2,689 research outputs found
Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors
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 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
We present and analyze a pollution-free Petrov-Galerkin multiscale finite
element method for the Helmholtz problem with large wave number as a
variant of [Peterseim, ArXiv:1411.1944, 2014]. We use standard continuous
finite elements at a coarse discretization scale as trial functions,
whereas the test functions are computed as the solutions of local problems at a
finer scale . The diameter of the support of the test functions behaves like
for some oversampling parameter . Provided is of the order of
and is sufficiently small, the resulting method is stable
and quasi-optimal in the regime where is proportional to . 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
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
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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