The aim of this paper is to analyze a formulation of the eddy current problem in terms of a time-primitive of the electric field in a bounded domain with input current intensities or voltage drops as source data. To this end, we introduce a Lagrange multiplier to impose the divergence-free condition in the dielectric domain. Thus, we obtain a time-dependent weak mixed formulation leading to a degenerate parabolic problem which we prove is well-posed. We propose a finite element method for space discretization based on Nédélec edge elements for the main variable and standard finite elements for the Lagrange multiplier, for which we obtain error estimates. Then, we introduce a backward Euler scheme for time discretization and prove error estimates for the fully discrete problem, too. Finally, the method is applied to solve a couple of test problems.

In this work we present new numerical methods to simulate the mechanics of head-tape magnetic storage devices. The elastohydrodynamic problem is formulated in terms of a coupled system which is governed by a nonlinear compressible Reynolds equation for the air pressure over the head, and a rod model for the tape displacement. A fixed point algorithm between the solutions of the elastic and hydrodynamic problems is proposed. For the nonlinear Reynolds equation, a characteristics method and a duality algorithm are developed to cope with the convection dominating and nonlinear diffusion features, respectively. Furthermore, in the duality method the convergence and optimal choice of the parameters are analyzed. At each fixed point iteration, in the elastic model a complementarity formulation is required and appropriate numerical techniques are used. For the spatial discretization different finite element spaces are chosen. Finally, numerical test examples illustrate the theoretical results, as well as the good performance in the simulation of real devices.

This work represents a first step towards the simulation of the motion of water in a complex hydrodynamic configuration, such asa channel network or a river delta, by means of a suitable“combination” of different mathematicalmodels. In this framework a wide spectrum of space and time scales is involveddue to the presence of physical phenomena ofdifferent nature. Ideally, moving from a hierarchy of hydrodynamic models, one should solvethroughout the whole domain the most complex model (with solution $u_{\rm{fine}}$ ) to accuratelydescribe all the physical features of the problem at hand. In our approach instead,for a user-defined output functional ${\cal F}$ , we aim to approximate,within a prescribed tolerance τ, the value ${\cal F}(u_{\rm{fine}})$ by means of the quantity ${\cal F}(u_{\rm{adapted}})$ , $u_{\rm{adapted}}$ being the so-called adapted solution solving thesimpler models on most of the computational domain while confining the complex ones only on a restricted region. Moving fromthe simplified setting where only two hydrodynamic models, fine and coarse, are considered, weprovide an efficient tool able to automatically selectthe regions of the domain where the coarsemodel rather than the fine one are to be solved, while guaranteeing $|{\cal F}(u_{\rm{fine}})-{\cal F}(u_{\rm{adapted}})|$ below the tolerance τ.This goal is achieved via a suitable a posteriori modeling error analysisdeveloped in the framework of a goal-oriented theory.We extend the dual-based approach provided in [Braack and Ern,Multiscale Model Sim.1 (2003) 221–238],for steady equations to the case of a generic time-dependent problem.Then this analysis is specialized to the case we areinterested in, i.e. the free-surface flows simulation, by emphasizing the crucial issue of the time discretization for both the primal and the dualproblems. Finally, inthe last part of the paper a widespread numerical validation is carried out.

The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.

We present a numerical algorithm to solve the micromagnetic equations based on tangential-plane minimization for the magnetization update and a homothethic-layer decomposition of outer space for the computation of the demagnetization field. As a first application, detailed results on the flower-vortex transition in the cube of Micromagnetic Standard Problem number 3 are obtained, which confirm, with a different method, those already present in the literature, and validate our method and code. We then turn to switching of small cubic or almost-cubic particles, in the single-domain limit. Our data show systematic deviations from the Stoner-Wohlfarth model due to the non-ellipsoidal shape of the particle, and in particular a non-monotone dependence on the particle size.

The LBB condition is well-known to guarantee the stability of a finiteelement (FE) velocity - pressure pair in incompressible flow calculations.To ensure the condition to be satisfied a certain constant should be positive andmesh-independent. The paper studies the dependence of the LBB condition on thedomain geometry. For model domains such as strips and rings thesubstantial dependence of this constant on geometry aspect ratios is observed.In domains with highly anisotropic substructures this may require special care with numerics to avoid failures similar to those whenthe LBB condition is violated. In the core of the paperwe prove that for any FE velocity-pressure pair satisfying usual approximationhypotheses the mesh-independent limit in the LBB condition is not greater thanits continuous counterpart, the constant from the Nečas inequality.For the latter the explicit and asymptotically accurate estimates are proved. The analytic results are illustrated by several numerical experiments.

The standard discretization of the Stokes andNavier–Stokes equations in vorticity and stream function formulation by affine finiteelements is known for its bad convergence. We present here a modified discretization, weprove that the convergence is improved and we establish a priori error estimates.

The Navier–Stokes equations are approximated by means ofa fractional step, Chorin–Temam projection method; the time derivativeis approximated by a three-level backward finite difference, whereasthe approximation in space is performed by a Galerkin technique.It is shown that the proposed scheme yields an errorof ${\cal O}(\delta t^2 + h^{l+1})$ for the velocity in the norm of l 2(L2(Ω)d), where l ≥ 1 isthe polynomial degree of the velocity approximation. It is also shownthat the splitting error of projection schemes based on theincremental pressure correction is of ${\cal O}(\delta t^2)$ independent of theapproximation order of the velocity time derivative.