This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

A conservative level set method for two phase flow II

https://hal.archives-ouvertes.fr/hal-01445445/document

An accurate adaptive solver for surface-tension-driven interfacial flows

Boundary Control of PDEs

http://www.lmm.jussieu.fr/~zaleski/Papers/vof3D.pdf

Interface reconstruction with least-squares fit and split advection in three-dimensional Cartesian geometry

A mixed markers and volume-of-fluid method for the reconstruction and advection of interfaces in two-phase and free-boundary flows

We consider the mathematical formulation, analysis, and the numerical solution of a time-dependent optimal control problem associated with the tracking of the velocity of a Navier--Stokes flow in a bounded two-dimensional domain through the adjustment of a distributed control. The existence of optimal solutions is proved and the first-order necessary conditions for optimality are used to derive an optimality system of partial differential equations whose solutions provide optimal states and controls. Semidiscrete-in-time and fully discrete space-time approximations are defined and their convergence to the exact optimal solutions is shown. A gradient method for the solution of the fully discrete equations is examined, as are its convergence properties. Finally, the results of some illustrative computational experiments are presented.

/pdf/analysis-and-approximation-of-the-velocity-tracking-problem-2hyiaowai2.pdf

Analysis and Approximation of the Velocity Tracking Problem for Navier--Stokes Flows with Distributed Control

A geometrical area-preserving volume-of-fluid advection method

A new class of algorithms that preserve mass exactly for incompressible flows on a Cartesian mesh are presented. They amount to piecewise-linear, area-preserving mappings of tessellations of the plane. They are equivalent to Volume-of-Fluid (VOF) advection methods which are decomposed into an Eulerian implicit scheme in one direction followed by a Lagrangian explicit step in the other one. It is demonstrated that mass conservation is exact for incompressible flows and that there are no undershoots or overshoots of the volume fraction which thus always remains constrained between 0 and 1. 2003 Elsevier B.V. All rights reserved. AMS: 65M05; 76M20; 76T99

Sandro Manservisi

Papers

Interface reconstruction with least-squares fit and split advection in three-dimensional Cartesian geometry

A mixed markers and volume-of-fluid method for the reconstruction and advection of interfaces in two-phase and free-boundary flows

Analysis and Approximation of the Velocity Tracking Problem for Navier--Stokes Flows with Distributed Control

A geometrical area-preserving volume-of-fluid advection method

Short Note A geometrical area-preserving Volume-of-Fluid advection method