An Operator Formulation of the Multiscale Finite-Volume Method with Correction Function

Abstract

Abstract. The multiscale finite-volume (MSFV) method has been derived to efficiently solve large problems with spatially varying coefficients. The fine-scale problem is subdivided into local problems that can be solved separately and are coupled by a global problem. This algorithm, in consequence, shares some characteristics with two-level domain decomposition (DD) methods. However, the MSFV algorithm is different in that it incorporates a flux reconstruction step, which delivers a fine-scale mass conservative flux field without the need for iterating. This is achieved by the use of two overlapping coarse grids. The recently introduced correction function allows for a consistent handling of source terms, which makes the MSFV method a flexible algorithm that is applicable to a wide spectrum of problems. It is demonstrated that the MSFV operator, used to compute an approximate pressure solution, can be equivalently constructed by writing the Schur complement with a tangential approximation of a single-cell overlapping grid and incorporation of appropriate coarse-scale mass-balance equations.

An adaptive multiscale method for density-driven instabilities

“…We refer to [29,30] for further details and for an extensive description of the construction of the conservative velocity field.…”

“…To describe the MsFV method, we use an operator formulation [29] and order the N components of the pressure vector in Eq. (6) such that first inner, then edge, and last node cells appear (Fig.…”

An iterative multiscale finite volume algorithm converging to the exact solution

“…In this section, we briefly recall the general formulation of the MsFV method with correction functions [19,21]; we closely follow the operator-based formulation presented by Lunati and Lee [22] and refer to their paper for further details. Let us consider a fine-scale problem on the domain X of the form…”

“…Recently, an iterative algorithm has been introduced in which the solution is smoothed by applying line relaxation in all spatial directions [8]. However, as the MsFV solution is allowed to be iterated, this method has a strong resemblance to domain decomposition techniques [22,23] and theory and practice of iterative linear solvers can be adopted [25]. In this paper, we employ a natural and elegant approach to iteratively improve the quality of the localization assumption: we construct an iterative method based on the MsFV operator only, which can be stabilized by use of a Krylov-space accelerator.…”

“…The Multiscale Finite-Volume (MsFV) method [12,13,22] has been developed to offer a computationally efficient alternative to the direct solution of such large (fine-scale) problems. The method has been extended to solve physically complex flow, which includes compressibility effects [18]; gravity and capillarity [21]; complex wells [14,31]; and interphase mass exchange [17].…”