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On Multiscale Methods in PetrovGalerkin formulation
, 2014
"... In this work we investigate the advantages of multiscale methods in PetrovGalerkin (PG) formulation in a general framework. The framework is subject to a localized orthogonal decomposition of a high dimensional solution space into a low dimensional multiscale space and a high dimensional remainder ..."
Abstract

Cited by 3 (2 self)
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In this work we investigate the advantages of multiscale methods in PetrovGalerkin (PG) formulation in a general framework. The framework is subject to a localized orthogonal decomposition of a high dimensional solution space into a low dimensional multiscale space and a high dimensional remainder space with negligible fine scale information. As a model problem we consider the Poisson problem. We prove that the PetrovGalerkin formulation does not suffer from a relevant loss of accuracy, still preserving the convergence order of the original multiscale method. We also prove infsup stability of a PG Continuous Galerkin Finite Element multiscale method. Furthermore, we demonstrate that the PetrovGalerkin method can decrease the computational complexity significantly, allowing for more efficient solution algorithms. As another application of the framework, we show how the PetrovGalerkin framework can be used to construct a locally mass conservative solver for the BuckleyLeverett equation. To achieve this, we couple a PG Discontinuous Galerkin Finite Element method with an upwing scheme for a hyperbolic conservation law.
ADAPTIVE HETEROGENEOUS MULTISCALE METHODS FOR IMMISCIBLE TWOPHASE FLOW IN POROUS MEDIA
, 2013
"... In this contribution we present the first formulation of a heterogeneous multiscale method for an incompressible immiscible twophase flow system with degenerate permeabilities. The method is in a general formulation which includes oversampling. We do not specify the discretization of the derived ma ..."
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Cited by 2 (0 self)
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In this contribution we present the first formulation of a heterogeneous multiscale method for an incompressible immiscible twophase flow system with degenerate permeabilities. The method is in a general formulation which includes oversampling. We do not specify the discretization of the derived macroscopic equation, but we give two examples of possible realizations, suggesting a finite element solver for the fine scale and a vertex centered finite volume method for the effective coarse scale equations. Assuming periodicity, we show that the method is equivalent to a discretization of the homogenized equation. We provide an aposteriori estimate for the error between the homogenized solutions of the pressure and saturation equations and the corresponding HMM approximations. The error estimate is based on the results recently achieved in [C. Cancès, I. S. Pop, and M. Vohralík. An a posteriori error estimate for vertexcentered finite volume discretizations of immiscible incompressible twophase flow. Math. Comp., in press, 2013].
1On Multiscale Methods in PetrovGalerkin formulation ∗
, 2014
"... In this work we investigate the advantages of multiscale methods in PetrovGalerkin (PG) formulation in a general framework. The framework is subject to a localized orthogonal decomposition of a high dimensional solution space into a low dimensional multiscale space and a high dimensional remainde ..."
Abstract
 Add to MetaCart
(Show Context)
In this work we investigate the advantages of multiscale methods in PetrovGalerkin (PG) formulation in a general framework. The framework is subject to a localized orthogonal decomposition of a high dimensional solution space into a low dimensional multiscale space and a high dimensional remainder space with negligible fine scale information. As a model problem we consider the Poisson problem. We prove that the PetrovGalerkin formulation does not suffer from a relevant loss of accuracy, still preserving the convergence order of the original multiscale method. We also prove infsup stability of a PG Continuous Galerkin Finite Element multiscale method. Furthermore, we demonstrate that the PetrovGalerkin method can decrease the computational complexity significantly, allowing for more efficient solution algorithms. As another application of the framework, we show how the PetrovGalerkin framework can be used to construct a locally mass conservative solver for the BuckleyLeverett equation. To achieve this, we couple a PG Discontinuous Galerkin Finite Element method with an upwing scheme for a hyperbolic conservation law.