260 research outputs found

    Stable Multiscale Petrov-Galerkin Finite Element Method for High Frequency Acoustic Scattering

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    We present and analyze a pollution-free Petrov-Galerkin multiscale finite element method for the Helmholtz problem with large wave number Îș\kappa as a variant of [Peterseim, ArXiv:1411.1944, 2014]. We use standard continuous Q1Q_1 finite elements at a coarse discretization scale HH as trial functions, whereas the test functions are computed as the solutions of local problems at a finer scale hh. The diameter of the support of the test functions behaves like mHmH for some oversampling parameter mm. Provided mm is of the order of log⁥(Îș)\log(\kappa) and hh is sufficiently small, the resulting method is stable and quasi-optimal in the regime where HH is proportional to Îș−1\kappa^{-1}. In homogeneous (or more general periodic) media, the fine scale test functions depend only on local mesh-configurations. Therefore, the seemingly high cost for the computation of the test functions can be drastically reduced on structured meshes. We present numerical experiments in two and three space dimensions.Comment: The version coincides with v3. We only resized some figures which were difficult to process for certain printer

    Numerical homogenization of H(curl)-problems

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    If an elliptic differential operator associated with an H(curl)\mathbf{H}(\mathrm{curl})-problem involves rough (rapidly varying) coefficients, then solutions to the corresponding H(curl)\mathbf{H}(\mathrm{curl})-problem admit typically very low regularity, which leads to arbitrarily bad convergence rates for conventional numerical schemes. The goal of this paper is to show that the missing regularity can be compensated through a corrector operator. More precisely, we consider the lowest order N\'ed\'elec finite element space and show the existence of a linear corrector operator with four central properties: it is computable, H(curl)\mathbf{H}(\mathrm{curl})-stable, quasi-local and allows for a correction of coarse finite element functions so that first-order estimates (in terms of the coarse mesh-size) in the H(curl)\mathbf{H}(\mathrm{curl}) norm are obtained provided the right-hand side belongs to H(div)\mathbf{H}(\mathrm{div}). With these four properties, a practical application is to construct generalized finite element spaces which can be straightforwardly used in a Galerkin method. In particular, this characterizes a homogenized solution and a first order corrector, including corresponding quantitative error estimates without the requirement of scale separation

    Adaptive Nonconforming Finite Element Approximation of Eigenvalue Clusters

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    This paper analyses an adaptive nonconforming finite element method for eigenvalue clusters of self-adjoint operators and proves optimal convergence rates (with respect to the concept of nonlinear approximation classes) for the approximation of the invariant subspace spanned by the eigenfunctions of the eigenvalue cluster. Applications include eigenvalues of the Laplacian and of the Stokes system

    Education and Environment Dementia Risk Factors: A Literature Review

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    Introduction and Background: Dementia has many different causes. Dementia is considered a multifactorial disease; hence the interplay between factors for every case is a complex study. Early and recent reviews had recognised several risk factors associated with dementia in general and with AD specifically. The most studied risk factors include genetics, increasing age, education, environment, and brain injuries. This review aims to give readers access to the latest research on the educational and environmental risk factors of dementia by selecting recent high-quality resources and summarising them in this review. Methods: The current article is a narrative review of broad literature research. Results and Discussion: The comprehensive examination of evidence supports the following. First, low education can be considered a relevant risk factor for developing dementia, although the operationalisation of "low education" is still unclear in many studies. The mechanisms of "cognitive reserve" have an important implication in the relationships between education and dementia, and this has been studied with limitations in people with intellectual disorders. Second, air pollution is now considered a dementia risk factor with plenty of evidence concerning PM2.5 but less conclusive evidence regarding single gaseous pollutants because of the “multi-exposure response.” Conclusion: The considerable body of research points towards an association between these risk factors and dementia prevalence. Low- and middle-income countries will benefit from prioritising child education for all since education is one of the major risk factors for dementia and a wide variety of health disparities

    The adaptive finite element method

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    Computer simulations of many physical phenomena rely on approximations by models with a finite number of unknowns. The number of these parameters determines the computational effort needed for the simulation. On the other hand, a larger number of unknowns can improve the precision of the simulation. The adaptive finite element method (AFEM) is an algorithm for optimizing the choice of parameters so accurate simulation results can be obtained with as little computational effort as possible

    Multiscale Petrov-Galerkin Method for High-Frequency Heterogeneous Helmholtz Equations

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    This paper presents a multiscale Petrov-Galerkin finite element method for time-harmonic acoustic scattering problems with heterogeneous coefficients in the high-frequency regime. We show that the method is pollution- free also in the case of heterogeneous media provided that the stability bound of the continuous problem grows at most polynomially with the wave number k. By generalizing classical estimates of [Melenk, Ph.D. Thesis 1995] and [Hetmaniuk, Commun. Math. Sci. 5 (2007)] for homogeneous medium, we show that this assumption of polynomially wave number growth holds true for a particular class of smooth heterogeneous material coefficients. Further, we present numerical examples to verify our stability estimates and implement an example in the wider class of discontinuous coefficients to show computational applicability beyond our limited class of coefficients

    Numerical approximation of planar oblique derivative problems in nondivergence form

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    A numerical method for approximating a uniformly elliptic oblique derivative problem in two-dimensional simply-connected domains is proposed. The numerical scheme employs a mixed formulation with piecewise affine functions on curved finite element domains. The direct approximation of the gradient of the solution turns the oblique derivative boundary condition into an oblique direction condition. A priori and a posteriori error estimates as well as numerical computations on uniform and adaptive meshes are provided
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