In the space-time (ST) computational analysis, the discretization methods, such as those with finite elements, have appeared in a number of variations over the past years. Most of the current methods employ discontinuous functions in time. The motivation for that is to avoid a cost associated with fully 4D computations. Consequently, the increase in the number of unknowns in time remains under control despite using ST elements. Such methods have been applied over the past decades to a very large number of fluid dynamics and advection-diffusion problems, which are governed by equations with first-order time derivatives, and to a far lesser extent to elastodynamics and other solid mechanics problems, which are governed by equations with second-order time derivatives. In both contexts, especially when the functions are continuous in space, the ST methods are often formulated in the framework of stabilized methods, such as the Streamline-Upwind/Petrov-Galerkin, Galerkin/Least-Squares, Pressure-Stabilizing/Petrov-Galerkin, and variational multiscale methods. These are sometimes supplemented with discontinuity-capturing techniques. With the widespread adoption of isogeometric analysis (IGA), the ST methods have also been synthesized with IGA and applied to a large number of problems, mostly in fluid dynamics. The synthesis, ST-IGA, enables and encourages the use of higher-order functions in time. Within the ST domain, increasing the polynomial order in time leads to the expected improvement in solution accuracy. At the lateral boundaries with Dirichlet condition, however, the fluxes do not gain increased accuracy with higher-order polynomials in time. Motivated by this concern, we focus here on 1D elastodynamics and how to treat the Dirichlet boundaries. We introduce two new stabilization methods and show, with test computations, how the stabilized equations used in computing the fluxes perform. The methods are consistent in the way the fluxes are computed and can be extended to 2D and 3D problems.
In this paper, we study asymptotic consensus and finite-time reachability to Pareto optimal set for time-discrete multi-objective consensus-based optimization (M-CBO) algorithm in particle regime without resorting to the mean-field approximation. While the original CBO algorithm [J. A. Carrillo, Y.-P. Choi, C. Totzeck and O. Tse, An analytical framework for consensus-based global optimization method, Math. Models Methods Appl. Sci. 28 (2018) 1037–1066] assumes that all particles share common single objective function, the M-CBO algorithm assigns each particle to a distinct sub-objective function obtained by the method of scalarization. In this paper, we show that for a sufficiently small noise, the expected value of the state diameter decays to zero exponentially fast. This results in almost sure asymptotic consensus. We also show that the CBO particle system approximates a Pareto optimal set in finite time. Moreover, the approximation becomes increasingly accurate, as the noise strength vanishes, while the number of particles and the inverse temperature parameter tend to infinity. These results extend our theoretical understanding for CBO-type algorithms to the multi-objective regime by exhibiting asymptotic behaviors that are not observed in the previous mean-field analysis for M-CBO in [G. Borghi, M. Herty and L. Pareschi, An adaptive consensus based method for multi-objective optimization with uniform Pareto front approximation, Appl. Math. Optim. 88 (2023) 58].
This paper is concerned with a mass-conservative multiscale finite element framework for single-phase flow in highly heterogeneous porous media. The method combines the local mass conservation and low computational cost of the Enriched Galerkin method within the Generalized Multiscale Finite Element Method (GMsFEM) framework, resulting in the Generalized Multiscale Enriched Galerkin (GMsEG) method. The GMsEG method embeds the fine-scale heterogeneity into the continuous multiscale basis functions by solving local eigen-problems, and the continuous multiscale finite element space is enriched by the piecewise constant space on the coarse grid. The resulting method preserves the mass conservation property and can resolve fine-grid heterogeneity while remaining computationally efficient. Well-posedness and convergence estimates of the method are established, and numerical experiments in various heterogeneous porous media demonstrate the efficiency and accuracy of the method, as well as highlight the significance of employing a mass conservative method when flow and transport problems are coupled together.
This editorial paper introduces the articles published in a special issue focused on essays and reviews, which offer a forward-looking perspectives on new frontiers of applied mathematics. First, a brief description of the scientific contributions of the papers published in this issue is provided. Then, based on the papers’ contents, a forward look at research perspectives is brought to the reader’s attention. These perspectives focus on new developments of the kinetic theory of classical and active particles, behavioral swarms, and numerical and analytical study of hyperbolic systems. Finally, our editorial explores how the contents of this issue can have an impact on mathematical tools of Scientific Machine learning Applied to the study of collective dynamics of living systems.
We establish the local-in-time well-posedness of classical solutions to the vacuum free boundary problem of the viscous Saint-Venant system for shallow waters in 1D derived rigorously from incompressible Navier–Stokes system with a moving free surface by Gerbeau and Perthame. Our solutions are uniformly smooth up to the moving boundary, although the depth degenerates as a singularity of the distance to the vacuum boundary. The main proof is built on some elaborate higher-order weighted energy functional and weighted estimates associated to the degenerate structure of the momentum equation.
