Various applications ranging from spintronic devices, giant magnetoresistance sensors, and magnetic storage devices, include magnetic parts on very different length scales. Since the consideration of the Landau-Lifshitz-Gilbert equation (LLG) constrains the maximum element size to the exchange length within the media, it is numerically not attractive to simulate macroscopic parts with this approach. On the other hand, the magnetostatic Maxwell equations do not constrain the element size, but cannot describe the short-range exchange interaction accurately. A combination of both methods allows to describe magnetic domains within the micromagnetic regime by use of LLG and also considers the macroscopic parts by a non-linear material law using the Maxwell equations. In our work, we prove that under certain assumptions on the non-linear material law, this multiscale version of LLG admits weak solutions. Our proof is constructive in the sense that we provide a linear-implicit numerical integrator for the multiscale model such that the numerically computable finite element solutions admit weak $H^1$-convergence (at least for a subsequence) towards a weak solution.
Meshfree GMsFEM-based exponential integration for multiscale 3D advection-diffusion problems
North-Eastern Federal University（东北联邦大学）；Wenzhou-Kean University（温州肯恩大学）；Khalifa University of Science and Technology（哈利法科学技术大学）；Universidad Nacional Autónoma de Honduras（洪都拉斯国立自治大学）；Universidad Nacional de Colombia（哥伦比亚国立大学）
University of Exeter（埃克塞特大学）；University of Vienna（维也纳大学）；TU Wien（维也纳工业大学）；Wolfgang Pauli Institute c/o Faculty of Mathematics, University of Vienna（沃尔夫冈·泡利研究所（挂靠维也纳大学数学系））
Fourier series (based) multiscale method for computational analysis in science and engineering: III. Fourier series multiscale method for linear differential equation with constant coefficients
A Nodal Discontinuous Galerkin Method with Rank-Adaptive Velocity Space Representation for the Multiscale BGK Model
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Swarthmore College（斯沃斯莫尔学院）；Virginia Polytechnic and State University（弗吉尼亚理工大学）；University of Delaware（特拉华大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
Optimal solutions employing an algebraic Variational Multiscale approach Part II: Application to Navier-Stokes
ETSIAE-UPM-School of Aeronautics, Universidad Politécnica de Madrid（马德里理工大学航空学院）；Delft University of Technology, Faculty of Aerospace Engineering（代尔夫特理工大学航空航天学院）；Center for Computational Simulation, Universidad Politécnica de Madrid（马德里理工大学计算模拟中心）
High-order stroboscopic averaging methods for highly oscillatory delay problems
Universidad de Valladolid（巴利亚多利德大学）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学与应用科学中心）
Scale-Adaptive Generative Flows for Multiscale Scientific Data
Department of Mathematics, University of California, Los Angeles（加州大学洛杉矶分校数学系）；Machine Learning Lab, Capital Fund Management（资本基金管理有限公司机器学习实验室）；Courant Institute, New York University（纽约大学柯朗研究所）
Explicit form of simplified Grad's 13 moments distribution function-based moment gas kinetic solver with unstructured meshes for the multiscale rarefied flow
