4,573 research outputs found
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
Postprocessing of Non-Conservative Flux for Compatibility with Transport in Heterogeneous Media
A conservative flux postprocessing algorithm is presented for both
steady-state and dynamic flow models. The postprocessed flux is shown to have
the same convergence order as the original flux. An arbitrary flux
approximation is projected into a conservative subspace by adding a piecewise
constant correction that is minimized in a weighted norm. The application
of a weighted norm appears to yield better results for heterogeneous media than
the standard norm which has been considered in earlier works. We also
study the effect of different flux calculations on the domain boundary. In
particular we consider the continuous Galerkin finite element method for
solving Darcy flow and couple it with a discontinuous Galerkin finite element
method for an advective transport problem.Comment: 34 pages, 17 figures, 11 table
Multiscale simulations of porous media flows in flow-based coordinate system
In this paper, we propose a multiscale technique for the simulation of porous media flows in a flow-based coordinate system. A flow-based coordinate system allows us to simplify the scale interaction and derive the upscaled equations for purely hyperbolic transport equations. We discuss the applications of the method to two-phase flows in heterogeneous porous media. For two-phase flow simulations, the use of a flow-based coordinate system requires limited global information, such as the solution of single-phase flow. Numerical results show that one can achieve accurate upscaling results using a flow-based coordinate system
Numerical computation of transonic flows by finite-element and finite-difference methods
Studies on applications of the finite element approach to transonic flow calculations are reported. Different discretization techniques of the differential equations and boundary conditions are compared. Finite element analogs of Murman's mixed type finite difference operators for small disturbance formulations were constructed and the time dependent approach (using finite differences in time and finite elements in space) was examined
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