A posteriori error estimation and related adaptive mesh-refining algorithms have themselves proven to be powerful tools in nowadays scientific computing. Contrary to adaptive finite element methods, convergence of adaptive boundary element schemes is, however, widely open. We propose a relaxed notion of convergence of adaptive boundary element schemes. Instead of asking for convergence of the error to zero, we only aim to prove estimator convergence in the sense that the adaptive algorithm drives the underlying error estimator to zero. We observe that certain error estimators satisfy an estimator reduction property which is sufficient for estimator convergence. The elementary analysis is only based on Dörfler marking and inverse estimates, but not on reliability and efficiency of the error estimator at hand. In particular, our approach gives a first mathematical justification for the proposed steering of anisotropic mesh-refinements, which is mandatory for optimal convergence behavior in 3D boundary element computations.

The h-h/2-strategy is one very basic and well-known technique for the a posteriori error estimation for Galerkin discretizations of energy minimization problems. Let φ denote the exact solution. One then considers

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We discuss several adaptive mesh-refinement strategies based on (h − h/2)-error estimation. This class of adaptive methods is particularly popular in practise since it is problem independent and requires virtually no implementational overhead. We prove that, under the saturation assumption, these adaptive algorithms are convergent. Our framework applies not only to finite element methods, but also yields a first convergence proof for adaptive boundary element schemes. For a finite element model problem, we extend the proposed adaptive scheme and prove convergence even if the saturation assumption fails to hold in general.

For a boundary integral formulation of the 2D Laplace equation with mixed boundary conditions, we consider an adaptive Galerkin BEM based on an false(h−hfalse/2false)-type error estimator. We include the resolution of the Dirichlet, Neumann, and volume data into the adaptive algorithm. In particular, an implementation of the developed algorithm has only to deal with discrete integral operators. We prove that the proposed adaptive scheme leads to a sequence of discrete solutions, for which the corresponding error estimators tend to zero. Under a saturation assumption for the non-perturbed problem which is observed empirically, the sequence of discrete solutions thus converges to the exact solution in the energy norm.

We report on the MATLAB program package HILBERT. It provides an easily-accessible implementation of lowest-order adaptive Galerkin boundary element methods for the numerical solution of the Poisson equation in 2D. The library was designed to serve several purposes: The stable implementation of the integral operators may be used in research code. The framework of MATLAB ensures usability in lectures on boundary element methods or scientific computing. Finally, we emphasize the use of adaptivity as general concept and for boundary element methods in particular. In this work, we summarize recent analytical results on adaptivity in the context of BEM and illustrate the use of HILBERT. Various benchmarks are performed to empirically analyze the performance of the proposed adaptive algorithms and to compare adaptive and uniform mesh-refinements. In particular, we do not only focus on mathematical convergence behavior but also on the usage of critical system resources such as memory consumption and computational time. In any case, the superiority of the proposed adaptive approach is empirically supported.

This note is about the prediction of the domain and wall configuration formed by the magnetization m in a soft ferromagnetic film under an external field. We propose a modification of the reduced model introduced by DeSimone et al. (2001), that is a variational model for a 2-d in-plane magnetization m on the cross section ω. The modification consists of two parts: 1.) An unphysical but computationally convenient selection criterion in DeSimone et al. (2001) is replaced by a selection criterion based on physical wall energy, 2.) The set of admissible configurations is restricted by imposing an additional boundary condition at the boundary ∂ω of the cross section. The specific wall energy is computed off-line based on a 2-d reduction of the 3-d model from Döring et al. (2012b), and then fed into a diffuse interface approximation for the selection criterion 1.). This retains some of the low complexity of the original model DeSimone et al. (2001). Modification 2.) allows to model hysteresis as observed in experiments by van den Berg and Vatvani (1982); preliminary numerical simulations reveal good qualitative agreement.

We analyze the reduced model for thin-film devices in stationary micromagnetics proposed in DeSimone et al. (R Soc Lond Proc Ser A Math Phys Eng Sci 457(2016):2983-2991). We introduce an appropriate functional analytic framework and prove well-posedness of the model in that setting. The scheme for the numerical approximation of solutions consists of two ingredients: The energy space is discretized in a conforming way using Raviart-Thomas finite elements; the non-linear but convex side constraint is treated with a penalty method. This strategy yields a convergent sequence of approximations as discretization and penalty parameter vanish. The proof generalizes to a large class of minimization problems and is of interest beyond the scope of thin-film micromagnetics.Mathematics Subject Classification (2010) 65K05 · 65K15 · 49M20 1 Introduction and abstract setting