Abstract. The Multiscale Hybrid-Mixed (MHM) method is a multiscale finite element method based on a hybrid weak formulation, originally proposed for problems linked to flow in porous media. Its starting point is a variational formulation that guarantees that the flux variable is [Formula: see text]-conforming. This property is lost when the local problems, in their elliptic form, are discretized using primal finite element methods. In this work, we close that gap by proposing and analyzing a new flux reconstruction for the MHM method, computed elementwise on submeshes, that belongs to [Formula: see text]. The reconstruction converges optimally in the [Formula: see text]-norm. Furthermore, the divergence of the reconstructed flow is the piecewise continuous polynomial projection of the source term onto submeshes and thus it also converges optimally in the [Formula: see text]-norm. As a by-product of the reconstruction technique, a fully computable a posteriori error estimator is presented and analyzed. These theoretical results are validated experimentally via numerical computations.
Abstract. A variational multiscale stabilized finite element (VMSFE) scheme for the transient magnetohydrodynamic Navier–Stokes (MHD-NS) equations has been derived. The method involves the decomposition of the unknown solution into resolvable (coarse) and subgrid (fine) scales. The subgrid scales are time-dependent, and the VMSFE formulation is generated by eliminating fine scales in terms of coarse scales. Using Fourier analysis, appropriate stabilization parameters have been derived to achieve the optimal order of convergence. The subgrid scales are taken as orthogonal to the coarse-scale finite element space. This approach ensures a unique decomposition of the solution and prevents the fine scales from polluting the coarse-scale solution. For time discretization, a fully implicit backward Euler scheme has been used. The stability analysis of the VMSFE scheme has been carried out. A priori error estimates have been derived to establish the theoretical convergence of the VMSFE scheme in the [Formula: see text] norm for velocity and in the [Formula: see text] norm for pressure. We also derive the optimal error estimate in the [Formula: see text] norm for the velocity. Numerical experiments on multiply driven cavity flow have been carried out, and the credibility of the proposed VMSFE scheme has been established successfully. The flow pattern is traced for various Hartmann, Reynolds, and magnetic force inclination angle values to visualize the dynamics of the multiply driven flow in a cavity. Moreover, the transient MHD-NS flow behavior has been thoroughly analyzed in the flow past a square cylinder. A complete derivation for their existence and uniqueness is given for the MHD-NS equations with Dirichlet boundary conditions in a bounded domain.
We develop a novel mechanochemical modeling framework to investigate shape deformation and pattern formation in multicomponent vesicle membranes. In this approach, membrane morphology is described by a diffuse-interface elastic bending model, while the microphase separation of membrane-associated proteins is captured via an Ohta–Kawasaki (OK) model defined on the evolving membrane surface. The coupled dynamics are governed by an overdamped force-balance equation for the membrane and an OK-type advection–reaction––diffusion equation, enabling the interplay between curvature elasticity and protein-driven active forces. We implement efficient spectral methods to simulate these dynamics in both two- and three-dimensions. Remarkably, our model successfully reproduces a wide spectrum of complex membrane morphologies and microdomain patterns observed in the landmark Nature experiments of Baumgart, Hess, and Webb [ Nature, 425 (2003), pp. 821–824]. This framework integrates curvature mechanics, microphase separation, and active biochemical forcing, providing new insights into the mechanisms of multicomponent vesicles, and offering a practical platform for predictive modeling of biomembrane dynamics.
Abstract. Collective variables (CVs) play a crucial role in capturing rare events in high-dimensional systems, motivating the continual search for principled approaches to their design. In this work, we revisit the framework of quantitative coarse graining and identify the orthogonality condition from Legoll and Lelievre (2010) as a key criterion for constructing CVs that accurately preserve the statistical properties of the original process. We establish that satisfaction of the orthogonality condition enables error estimates for both relative entropy and pathwise distance to scale proportionally with the degree of scale separation. Building on this foundation, we introduce a general numerical method for designing neural network–based CVs that integrates tools from manifold learning with group-invariant featurization. To demonstrate the efficacy of our approach, we construct CVs for butane and achieve a CV that reproduces the antigauche transition rate with less than ten percent relative error and within two standard deviations of the empirically measured rate. Additionally, we provide empirical evidence challenging the necessity of uniform positive definiteness in diffusion tensors for transition rate reproduction and highlight the critical role of light atoms in CV design for molecular dynamics.
