On multiscale methods in Petrov–Galerkin formulation

Abstract

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.