Abstract. An enriched nonconforming Multiscale Finite Element Method (MsFEM) to solve viscous incompressible flow problems in genuine heterogeneous or porous media was proposed in [Q. Feng, G. Allaire, and P. Omnes, Multiscale Model. Simul., 20 (2022), pp. 462–492]. The main feature of this MsFEM is the consideration of high-order sets of weighting functions: for the velocity, they are polynomials of order [Formula: see text] on the faces and of order [Formula: see text] in the volume of the elements; for the pressure they are polynomials of order [Formula: see text] in the element volume. In the previously cited reference, only the case [Formula: see text] was numerically tested. The present paper proposes the first implementation for the case [Formula: see text] in two and three dimensions. Furthermore, a discrete analysis of this MsFEM applied to the Stokes problem in heterogeneous media is performed. In particular, a first error estimate is obtained, proving the convergence of this MsFEM for the Stokes problem in periodic perforated media. In addition, it has been shown in the previously cited reference that the continuous local problems involved in this MsFEM are well-posed. Here, their discrete counterparts are also proved to be well-posed, for any [Formula: see text] in two dimensions and for [Formula: see text] equal to 1 and 2 in three dimensions, with a judicious choice of nonconforming pairs of finite elements.
Abstract. We analyze a system of equations that describes the propagation of sound waves. We are interested in complex constructions along a part of the boundary of the domain, for example constructions with small chambers that are connected to the domain. We also allow that different flow equations are used in the chambers, e.g., modelling a damping material. In addition to the complex geometry, we assume that the viscosity vanishes in the limit. The limiting system is given by wave equations, we derive these equations and determine the effective boundary conditions. The effective boundary conditions replace the large number of small chambers. We provide examples for sound absorbing constructions and the resulting effective boundary conditions.
Abstract. The Multiscale Hybrid-Mixed (MHM) method is a multiscale finite element method based on a hybrid weak formulation, originally proposed for problems linked to flow in porous media. Its starting point is a variational formulation that guarantees that the flux variable is [Formula: see text]-conforming. This property is lost when the local problems, in their elliptic form, are discretized using primal finite element methods. In this work, we close that gap by proposing and analyzing a new flux reconstruction for the MHM method, computed elementwise on submeshes, that belongs to [Formula: see text]. The reconstruction converges optimally in the [Formula: see text]-norm. Furthermore, the divergence of the reconstructed flow is the piecewise continuous polynomial projection of the source term onto submeshes and thus it also converges optimally in the [Formula: see text]-norm. As a by-product of the reconstruction technique, a fully computable a posteriori error estimator is presented and analyzed. These theoretical results are validated experimentally via numerical computations.
Abstract. A variational multiscale stabilized finite element (VMSFE) scheme for the transient magnetohydrodynamic Navier–Stokes (MHD-NS) equations has been derived. The method involves the decomposition of the unknown solution into resolvable (coarse) and subgrid (fine) scales. The subgrid scales are time-dependent, and the VMSFE formulation is generated by eliminating fine scales in terms of coarse scales. Using Fourier analysis, appropriate stabilization parameters have been derived to achieve the optimal order of convergence. The subgrid scales are taken as orthogonal to the coarse-scale finite element space. This approach ensures a unique decomposition of the solution and prevents the fine scales from polluting the coarse-scale solution. For time discretization, a fully implicit backward Euler scheme has been used. The stability analysis of the VMSFE scheme has been carried out. A priori error estimates have been derived to establish the theoretical convergence of the VMSFE scheme in the [Formula: see text] norm for velocity and in the [Formula: see text] norm for pressure. We also derive the optimal error estimate in the [Formula: see text] norm for the velocity. Numerical experiments on multiply driven cavity flow have been carried out, and the credibility of the proposed VMSFE scheme has been established successfully. The flow pattern is traced for various Hartmann, Reynolds, and magnetic force inclination angle values to visualize the dynamics of the multiply driven flow in a cavity. Moreover, the transient MHD-NS flow behavior has been thoroughly analyzed in the flow past a square cylinder. A complete derivation for their existence and uniqueness is given for the MHD-NS equations with Dirichlet boundary conditions in a bounded domain.
