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 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 Petrov-Galerkin formulation does not suffer from a relevant loss of accuracy, still preserving the convergence order of the original multiscale method. We also prove inf-sup stability of a PG Continuous 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 the Buckley-Leverett equation. To achieve this, we couple a PG Discontinuous Galerkin Finite Element method with an upwing scheme for a hyperbolic conservation law.