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We present a sharp collocated projection method for solving the immiscible, two-phase Navier-Stokes equations in two- and three-dimensions. Our method is built using non-graded adaptive quadtree and octree grids, where all of the fluid…

Numerical Analysis · Mathematics 2025-08-18 Adam L. Binswanger , Matthew Blomquist , Scott R. West , Shilpa Khatri , Maxime Theillard

We introduce a collection of benchmark problems in 2D and 3D (geometry description and boundary conditions), including simple cases with known analytic solution, classical experimental setups, and complex geometries with fabricated…

Computational Engineering, Finance, and Science · Computer Science 2021-12-13 Zizhou Huang , Teseo Schneider , Minchen Li , Chenfanfu Jiang , Denis Zorin , Daniele Panozzo

In this paper, we analyse a Vector Penalty Projection Scheme (see [1]) to treat the displacement of a moving body in incompressible viscous flows in the case where the interaction of the fluid on the body can be neglected. The presence of…

Numerical Analysis · Mathematics 2016-06-15 Vincent Bruneau , Adrien Doradoux , Pierre Fabrie

We introduce a surface finite element method for the numerical solution of Navier-Stokes equations on evolving surfaces with a prescribed deformation of the surface in normal direction. The method is based on approaches for the full surface…

Numerical Analysis · Mathematics 2023-06-16 Veit Krause , Eric Kunze , Axel Voigt

We present a space-time Cut Finite Element Method (CutFEM) for the time-dependent Navier-Stokes equations involving two immiscible incompressible fluids with different viscosities, densities, and with surface tension. The numerical method…

Numerical Analysis · Mathematics 2019-03-27 Thomas Frachon , Sara Zahedi

We study a colocated cell centered finite volume method for the approximation of the incompressible Navier-Stokes equations posed on a 2D or 3D finite domain. The discrete unknowns are the components of the velocity and the pressures, all…

Numerical Analysis · Mathematics 2008-04-30 Robert Eymard , Raphaele Herbin , Jean-Claude Latché

We propose first-order pressure-correction scheme for the incompressible Navier-Stokes equations, incorporating the recently developed the Dynamically Regularized Lagrange Multiplier (DRLM) methods. The resulting algorithms are fully…

Numerical Analysis · Mathematics 2026-03-18 Yi Shen , Rihui Lan , Hua Wang

We propose and analyse an augmented mixed finite element method for the Navier--Stokes equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and no-slip boundary conditions. The weak formulation…

Numerical Analysis · Mathematics 2023-06-27 Veronica Anaya , Ruben Caraballo , Ricardo Ruiz-Baier , Hector Torres

We consider a finite volume scheme for the two-dimensional incompressible Navier-Stokes equations. We use a triangular mesh. The unknowns for the velocity and pressure are respectively piecewise constant and affine. We use a projection…

Numerical Analysis · Mathematics 2007-05-23 Sebastien Zimmermann

In this work we propose a novel Hybrid High-Order method for the incompressible Navier--Stokes equations based on a formulation of the convective term including Temam's device for stability. The proposed method has several advantageous…

Numerical Analysis · Mathematics 2018-10-11 Lorenzo Botti , Daniele Di Pietro , Jérôme Droniou

In this work we introduce and analyze a novel Hybrid High-Order method for the steady incompressible Navier-Stokes equations. The proposed method is inf-sup stable on general polyhedral meshes, supports arbitrary approximation orders, and…

Numerical Analysis · Mathematics 2018-02-26 Daniele A. Di Pietro , Stella Krell

There has been an increasing interest in developing efficient immersed boundary method (IBM) based on Cartesian grids, recently in the context of high-order methods. IBM based on volume penalization is a robust and easy to implement method…

Numerical Analysis · Mathematics 2021-07-22 Jiaqing Kou , Esteban Ferrer

We present a numerical method for the solution of interfacial growth governed by the Stefan model coupled with incompressible fluid flow. An algorithm is presented which takes special care to enforce sharp interfacial conditions on the…

Fluid Dynamics · Physics 2022-10-19 Elyce Bayat , Raphael Egan , Daniil Bochkov , Alban Sauret , Frederic Gibou

We present a second-order monolithic method for solving incompressible Navier--Stokes equations on irregular domains with quadtree grids. A semi-collocated grid layout is adopted, where velocity variables are located at cell vertices, and…

Numerical Analysis · Mathematics 2022-06-01 Hyuntae Cho , Yesom Park , Myungjoo Kang

This paper investigates a modification of the fictitious domain method with continuation in the lower-order coefficients for the unsteady Navier-Stokes equations governing the motion of an incompressible homogeneous fluid in a bounded 2D or…

Numerical Analysis · Mathematics 2025-12-23 Zhanybek Baitulenov , Maxim Olshanskii , Almas Temirbekov , Nurlan Temirbekov , Syrym Kasenov

A constructive numerical approximation of the two-dimensional unsteady stochastic Navier-Stokes equations of an incompressible fluid is proposed via a pseudo-compressibility technique involving a parameter $\epsilon$. Space and time are…

Numerical Analysis · Mathematics 2022-05-02 Jad Doghman

Stabilised mixed velocity-pressure formulations are one of the widely-used finite element schemes for computing the numerical solutions of laminar incompressible Navier-Stokes. In these formulations, the Newton-Raphson scheme is employed to…

Computational Engineering, Finance, and Science · Computer Science 2020-07-15 Chennakesava Kadapa , Wulf G Dettmer , Djordje Peric

We derive an a priori error estimate for the numerical solution obtained by time and space discretization by the finite volume/finite element method of the barotropic Navier--Stokes equations. The numerical solution on a convenient…

Numerical Analysis · Mathematics 2015-08-27 Eduard Feireisl , Radim Hošek , David Maltese , Antonín Novotný

We consider surface finite elements and a semi-implicit time stepping scheme to simulate fluid deformable surfaces. Such surfaces are modeled by incompressible surface Navier-Stokes equations with bending forces. Here, we consider closed…

Numerical Analysis · Mathematics 2023-05-03 Veit Krause , Axel Voigt

We investigate artificial compressibility (AC) techniques for the time discretization of the incompressible Navier-Stokes equations. The space discretization is based on a lowest-order face-based scheme supporting polytopal meshes, namely…

Numerical Analysis · Mathematics 2021-05-04 Riccardo Milani , Jérôme Bonelle , Alexandre Ern