We prove convergence with optimal algebraic rates for an adaptive finite element method for nonlinear equations with strongly monotone operator. Unlike prior works, our analysis also includes the iterative and inexact solution of the arising nonlinear systems by means of the Picard iteration. Using nested iteration, we prove, in particular, that the number of of Picard iterations is uniformly bounded in generic cases, and the overall computational cost is (almost) optimal. Numerical experiments confirm the theoretical results.Date: November 20, 2018. 1991 Mathematics Subject Classification. 65N30, 65N12, 65N50, 65M22, 65J15. Key words and phrases. quasilinear elliptic PDE, finite element method, adaptive mesh-refinement, adaptive solution of nonlinear algebraic system, optimal convergence rates, Banach fixed point theorem.

Based on lowest-order finite elements in space, we consider the numerical integration of the Landau-Lifschitz-Gilbert equation (LLG). The dynamics of LLG is driven by the so-called effective field which usually consists of the exchange field, the external field, and lower-order contributions such as the stray field. The latter requires the solution of an additional partial differential equation in full space. Following Bartels and Prohl (2006) (Convergence of an implicit finite element method for the Landau-Lifschitz-Gilbert equation. SIAM J. Numer. Anal. 44(4):1405-1419), we employ the implicit midpoint rule to treat the exchange field. However, in order to treat the lowerorder terms effectively, we combine the midpoint rule with an explicit Adams-Bashforth scheme. The resulting integrator is formally of second-order in time, and we prove unconditional convergence towards a weak solution of LLG. Numerical experiments underpin the theoretical findings.

Combining ideas from [Alouges et al. (Numer. Math., 128, 2014)] and [Praetorius et al. (Comput. Math. Appl., 2017)], we propose a numerical algorithm for the integration of the nonlinear and time-dependent Landau-Lifshitz-Gilbert (LLG) equation which is unconditionally convergent, formally (almost) second-order in time, and requires only the solution of one linear system per time-step. Only the exchange contribution is integrated implicitly in time, while the lower-order contributions like the computationally expensive stray field are treated explicitly in time. Then, we extend the scheme to the coupled system of the Landau-Lifshitz-Gilbert equation with the eddy current approximation of Maxwell equations (ELLG). Unlike existing schemes for this system, the new integrator is unconditionally convergent, (almost) second-order in time, and requires only the solution of two linear systems per time-step. Recently, the numerical integration of LLG has been the subject of many mathematical studies; see, e.g., the review articles [KP06, GC07, Cim08], the monograph [Pro01] and the references therein. The main challenges concern the strong nonlinearity of the problem, a nonconvex unit-length constraint which enforces the solutions to take their pointwise values in the unit sphere, an intrinsic energy law, which combines conservative and dissipative effects and should be preserved by the numerical scheme, as well as the presence of nonlocal field contributions, which prescribe the (possibly nonlinear) coupling 2010 Mathematics Subject Classification. 35K55, 65M12, 65M60, 65Z05.• During the design and the implementation of a micromagnetic code, one of the main issues concerns the computation of the nonlocal magnetostatic interactions. In many situations, it turns out to be the most time-consuming part of micromagnetic simulations [AES + 13]. To cope with this problem, we follow the approach of [PRS17] and propose an implicit-explicit treatment for the lower-order effective field contributions. Then, only one expensive stray field computation per time-step needs to be carried out. Nevertheless, our time-stepping preserves the (almost) second-order convergence in time of the scheme as well as the unconditional convergence result.• The discovery of the giant magnetoresistance (GMR) effect in [BBF + 88, BGSZ89] determined a breakthrough in magnetic hard disk storage capacity and encouraged several extensions of the micromagnetic model, which aim to describe the effect of spin-polarized currents on magnetic materials. The most used approaches involve extended forms of

We consider the numerical approximation of the Landau-Lifshitz-Gilbert equation, which describes the dynamics of the magnetization in ferromagnetic materials. In addition to the classical micromagnetic contributions, the energy comprises the Dzyaloshinskii-Moriya interaction, which is the most important ingredient for the enucleation and the stabilization of chiral magnetic skyrmions. We propose and analyze three tangent plane integrators, for which we prove (unconditional) convergence of the finite element solutions towards a weak solution of the problem. The analysis is constructive and also establishes existence of weak solutions. Numerical experiments demonstrate the applicability of the methods for the simulation of practically relevant problem sizes.

The tangent plane scheme is a time-marching scheme for the numerical solution of the nonlinear parabolic Landau-Lifshitz-Gilbert equation (LLG), which describes the time evolution of ferromagnetic configurations. Exploiting the geometric structure of LLG, the tangent plane scheme requires only the solution of one linear variational form per time-step, which is posed in the discrete tangent space determined by the nodal values of the current magnetization. We develop an effective solution strategy for the arising constrained linear systems, which is based on appropriate Householder reflections. We derive possible preconditioners, which are (essentially) independent of the time-step, and prove that the preconditioned GMRES algorithm leads to linear convergence. Numerical experiments underpin the theoretical findings.2010 Mathematics Subject Classification. 35K35, 65M60, 65F08, 65F10, 65M22, 65Z05.