Characterizing the local voltage distribution within nanophysiological domains, driven by ionic currents through membrane channels, is crucial for studying cellular activity in modern biophysics, yet it presents significant experimental and theoretical challenges. Theoretically, the complexity arises from the difficulty of solving electro diffusion equations in three-dimensional domains. Currently, there are no general methods available for obtaining asymptotic computations or approximate solutions of these nonlinear equations, and, numerically, it is challenging to explore solutions across both small and large spatial scales. In this work, we develop a method for solving the Poisson-Nernst-Planck equations with ionic currents entering and exiting through two narrow circular window channels located on the boundary. The inflow through the first window is composed of a single cation species, while the outflow maintains a constant ionic density satisfying local electroneutrality conditions. Employing regular expansions and Green's function representations, we derive steady-state solutions of ionic profiles and voltage drops in both small and large ionic charge regimes. We explore how the local surface curvature and size of windows influence voltage dynamics and validate our theoretical predictions through numerical simulations, assessing the accuracy of our asymptotic computations. These novel relationships between current, voltage, concentrations, and geometry can enhance the characterization of physiological behaviors of nano domains.
The Multiscale Hybrid-Mixed (MHM) method is a multiscale finite element method based on a hybrid weak formulation, originally proposed for problems linked to flow in porous media. Its starting point is a variational formulation that guarantees that the flux variable is H(div; Q)-conforming. This property is lost when the local problems, in their elliptic form, are discretized using primal finite element methods. In this work, we close that gap by proposing and analyzing a new flux reconstruction for the MHM method, computed elementwise on submeshes, that belongs to H(div; Q). The reconstruction converges optimally in the L-2(Q)-norm. Furthermore, the divergence of the reconstructed flow is the piecewise continuous polynomial projection of the source term onto submeshes and thus it also converges optimally in the L-2(Q)-norm. As a by-product of the reconstruction technique, a fully computable a posteriori error estimator is presented and analyzed. These theoretical results are validated experimentally via numerical computations.
We investigate the competing mechanisms of localization in one-dimensional block disordered subwavelength resonator systems subject to nonrecipro cal damping, induced by an imaginary gauge potential. Using a symmetrization approach to enable the adaptation of tools from Hermitian systems, we derive the limiting spectral distribution of these systems as the number of blocks goes to infinity and characterize their spectral properties in terms of the spectral properties of their constituent blocks. By employing a transfer matrix approach, we then clarify, in terms of Lyapunov exponents, the competition between the edge localization due to imaginary gauge potentials and the bulk localization due to disorder. In particular, we demonstrate how the disorder acts as insulation against the non-Hermitian skin effect, preventing edge localization for small imaginary gauge potentials.
In this paper, we consider the time-harmonic acoustic scattering by a locally perturbed interface. The above problem is reduced to an equivalent Lippmann--Schwinger-type integral equation defined in a bounded domain, which preserves all the information of the unbounded interface. A direct discrete high-precision algorithm is developed to solve the above Lippmann--Schwinger equations by introducing a variable discretization step property and a smoothed integration kernel technique, which is not only computationally inexpensive but also fast converging. Moreover, the algorithm can be readily extended to the case of interfaces with multiple corners. Finally, numerical results are presented to illustrate the effectiveness of the algorithm even in the presence of corners.
The main concern of the present paper is the mathematical modeling and numerical simulation of thermonuclear fusion plasmas, constituted of two different kinds of particles, namely, the thermal electron/ion bulk and an energetic \alpha -particle population, created by the fusion reaction at very high speeds (3.5 MeV ). This \alpha -particle population behaves differently than the thermal plasma bulk, and due to its high energy content, can have a considerable impact on the stability and thus confinement of the plasma bulk. An adequate modeling of this \alpha -population is therefore crucial, and the present paper intends to make use of kinetic Fokker--Planck models to describe the evolution of these energetic particles experiencing Coulomb collisions. The main mathematical and numerical difficulties encountered in this study are related to the multiscale nature of the overall problem, involving different types of particles and their multiscale dynamics.
In this paper, we discuss an optimal control problem governed by the backward fractional Feynman--Kac equation, which depicts the probability density function of functionals of widespread diffusion phenomena in various multiscale fields. In the constrained equation, the timespace coupled nonlo cal operator and its noncommutativity with Laplacian bring significant difficulties. The well-posedness, optimality conditions, and solution's regularity of the continuous optimal control problem are established. Then, we propose a temporal semidiscrete scheme for the optimal control problem by applying the backward Euler convolution quadrature formula to discretize the Riemann--Liouville fractional substantial derivative in the equation. In addition, the error estimate of the proposed semidiscrete scheme is rigorously established, and almost optimal convergence of O(\tau | ln\tau | ) is obtained depending only on the regularity assumptions on the data and without extra assumptions on the solution. Some numerical examples are illustrated to verify the theoretical results.
