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Open Access
January 1, 2011
Abstract
We study an efficient finite element method for the NS-ω model, that uses van Cittert approximate deconvolution to improve accuracy and Scott-Vogelius elements to provide pointwise mass conservative solutions and remove the dependence of the (often large) Bernoulli pressure error on the velocity error. We provide a complete numerical analysis of the method, including well-posedness, unconditional stability, and optimal convergence. Additionally, an improved choice of filtering radius (versus the usual choice of the average mesh width) for the scheme is found, by identifying a connection with a scheme for the velocity-vorticity-helicity NSE formulation. Several numerical experiments are given that demonstrate the performance of the scheme, and the improvement offered by the new choice of the filtering radius.
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Open Access
January 1, 2011
Abstract
Mimetic discretization methods for the numerical solution of continuum mechanics problems directly use vector calculus and differential forms identities for their derivation and analysis. Fully mimetic discretizations satisfy discrete analogs of the continuum theory results used to derive energy inequalities. Consequently, continuum arguments carry over and can be used to show that discrete problems are well-posed and discrete solutions converge. A fully mimetic discrete vector calculus on three dimensional tensor product grids is derived and its key properties proven. Opinions regarding the future of the field are stated.
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Open Access
January 1, 2011
Abstract
A method for calculating all nonmultiple zeros of the complex function in a given rectangle is proposed. The main idea of the method is to construct a covering of the initial rectangle by subsets where either there no solutions or there is only one solution. The algorithm for the construction of such a covering is presented and its convergence is proved. The implementation of the method is shown using different examples.
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Open Access
January 1, 2011
Abstract
We consider the classical ill-posed problem of the recovery of continuous functions from noisy Fourier coefficients. For the classes of functions given in terms of generalized smoothness, we present a priori and a posteriori regularization parameter choice realizing an order-optimal error bound.
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Open Access
January 1, 2011
Abstract
We deal with the inverse scattering problem by an obstacle at a fixed frequency. The obstacle is characterized by its shape, the type of boundary conditions on its surface and the eventual coefficients distributed on this surface. In this paper, we assume that the surface ∂D of the obstacle D is Lipschitz and the surface impedance, λ, is given by a complex valued, measurable and bounded function. We prove uniqueness of (∂D,λ) from the far field map under these regularity conditions. The usual proof of uniqueness for obstacles, based on the use of singular solutions, is divided into two steps. The first one consists of the use of Rellich type lemma to go from the far fields to the near fields and then use the singularities of the singular solutions, via orthogonality relations, to show uniqueness of ∂D. The second step is to use the boundary conditions to prove uniqueness of λ on ∂D via the unique continuation property. This last step requires the surface impedance to be continuous. We propose an approach using layer potentials to transform the inverse problem to the invertibility of integral equations of second kind involving the unknowns ∂D and λ. This enables us to weaken the required regularity conditions by assuming ∂D to be Lipschitz and λ to be only bounded. The procedure of the proof is reconstructive and provides a method to compute the complex valued and bounded surface impedance λ on ∂D by inverting an invertible integral equation. In addition, assuming ∂D to be C^2 regular and λ to be of class C^{0,α}, with α>0, we give a direct and stable formula as another method to reconstruct the surface impedance on ∂D.