A combined method for computing the field of the point source in a waveguide (plane-layered media).
A comparison of the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method for solutions of partial differential equations
We compare numerical experiments from the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method, applied to three benchmark problems based on two different partial differential equations. Both methods are described in detail and we highlight some strengths and weaknesses of each method via the numerical comparisons. The two equations used in the benchmark problems are the viscous Burgers’ equation and the porous medium equation, both...
A Computer Algebra Application to Determination of Lie Symmetries of Partial Differential Equations
The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006A MATHEMATICA package for finding Lie symmetries of partial differential equations is presented. The package is designed to create and solve the associated determining system of equations, the full set of solutions of which generates the widest permissible local Lie group of point symmetry transformations. Examples illustrating the functionality of the package's tools...
A confinement result for axisymmetric fluids
A conservative finite difference scheme for static diffusion equation.
A construction of quasi-modes using coherent states
A Constructive Method for Deriving Finite Elements of Nodal Type.
A continuity property for the inverse of Mañé's projection
Let be a compact subset of a separable Hilbert space with finite fractal dimension , and an orthogonal projection in of rank greater than or equal to . For every , there exists an orthogonal projection in of the same rank as , which is injective when restricted to and such that . This result follows from Mañé’s paper. Thus the inverse of the restricted mapping is well defined. It is natural to ask whether there exists a universal modulus of continuity for the inverse of Mañé’s...
A controllability result for the -D isentropic Euler equation
A counterexample to the smoothness of the solution to an equation arising in fluid mechanics
We analyze the equation coming from the Eulerian-Lagrangian description of fluids. We discuss a couple of ways to extend this notion to viscous fluids. The main focus of this paper is to discuss the first way, due to Constantin. We show that this description can only work for short times, after which the ``back to coordinates map'' may have no smooth inverse. Then we briefly discuss a second way that uses Brownian motion. We use this to provide a plausibility argument for the global regularity for...
A diffuse interface fractional time-stepping technique for incompressible two-phase flows with moving contact lines
For a two phase incompressible flow we consider a diffuse interface model aimed at addressing the movement of three-phase (fluid-fluid-solid) contact lines. The model consists of the Cahn Hilliard Navier Stokes system with a variant of the Navier slip boundary conditions. We show that this model possesses a natural energy law. For this system, a new numerical technique based on operator splitting and fractional time-stepping is proposed. The method is shown to be unconditionally stable. We present...
A direct approach to the Weiss conjecture for bounded analytic semigroups
We give a new proof of the Weiss conjecture for analytic semigroups. Our approach does not make any recourse to the bounded -calculus and is based on elementary analysis.
A direct boundary integral method for a mobility problem.
A direct proof of the Caffarelli-Kohn-Nirenberg theorem
In the present paper we give a new proof of the Caffarelli-Kohn-Nirenberg theorem based on a direct approach. Given a pair (u,p) of suitable weak solutions to the Navier-Stokes equations in ℝ³ × ]0,∞[ the velocity field u satisfies the following property of partial regularity: The velocity u is Lipschitz continuous in a neighbourhood of a point (x₀,t₀) ∈ Ω × ]0,∞ [ if for a sufficiently small .
A discrete variational approach for investigation of stationary localized states in a discrete nonlinear Schrödinger equation, named IN-DNLS.
A few symmetry results for nonlinear elliptic PDE on noncompact manifolds
A field equation defined by a Hurwitz pair
A finite element convergence analysis for 3D Stokes equations in case of variational crimes
We investigate a finite element discretization of the Stokes equations with nonstandard boundary conditions, defined in a bounded three-dimensional domain with a curved, piecewise smooth boundary. For tetrahedral triangulations of this domain we prove, under general assumptions on the discrete problem and without any additional regularity assumptions on the weak solution, that the discrete solutions converge to the weak solution. Examples of appropriate finite element spaces are given.
A finite element method for approximating the time-harmonic Maxwell equations.
A Fortin operator for two-dimensional Taylor-Hood elements
A standard method for proving the inf-sup condition implying stability of finite element approximations for the stationary Stokes equations is to construct a Fortin operator. In this paper, we show how this can be done for two-dimensional triangular and rectangular Taylor-Hood methods, which use continuous piecewise polynomial approximations for both velocity and pressure.