This project focuses on the formal derivation of a unified model for groundwater and surface water flow, combining the shallow-water model and the Dupuit--Forchheimer model. The primary goal of the model is to describe the dynamics of the water table, which represents the level below which the medium is fully saturated, while disregarding unsaturated regions. The unified shallow-water/Dupuit--Forchheimer model provides a single set of equations across the domain. It transitions to the shallow-water model in areas where the porous medium is absent and to the Dupuit-Forchheimer model when the water table remains below the surface. The derivation is carried out in two steps: first, a vertical integration separates the ground and surface media, followed by the application of a low-permeability limit in the ground layer. The model ensures conservation of both mass and energy. A numerical scheme that preserves these properties at the discrete level, along with several numerical experiments, demonstrates the robustness and efficiency of the approach.
In this work, we analyze the asymptotic consistent formulation (ACF) for the periodic Zakharov system (ZS) in the subsonic limit, where the dimensionless parameter \varepsilon approaches zero, signifying an increase in acoustic speed. The primary focus is to determine the optimal asymptotic order of the functions involved in the ACF as \varepsilon \rightarrow 0 for various types of initial data. A highorder extension of the ACF is proposed. Additionally, multiscale integrators based on the ACF are proposed, and their uniform accuracy and asymptotic-preserving properties are discussed. Numerical experiments are conducted to illustrate the performance of these schemes and explore the convergence of the ZS in the subsonic limit. To the best of our knowledge, this is the first time that the asymptotic behavior of the periodic ZS has been rigorously analyzed. Our results show that the behavior of the periodic ZS in the subsonic limit is different from that of the ZS in the whole space.
We present a bottom-up multiscale model, termed sequential and intelligent dissipative particle dynamics (SIDPD), which bridges the gap between micro-and meso-scales with high accuracy and computational efficiency. This model employs machine learning techniques to predict the mesoscopic force fields directly from microscopic data, thus maintaining microscopic accuracy. Moreover, it matches force and momentum sequences rather than using memory kernels, an essential but computationally expensive component for bridging micro-and mesoscales, further enhancing its efficiency. The resulting force fields are applied in dissipative particle dynamics simulations to analyze static and dynamic properties. We consider three distinct microscopic systems with different densities, star polymer, methane and water, to evaluate the model performance. It is shown that SIDPD accurately preserve both static and dynamic properties with high efficiency, even for systems with high density and strong long-range interactions.
This is our third work in the series on constructing boundary conditions for hyperbolic relaxation systems. Here, we focus on the linearized Suliciu model with characteristic boundaries. Assuming that appropriate initial and boundary conditions for the corresponding equilibrium system are given, we construct certain boundary conditions for the Suliciu model by means of a threescale asymptotic expansion so that the resultant initial boundary value problems are asymptotically equivalent to the given problem. The constructed boundary conditions are demonstrated to be strictly dissipative. Moreover, we choose initial data compatible with the boundary conditions. Finally, we validate the constructed boundary conditions by showing the convergence of the solutions to the resultant initial boundary value problems, as the relaxation time tends to zero, toward the solution to the given problem.
Motivated by Hadamard integrators that were originally designed for pure initialvalue problems of wave equations, we develop novel Hadamard integrators for initial-boundary value problems of wave equations in bounded domains in the high-frequency regime, where inhomogeneous Dirichlet boundary conditions may be imposed. To capture high-frequency wave motion near the boundary, we need to deal with two time-boundary integrals in the Huygens--Kirchhoff representation of the wave solution: the first involves the Green's function itself, and the second involves its normal derivative. To eliminate the first time-boundary integral, we construct Hadamard ansatz--based mirror Green's functions that locally satisfy the homogeneous Dirichlet boundary condition so that this integral vanishes. To treat the second time-boundary integral, we use the properties of the Green's function to derive some computable time-boundary singular integrals. To handle these singular integrals numerically, we develop tensorized Gauss--Legendre quadrature formulas to evaluate these integrals to high-order accuracy. Based on these novel treatments of time-boundary integrals, we employ the ray tracing--based Lagrangian Hadamard integrator to update the wavefield in the rest of the computational domain away from the boundary. Consequently, the proposed method can accurately propagate high-frequency waves in bounded domains, either forward in time or backward in time, on relatively coarse grids that scale linearly with respect to the wavenumber, where caustics are implicitly handled with ease. Numerical examples demonstrate the accuracy and efficiency of the proposed method.
