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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

Abigail Wacher (2013)

Open Mathematics

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 conservative spectral element method for the approximation of compressible fluid flow

Kelly Black (1999)

Kybernetika

A method to approximate the Euler equations is presented. The method is a multi-domain approximation, and a variational form of the Euler equations is found by making use of the divergence theorem. The method is similar to that of the Discontinuous-Galerkin method of Cockburn and Shu, but the implementation is constructed through a spectral, multi-domain approach. The method is introduced and is shown to be a conservative scheme. A numerical example is given for the expanding flow around a point...

A continuity property for the inverse of Mañé's projection

Zdeněk Skalák (1998)

Applications of Mathematics

Let X be a compact subset of a separable Hilbert space H with finite fractal dimension d F ( X ) , and P 0 an orthogonal projection in H of rank greater than or equal to 2 d F ( X ) + 1 . For every δ > 0 , there exists an orthogonal projection P in H of the same rank as P 0 , which is injective when restricted to X and such that ∥ P - P 0 ∥ < δ . This result follows from Mañé’s paper. Thus the inverse ( P | X ) - 1 of the restricted mapping P | X X → P X is well defined. It is natural to ask whether there exists a universal modulus of continuity for the inverse of Mañé’s...

A continuous finite element method with face penalty to approximate Friedrichs' systems

Erik Burman, Alexandre Ern (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

A continuous finite element method to approximate Friedrichs' systems is proposed and analyzed. Stability is achieved by penalizing the jumps across mesh interfaces of the normal derivative of some components of the discrete solution. The convergence analysis leads to optimal convergence rates in the graph norm and suboptimal of order ½ convergence rates in the L2-norm. A variant of the method specialized to Friedrichs' systems associated with elliptic PDE's in mixed form and reducing the number...

A counterexample to the smoothness of the solution to an equation arising in fluid mechanics

Stephen Montgomery-Smith, Milan Pokorný (2002)

Commentationes Mathematicae Universitatis Carolinae

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 coupled well-balanced and random sampling scheme for computing bubble oscillations*

Philippe Helluy, Jonathan Jung (2012)

ESAIM: Proceedings

We propose a finite volume scheme to study the oscillations of a spherical bubble of gas in a liquid phase. Spherical symmetry implies a geometric source term in the Euler equations. Our scheme satisfies the well-balanced property. It is based on the VFRoe approach. In order to avoid spurious pressure oscillations, the well-balanced approach is coupled with an ALE (Arbitrary Lagrangian Eulerian) technique at the interface and a random sampling remap.

A degenerate parabolic system for three-phase flows in porous media

Vladimir Shelukhin (2007)

Annales mathématiques Blaise Pascal

A classical model for three-phase capillary immiscible flows in a porous medium is considered. Capillarity pressure functions are found, with a corresponding diffusion-capillarity tensor being triangular. The model is reduced to a degenerate quasilinear parabolic system. A global existence theorem is proved under some hypotheses on the model data.

A diffuse interface fractional time-stepping technique for incompressible two-phase flows with moving contact lines

Abner J. Salgado (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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 proof of the Caffarelli-Kohn-Nirenberg theorem

Jörg Wolf (2008)

Banach Center Publications

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 l i m s u p R → 0 ⁺ 1 / R ∫ Q R ( x ₀ , t ₀ ) | c u r l u × u / | u | | ² d x d t ≤ ε * for a sufficiently small ε * > 0 .

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