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Publication# Interior penalty finite element approximation of Navier-Stokes equations and application to free surface flows

Abstract

In the present work, we investigate mathematical and numerical aspects of interior penalty finite element methods for free surface flows. We consider the incompressible Navier-Stokes equations with variable density and viscosity, combined with a front capturing model using the level set method. We formulate interior penalty finite element methods for both the Navier-Stokes equations and the level set advection equation. For the two-fluid Stokes equations, we propose and analyze an unfitted finite element scheme with interior penalty. Optimal a priori error estimates for the velocity and the pressure are proved in the energy norm. A preconditioning strategy with adaptive reuse of incomplete factorizations as preconditioners for Krylov subspace methods is introduced and applied for solving the linear systems. Different and complementary solutions for reducing the matrix assembly time and the memory consumption are proposed and tested, each of which is applicable in general in the context of either multiphase flow or interior penalty stabilization. As level set reinitialization method, we apply a combination of the interface local projection and a fast marching scheme. We provide for the latter a reformulation of the distance computation algorithm on unstructured simplicial meshes in any spatial dimension, allowing for both an efficient implementation and geometric insight. We present and discuss numerical solutions of reference problems for the one-fluid Navier-Stokes equations and for the level set advection problem. Solutions of benchmark problems in two and three dimensions involving one or two fluids are then approximated, and the results are compared to literature values. Finally, we describe software design techniques and abstractions for the efficient and general implementation of the applied methods.

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Navier–Stokes equations

The Navier–Stokes equations (nævˈjeː_stəʊks ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842-1850 (Stokes). The Navier–Stokes equations mathematically express momentum balance and conservation of mass for Newtonian fluids.

Linear system

In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be modeled by linear systems.

Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it - for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it - for example, both a latitude and longitude are required to locate a point on the surface of a sphere.