We consider two-scale, linear spectral fractional diffusion of order 2s \in (0,2) with homogeneous Dirichlet boundary condition and locally periodic, two-scale coefficients in a bounded domain D \subset Rd, with fundamental period Y = (0, 1)d \subset Rd. We derive a local limiting two-scale homogenized equation for the so-called Caffarelli--Silvestre (CS) extension (we refer to the corresponding boundary value problem simply as the ``extended boundary value problem"" or ``extended BVP"") in the tensorized domain D \times Y \times (0, oo) \subset R2d+1, by applying the two-scale convergence approach of Nguetseng [SIAM J. Math. Anal., 20 (1989), pp. 608--623] and Allaire [SIAM J. Math. Anal., 23 (1992), pp. 1482--1518], to the local, elliptic PDE in the extended BVP. Based on the two-scale homogenized equation of the CS extension, we show that the homogenized equation of the nonlo cal two-scale spectral fractional diffusion problem is the spectral fractional diffusion corresponding to the limiting diffusion operator from classical homogenization theory for local, elliptic diffusion in D. We study anisotropic regularity of the solution of the local, limiting two-scale homogenized equation in D \times Y \times (0, oo). Using this, we develop the essentially optimal sparse tensor product finite element discretizations using continuous, piecewise linear Lagrangian finite elements for each, the slow variable x\prime \in D, the fast variable y \in Y, and the extended variable z \in (0, oo). As the solution of the two-scale homogenized equation is analytic with respect to the extended variable z in weighted Sobolev spaces, we develop a second, likewise essentially optimal approach using the full tensor product of hp finite element spaces in z \in (0, oo) and a sparse tensor product finite element space in D \times Y using continuous, piecewise linear Lagrangian finite element basis functions for the variables x\prime and y. From the finite element solution of this extended two-scale homogenized equation, we construct novel numerical correctors for the two-scale CS extended equation. This results in novel numerical correctors for the solution of the original nonlo cal two-scale spectral fractional diffusion problem. Error estimates in terms of the microscopic scale \varepsilon and the macroscopic finite element mesh size h are rigorously derived for these numerical correctors. Numerical experiments confirm the theoretical error estimates of the sparse tensor product finite element schemes.
We consider a kinetic-fluid model, derived previously by Helzel and Tzavaras suspensions of rod-like particles. The kinetic equation describes the evolution of a density distribution function for the suspended particles as a function of space, orientation, and time. It has the form of a drift-diffusion equation on the sphere (S2) for each point in physical space. In addition, the density distribution of the rod-like particles is translated and diffused in physical space in a way that depends on the particle orientation as well as the flow velocity. The fluid is modeled by a Navier-Stokes equation with a buoyancy term and additional stress tensor that is imposed by the suspended particles. To approximate this high-dimensional system, moment equations are derived, approximating the high-dimensional scalar kinetic equation by a lower-dimensional system of hyperbolic PDEs which only depends on the spatial variables as well as on time. The derivation of the system of moment equations is based on an expansion of the density distribution function using spherical harmonic basis functions. Numerical simulations validate all approximations introduced in this paper. This paper extends previous work by Dahm and Helzel [Multiscale Model. Simul., 20 (2022), pp. 1002-1039], where a simplified version of the kinetic equation was considered that restricts the orientation of the rod-like particles to S1 instead of S2.
Ionic crystals, such as solid electrolytes and complex oxides, are central to modern technologies for energy storage, sensing, actuation, and other functional applications. An important fundamental issue in the atomic and quantum-scale modeling of these materials is defining the macroscopic polarization. In a periodic crystal, the usual definition of the polarization as the first moment of the charge density in a unit cell is found to depend qualitatively---allowing even a change in the sign!---and quantitatively on the choice of unit cell. We examine this issue using a rigorous approach based on the framework of two-scale convergence. By examining the continuum limit of when the lattice spacing is much smaller than the characteristic dimensions of the body, we show that the two-scale limit provides both a bulk polarization as well as a surface charge density supported on the boundary of the body. Further, different choices of the periodic unit cell of the body lead to correspondingly different partial unit cells at the boundary; these choices give different bulk polarization and surface charges but compensate such that the electric field and energy are independent of the choice of unit cell.
We propose a multiscale method for mixed-dimensional elliptic problems with highly heterogeneous coefficients arising, for example, in the modeling of fractured porous media. The method is based on the Localized Orthogonal Decomposition (LOD) framework and constructs locally supported, problem-adapted basis functions on a coarse mesh that does not need to resolve the coefficient oscillations. These basis functions are obtained in parallel by solving localized fine-scale problems. Our a priori error analysis shows that the method achieves optimal convergence with respect to the coarse mesh size, independent of the coefficient regularity, with an exponentially decaying localization error. Numerical experiments validate these theoretical findings and demonstrate the computational viability of the method.
