Multiscale modelling is a powerful technique, which allows for computational efficiency while retaining small-scale details when they are essential for understanding a finer behaviour of the studied system. In the case of materials modelling, one of the effective multiscaling concepts is domain partitioning, which implies the existence of an explicit interface between various material descriptions, for instance atomistic and continuum regions. When dynamic material behaviour is considered, the major problem for this technique is dealing with reflections of high frequency waves from the interface separating two scales. In this article, a new method is suggested, which overcomes this problem for the case of magnetisation dynamics. The introduction of a damping band at the interface between scales, which absorbs high frequency waves, is suggested. The idea is verified using a number of one-dimensional examples with fine/coarse scale discretisation of a continuum problem of spin wave propagation. This work is the first step towards establishing a reliable atomistic/continuum multiscale transition for the description of the evolution of magnetic properties of ferromagnets.

A finite-element multigrid scheme for elliptic Nash-equilibrium multiobjective optimal control problems with control constraints is investigated. The multigrid computational framework implements a nonlinear multigrid strategy with collective smoothing for solving the multiobjective optimality system discretized with finite elements. Error estimates for the optimal solution and two-grid local Fourier analysis of the multigrid scheme are presented. Results of numerical experiments are presented to demonstrate the effectiveness of the proposed framework.

For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the time-harmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge elements of Nédélec’s first family. The LMM features local hybrid Hiptmair smoothers of Jacobi and Gauss–Seidel type which are performed only on basis functions associated with newly created edges/nodal points or those edges/nodal points where the support of the corresponding basis function has changed during the refinement process. The adaptive mesh refinement is based on Dörfler marking for residual-type a posteriori error estimators and the newest vertex bisection strategy. Using the abstract Schwarz theory of multilevel iterative schemes, quasi-optimal convergence of the LMM is shown, i.e., the convergence rates are independent of mesh sizes and mesh levels provided the coarsest mesh is chosen sufficiently fine. The theoretical findings are illustrated by the results of some numerical examples.

A mesh-independent, robust, and accurate multigrid scheme to solve a linear state-constrained parabolic optimal control problem is presented. We first consider a Lavrentiev regularization of the state-constrained optimization problem. Then, a multi-grid scheme is designed for the numerical solution of the regularized optimality system. Central to this scheme is the construction of an iterative pointwise smoother which can be formulated as a local semismooth Newton iteration. Results of numerical experiments and theoretical twogrid local Fourier analysis estimates demonstrate that the proposed scheme is able to solve parabolic state-constrained optimality systems with textbook multigrid efficiency.

Iterative regularization multigrid methods have been successful applied to signal/image deblurring problems. When zero-Dirichlet boundary conditions are imposed the deblurring matrix has a Toeplitz structure and it is potentially full. A crucial task of a multilevel strategy is to preserve the Toeplitz structure at the coarse levels which can be exploited to obtain fast computations. The smoother has to be an iterative regularization method. The grid transfer operator should preserve the regularization property of the smoother. This paper improves the iterative multigrid method proposed in [11] introducing a wavelet soft-thresholding denoising post-smoother. Such post-smoother avoids the noise amplification that is the cause of the semi-convergence of iterative regularization methods and reduces ringing effects. The resulting iterative multigrid regularization method stabilizes the iterations so that and imprecise (over) estimate of the stopping iteration does not have a deleterious effect on the computed solution. Numerical examples of signal and image deblurring problems confirm the effectiveness of the proposed method.

An elliptic optimal control problem with constraints on the state variable is considered. The Lavrentiev-type regularization is used to treat the constraints on the state variable. To solve the problem numerically, the multigrid for optimization (MGOPT) technique and the collective smoothing multigrid (CSMG) are implemented. Numerical results are reported to illustrate and compare the efficiency of both multigrid strategies.