General 2D Domain Research Articles

Oscillatory translational instabilities of spot patterns in the Schnakenberg system on general 2D domains

For a bounded 2D planar domain Ω, we investigate the impact of domain geometry on oscillatory translational instabilities of N-spot equilibrium solutions for a singularly perturbed Schnakenberg reaction-diffusion system with activator-inhibitor diffusivity ratio. An N-spot equilibrium is characterized by an activator concentration that is exponentially small everywhere in Ω except in N well-separated localized regions of extent. We use the method of matched asymptotic analysis to analyze Hopf bifurcation thresholds above which the equilibrium becomes unstable to translational perturbations, which result in -frequency oscillations in the locations of the spots. We find that stability to these perturbations is governed by a nonlinear matrix-eigenvalue problem, the eigenvector of which is a 2N-vector that characterizes the possible modes (directions) of oscillation. The matrix contains terms associated with a certain Green’s function on Ω, which encodes geometric effects. For the special case of a perturbed disk with radius in polar coordinates with , , and 2π-periodic, we show that only the mode-2 coefficients of the Fourier series of f impact the bifurcation threshold at leading order in σ. We further show that when , the dominant mode of oscillation is in the direction parallel to the longer axis of the perturbed disk. Numerical investigations on the full Schnakenberg PDE are performed for various domains Ω and N-spot equilibria to confirm asymptotic results and also to demonstrate how domain geometry impacts thresholds and dominant oscillation modes.

Plasmonic eigenvalue problem for corners: Limiting absorption principle and absolute continuity in the essential spectrum

We consider the plasmonic eigenvalue problem for a general 2D domain with a curvilinear corner, studying the spectral theory of the Neumann–Poincaré operator of the boundary. A limiting absorption principle is proved, valid when the spectral parameter approaches the essential spectrum. Putting the principle into use, it is proved that the corner produces absolutely continuous spectrum of multiplicity 1. The embedded eigenvalues are discrete. In particular, there is no singular continuous spectrum.

Function Approximation on Arbitrary Domains Using Fourier Extension Frames

Fourier extension is an approximation scheme in which a function on an arbitrary bounded domain is approximated using classical Fourier series on a bounding box. On the domain of interest the Fourier series exhibits redundancy, and it has the mathematical structure of a frame rather than a basis. It is not trivial to construct approximations in this frame using function evaluations in points that belong to the domain only, but one way to do so is through a discrete least squares approximation. The corresponding system is extremely ill-conditioned, due to the redundancy in the frame, yet its solution via a regularized singular value decomposition is known to be accurate to very high (and nearly spectral) precision. Still, this computation requires $\mathcal{O}(N^3)$ operations. In this paper we describe an algorithm to compute such Fourier extension frame approximations in only $\mathcal{O}(N^2\log^2N)$ operations for general 2D domains. The cost improves to $\mathcal{O}(N\log^2N)$ operations for simpler tensor-product domains. The algorithm exploits a phenomenon called the plunge region in the analysis of time-frequency localization operators, which manifests itself here as a sudden drop in the singular values of the least squares matrix. It is known that the size of the plunge region scales like $\mathcal{O}(\log N)$ in 1D problems. In this paper we show that for most 2D domains in the fully discrete case the plunge region scales like $\mathcal{O}(N\log N)$, proving a discrete equivalent of a result that was conjectured by Widom for a related continuous problem. The complexity estimate depends on the Minkowski or box-counting dimension of the domain boundary, and as such it is larger than $\mathcal{O}(N\log N)$ for domains with fractal shape.

On the stability of medial axis transform

Medial axis transform (MAT) is very sensitive to noise, in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of the original shape and the perturbed one may be large. Recently, Choiet al. (2002) showed that MAT is stable for a class of 2D domains called weakly injective, if we view this phenomenon with the one-sided Hausdorff distance, rather than with the two-sided Hausdorff distance. In this paper, we extend this result to general 2D domains with natural boundary regularity. We also present explicit bounds for this general one-sided stability of the 2D MAT.

Asymptotic behavior of the Navier–Stokes flow in a general 2D domain

In this work, we consider the convergence of the Navier–Stokes flow to the Euler flow at small viscosity in a general 2D domain and derive some explicit asymptotic formulas for the convergence at small viscosity under some smoothness assumptions on the solution of the Euler flow.

On coupling wavelets with fictitious domain approaches

We define and analyse a numerical algorithm for the approximation of parabolic equations on a general 2D domain with Dirichlet boundary conditions. It couples wavelet approximations with fictitious domain surface Lagrange multiplier approaches. This algorithm turns out to be precise, fast and numerically efficient.

On the approximation of Maxwell's eigenproblem in general 2D domains

In this paper we review some finite element methods to approximate the eigenvalues of Maxwell equations. The numerical schemes we are going to consider are based on two different variational formulations. Our aim is to compare the performances of the methods depending on the shape of the domain. We shall see that the nodal elements can give good results only using the penalized formulation and only if the domain is a convex or smooth polygon. In the case of domains with reentrant corners it turns out that the edge elements are efficient. Moreover we propose a new non-standard finite element method in order to deal with the penalized formulation in presence of reentrant corners: a biquadratic element with a suitable projection.

A simulation tool for the analysis of high speed flows

The availability of an efficient, low-cost numerical simulator is essential in the design of fluid dynamic systems, both to achieve a deep understanding of the flow field and to allow a quick and economical optimization of the technical characteristics of industrial devices. With the aim of developing an adequate simulation tool for the analysis of the fluid dynamical field in general 2d domains, a positive scheme, recently proposed as a simple and yet powerful method for the approximation of conservation laws, is extended in this paper to general curvilinear coordinates. The resulting algorithm is successfully applied to several test cases of increasing complexity.

Asymptotic analysis of the linearized Navier–Stokes equations in a general 2D domain

We continue our study of the asymptotic behavior of the Navier-Stokes equations linearized around the rest state as viscostiy approaches zero. We study the convergence as ! 0 to the inviscid type equations. Suitable correctors are obtained which resolve the boundary layer and we obtain convergence results valid up to the boundary. Explicit asymptotic expansion formulas are given which display the boundary layer phenomena. We improve our previous by treating here the general smooth bounded domain in R 2 instead of two-dimensional channels. Curvilinear coordinates are used to resolve the complex geometry.

L iSS — An environment for the parallel multigrid solution of partial differential equations on general 2D domains

L iSS is a very general package for the parallel multigrid solution of nonlinear partial differential equations. This paper gives the reader an impression of what L iSS can do. It includes the description of two important current perspectives, the implementation of adaptive structures and the generalization to 3D. An overview of numerical and parallel results is given.