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A recent paper [J. A. Evans, D. Kamensky, Y. Bazilevs, "Variational multiscale modeling with discretely divergence-free subscales", Computers & Mathematics with Applications, 80 (2020) 2517-2537] introduced a novel stabilized finite element…

Numerical Analysis · Mathematics 2021-12-21 Sajje Lee Calfy , John A. Evans , David Kamensky

We study a generalization of the Navier-Stokes-Fourier system for an incompressible fluid where the deviatoric part of the Cauchy stress tensor is related to the symmetric part of the velocity gradient via a maximal monotone 2-graph that is…

Analysis of PDEs · Mathematics 2017-05-02 Erika Maringová , Josef Žabenský

We consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14 259 degrees of freedom. The steady-state snapshot solu- tions…

Numerical Analysis · Mathematics 2017-12-19 Martin Hess , Gianluigi Rozza

We propose and analyze unfitted finite element approximations for the two-phase incompressible Navier--Stokes flow in an axisymmetric setting. The discretized schemes are based on an Eulerian weak formulation for the Navier--Stokes equation…

Numerical Analysis · Mathematics 2023-09-12 Harald Garcke , Robert Nürnberg , Quan Zhao

We present a novel framework based on semi-bounded spatial operators for analyzing and discretizing initial boundary value problems on moving and deforming domains. This development extends an existing framework for well-posed problems and…

Numerical Analysis · Mathematics 2023-02-14 Tomas Lundquist , Arnaud Malan , Jan Nordström

We derive a linear model of navigation in a two-layer fluid with a variable velocity of the ship. A spectral version of the model including a Rayleigh damping term is analyzed. We prove that the Cauchy problem has a unique solution if the…

Analysis of PDEs · Mathematics 2025-12-02 Zeina Rammal , Matthieu Brachet , Germain Rousseaux , Morgan Pierre

Both Newtonian and non-Newtonian fluids may exhibit complex slip behaviour at the boundary. We examine a broad class of slip boundary conditions that generalises the commonly used Navier slip, perfect slip, stick-slip and Tresca friction…

Numerical Analysis · Mathematics 2025-02-14 Pablo Alexei Gazca-Orozco , Franz Gmeineder , Erika Maringová Kokavcová , Tabea Tscherpel

The Navier--Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretization. We analyse the semi-discrete equations --…

Numerical Analysis · Mathematics 2024-12-20 Robert Altmann , Jan Heiland

An immersed-boundary method for the incompressible Navier--Stokes equations is presented. It employs discrete forcing for a sharp discrimination of the solid-fluid interface, and achieves second-order accuracy, demonstrated in examples with…

A conforming finite element scheme with mixed explicit-implicit time discretization for quasi-incompressible Navier-Stokes-Maxwell-Stefan systems in a bounded domain with periodic boundary conditions is presented. The system consists of the…

Numerical Analysis · Mathematics 2026-02-05 Aaron Brunk , Ansgar Jüngel , Maria Lukáčová-Medvid'ová

This paper presents robust discontinuous Galerkin methods for the incompressible Navier-Stokes equations on moving meshes. High-order accurate arbitrary Lagrangian-Eulerian formulations are proposed in a unified framework for both…

Computational Physics · Physics 2021-03-17 Niklas Fehn , Johannes Heinz , Wolfgang A. Wall , Martin Kronbichler

We study well-posedness of a velocity-vorticity formulation of the Navier--Stokes equations, supplemented with no-slip velocity boundary conditions, a no-penetration vorticity boundary condition, along with a natural vorticity boundary…

Analysis of PDEs · Mathematics 2017-08-09 Maxim A. Olshanskii , Leo G. Rebholz , Abner J. Salgado

In this study, a shape optimization problem for the two-dimensional stationary Navier--Stokes equations with an artificial boundary condition is considered. The fluid is assumed to be flowing through a rectangular channel, and the…

Optimization and Control · Mathematics 2021-08-10 John Sebastian H. Simon , Hirofumi Notsu

We propose a finite element discretization for the steady, generalized Navier-Stokes equations for fluids with shear-dependent viscosity, completed with inhomogeneous Dirichlet boundary conditions and an inhomogeneous divergence constraint.…

Numerical Analysis · Mathematics 2023-10-09 Julius Jeßberger , Alex Kaltenbach

We present a parametric finite element approximation of two-phase flow with insoluble surfactant. This free boundary problem is given by the Navier--Stokes equations for the two-phase flow in the bulk, which are coupled to the transport…

Numerical Analysis · Mathematics 2015-06-02 John W. Barrett , Harald Garcke , Robert Nürnberg

In this paper, we introduce a novel way to represent the interface for two-phase flows with phase change. We combine a level-set method with a Cartesian embedded boundary method and take advantage of both. This is part of an effort to…

Numerical Analysis · Mathematics 2022-12-28 A. Limare , S. Popinet , C. Josserand , Z. Xue , A. Ghigo

In this paper we developed a numerical methodology to study some incompressible fluid flows without free surface, using the curvilinear coordinate system and whose edge geometry is constructed via parametrized spline. First, we discussed…

Computational modeling and simulation of fluid-structure interactions constitute a fundamental cornerstone for advancing aerospace engineering endeavors. This paper addresses the notion and implementation of the immersed boundary method for…

Computational Physics · Physics 2025-12-24 Longqing Ge , Qingdong Cai , Yonghao Zhang , Tianbai Xiao

We develop an immersed boundary (IB) method for modeling flows around fixed or moving rigid bodies that is suitable for a broad range of Reynolds numbers, including steady Stokes flow. The spatio-temporal discretization of the fluid…

Numerical Analysis · Mathematics 2016-03-02 B. Kallemov , A. Pal Singh Bhalla , B. E. Griffith , A. Donev

The choice of the boundary conditions in mechanical problems has to reflect the interaction of the considered material with the surface, despite the assumption of the no-slip condition is preferred to avoid boundary terms in the analysis…

Analysis of PDEs · Mathematics 2020-09-22 Anna Abbatiello , Miroslav Bulíček , Erika Maringová