In the present work, we demonstrate how the pseudoinverse concept from linear algebra can be used to represent and analyze the boundary conditions of linear systems of partial differential equations. This approach has theoretical and practical implications; the theory applies even if the boundary operator is rank deficient, or near rank deficient. If desired, the pseudoinverse can be implemented directly using standard tools like Matlab. We also introduce a new and simplified version of the semidiscrete approximation of the linear PDE system, which completely avoids taking the time derivative of the boundary data. The stability results are valid for general, nondiagonal summation-by-parts norms. Another key result is the extension of summation-by-parts operators to multi-domains by means of carefully crafted embedding operators. No extra numerical boundary conditions are required at the grid interfaces. The aforementioned pseudoinverse allows for a compact representation of these multi-block operators, which preserves all relevant properties of the single-block operators. The embedding operators can be constructed for multiple space dimensions. Numerical results for the two-dimensional Maxwell's equations are presented, and they show very good agreement with theory.
Stability conditions for the explicit integration of projection based nonlinear reduced-order and hyper reduced structural mechanics finite element models
BMW Group, Research and Innovation Centre（宝马集团研究与创新中心）；Dynamore GmbH（Dynamore有限公司）；Queen Mary, University of London（伦敦大学女王玛丽学院）
Local L2-bounded commuting projections using discrete local problems on Alfeld splits
CERMICS, ENPC, Institut Polytechnique de Paris（CERMICS，国立巴黎高等路桥学院，巴黎理工学院）；Inria Paris（法国国家数字技术研究所巴黎分院）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）
Application of Stabilized Explicit Runge-Kutta Methods to the Incompressible Navier-Stokes Equations by means of a Projection Method and a Differential Algebraic Approach
Linear Stability Analysis of Physics-Informed Random Projection Neural Networks for ODEs
Modelling Engineering Risk and Complexity, Scuola Superiore Meridionale（工程风险与复杂性建模，南方高级学院）；Dept. of Electrical and Computer Engineering, Clarkson University（电气与计算机工程系，克林顿大学）；Department of Statistics, Athens University of Economics and Business（统计学系，雅典经济与商业大学）
