2,689 research outputs found

    Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors

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    This paper addresses the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local cell problems is rigorously justified. The stabilization eliminates scale-dependent pre-asymptotic effects as they appear for standard finite element discretizations of highly oscillatory problems, e.g., the poor L2L^2 approximation in homogenization problems or the pollution effect in high-frequency acoustic scattering

    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 κ\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(κ)\log(\kappa) and hh is sufficiently small, the resulting method is stable and quasi-optimal in the regime where HH is proportional to κ1\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

    First order least squares method with weakly imposed boundary condition for convection dominated diffusion problems

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    We present and analyze a first order least squares method for convection dominated diffusion problems, which provides robust L2 a priori error estimate for the scalar variable even if the given data f in L2 space. The novel theoretical approach is to rewrite the method in the framework of discontinuous Petrov - Galerkin (DPG) method, and then show numerical stability by using a key equation discovered by J. Gopalakrishnan and W. Qiu [Math. Comp. 83(2014), pp. 537-552]. This new approach gives an alternative way to do numerical analysis for least squares methods for a large class of differential equations. We also show that the condition number of the global matrix is independent of the diffusion coefficient. A key feature of the method is that there is no stabilization parameter chosen empirically. In addition, Dirichlet boundary condition is weakly imposed. Numerical experiments verify our theoretical results and, in particular, show our way of weakly imposing Dirichlet boundary condition is essential to the design of least squares methods - numerical solutions on subdomains away from interior layers or boundary layers have remarkable accuracy even on coarse meshes, which are unstructured quasi-uniform

    On Multiscale Methods in Petrov-Galerkin formulation

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    In this work we investigate the advantages of multiscale methods in Petrov-Galerkin (PG) formulation in a general framework. The framework is based on a localized orthogonal decomposition of a high dimensional solution space into a low dimensional multiscale space with good approximation properties and a high dimensional remainder space{, which only contains negligible fine scale information}. The multiscale space can then be used to obtain accurate Galerkin approximations. As a model problem we consider the Poisson equation. We prove that a Petrov-Galerkin formulation does not suffer from a significant loss of accuracy, and still preserve the convergence order of the original multiscale method. We also prove inf-sup stability of a PG Continuous and a Discontinuous Galerkin Finite Element multiscale method. Furthermore, we demonstrate that the Petrov-Galerkin method can decrease the computational complexity significantly, allowing for more efficient solution algorithms. As another application of the framework, we show how the Petrov-Galerkin framework can be used to construct a locally mass conservative solver for two-phase flow simulation that employs the Buckley-Leverett equation. To achieve this, we couple a PG Discontinuous Galerkin Finite Element method with an upwind scheme for a hyperbolic conservation law
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