Abstract. The numerical simulation of three-dimensional charged-particle dynamics (CPD) under a strong magnetic field is a basic and challenging algorithmic task in plasma physics. In this paper, we introduce a new methodology to design two-scale exponential integrators for three-dimensional CPD whose magnetic field’s strength is inversely proportional to a dimensionless and small parameter [Formula: see text]. By dealing with the transformed form of three-dimensional CPD, we linearize the magnetic field and put the residual component in a new nonlinear function which is shown to be uniformly bounded. Based on this foundation and the proposed two-scale exponential integrators, a class of novel integrators is formulated and studied. The corresponding uniform accuracy of the proposed [Formula: see text]th order integrator is shown to be [Formula: see text] in the approximation of the position, where [Formula: see text] and the constant symbolized by [Formula: see text], the time stepsize [Formula: see text], and the computation cost are all independent of [Formula: see text]. Moreover, in the case of a maximal ordering strong magnetic field, an improved error bound [Formula: see text] in the approximation of the position is obtained for the proposed [Formula: see text]th order integrator. A rigorous proof of these uniform and improved error bounds is presented, and a numerical test is performed to illustrate the error and efficiency behavior of the proposed integrators.
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.
