier-Stokes equations for incompressible flows with a stochastic viscosity. The equations are discretized using the stochastic Galerkin method, and we consider an all-at-once formulation where the algebraic systems at all the time steps are collected and solved simultaneously. The problem is linearized with Picard's method. To efficiently solve the linear systems at each step, we use low-rank tensor representations within the Krylov subspace method, which leads to significant reductions in storage requirements and computational costs. Com…
A linearized energy preserving finite element method for the dynamical incompressible magnetohydrodynamics equations
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Department of Mathematics, City University of Hong Kong（香港城市大学数学系）
For the discretization of the convective term in the Navier-Stokes equations (NSEs), the commonly used convective formulation (CONV) does not preserve the energy if the divergence constraint is only weakly enforced. In this paper, we apply the skew-symmetrization technique in [B. Cockburn, G. Kanschat and D. Schötzau, Math. Comp., 74 (2005), pp. 1067-1095] to conforming finite element methods, which restores energy conservation for CONV. The crucial idea is to replace the discrete advective velocity with its a $H(\operatorname{div})$-conforming divergence-free approximation in CONV. We prove t…
Unconditionally optimal error Estimate of a linearized Second-order Fully Discrete Finite Element Method for the bioconvection flows with concentration dependent viscosity
School of Mathematical Sciences, East China Normal University（华东师范大学数学科学学院）；School of Mathematical Sciences, Ministry of Education Key Laboratory of Mathematics and Engineering Applications, Shanghai Key Laboratory of PMMP, East China Normal University（华东师范大学数学科学学院，教育部数学与工程应用重点实验室，上海市PMMP重点实验室）
A vectorial lattice Boltzmann scheme for the incompressible Navier-Stokes equations
Université de Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251（波尔多大学）；Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire EM2C & Fédération de Mathématiques de CentraleSupélec（巴黎萨克雷大学）；Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche（意大利国家研究委员会计算应用研究所）；Laboratoire J. A. Dieudonné, Université Côte d’Azur, CNRS（蔚蓝海岸大学）；Dipartimento di Matematica “Guido Castelnuovo”, Sapienza Università di Roma（罗马第一大学）
A decoupled energy-stable mixed finite element method for Poisson-Nernst-Planck-Navier-Stokes equations
School of Mathematics and Computational Science, Guilin University of Electronic Technology（桂林电子科技大学数学与 computational 科学学院）；Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation（广西高校数据分析与计算重点实验室）；Guangxi Applied Mathematics Center (GUET)（广西应用数学中心（桂林电子科技大学））
The flow regime of micro flow varies from collisionless regime to hydrodynamic regime according to the Knudsen number. On the kinetic scale, the dynamics of micro flow can be described by the linearized kinetic equation. In the continuum regime, hydrodynamic equations such as linearized Navier-Stokes equations and Euler equations can be derived from the linearized kinetic equation by the Chapman-Enskog asymptotic analysis. In this paper, based on the linearized kinetic equation we are going to propose a unified gas kinetic scheme scheme (UGKS) for micro flow simulation, which is an effective m…
