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In this paper we discretize the incompressible Navier-Stokes equations in the framework of finite element exterior calculus. We make use of the Lamb identity to rewrite the equations into a vorticity-velocity-pressure form which fits into the de Rham complex of minimal regularity. We propose a discretization on a large class of finite elements, including arbitrary order polynomial spaces readily available in many libraries. The main advantage of this discretization is that the divergence of the fluid velocity is pointwise zero at the discrete level. This exactness ensures pressure robustness. We focus the analysis on a class of linearized equations for which we prove well-posedness and provide a priori error estimates. The results are validated with numerical simulations. [ABSTRACT FROM AUTHOR]

In the last few decades, numerical simulation for nonlinear oscillators has received a great deal of attention, and manyresearchers have been concerned with the design and analysis of numerical methods for solving oscillatory problems. In this paper, from the perspective of the continuous finite element method, we propose and analyze new energy-preserving functionally fitted methods, in particular trigonometrically fitted methods of an arbitrarily high order for solving oscillatory nonlinear Hamiltonian systems with a fixed frequency. To implement these new methods in a widespread way, they are transformed into a class of continuous-stage Runge-Kutta methods. This paper is accompanied by numerical experiments on oscillatory Hamiltonian systems such as the FPU problem and nonlinear Schrödinger equation. The numerical results demonstrate the remarkable accuracy and efficiency of our new methods compared with the existing high-order energy-preserving methods in the literature. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with whether the almost sure exponential stability of stochastic differential equations (SDEs) is shared with that of a numerical method. Under the global Lipschitz condition, we first show that the SDE is pth moment exponentially stable (for p ? (0, 1)) if and only if the stochastic theta method is pth moment exponentially stable for a sufficiently small step size. We then show that the pth moment exponential stability of the SDE or the stochastic theta method implies the almost sure exponential stability of the SDE or the stochastic theta method, respectively. Hence, our new theory enables us to study the almost sure exponential stability of the SDEs using the stochastic theta method, instead of the method of the Lyapunov functions. That is, we can now carry out careful numerical simulations using the stochastic theta method with a sufficiently small step size ?t. If the stochastic theta method is pth moment exponentially stable for a sufficiently small p ? (0, 1), we can then infer that the underlying SDE is almost surely exponentially stable. Our new theory also enables us to show the ability of the stochastic theta method to reproduce the almost sure exponential stability of the SDEs. In particular, we give positive answers to two open problems, (P1) and (P2) listed in section 1. [ABSTRACT FROM AUTHOR]

Numerical schemes provably preserving the positivity of density and pressure are highly desirable for ideal magnetohydrodynamics (MHD), but the rigorous positivity-preserving (PP) analysis remains challenging. The difficulties mainly arise from the intrinsic complexity of the MHD equations as well as the indeterminate relation between the PP property and the divergence-free condition on the magnetic field. This paper presents the first rigorous PP analysis of conservative schemes with the Lax--Friedrichs (LF) flux for 1D and multidimensional ideal MHD. The significant innovation is the discovery of the theoretical connection between the PP property and a discrete divergence-free (DDF) condition. This connection is established through the generalized LF splitting properties, which are alternatives to the usually expected LF splitting property that does not hold for ideal MHD. The generalized LF splitting properties involve a number of admissible states strongly coupled by the DDF condition, making their derivation very difficult. We derive these properties via a novel equivalent form of the admissible state set and an important inequality, which is skillfully constructed by technical estimates. Rigorous PP analysis is then presented for finite volume and discontinuous Galerkin schemes with the LF flux on uniform Cartesian meshes. In the 1D case, the PP property is proved for the first-order scheme with proper numerical viscosity, and also for arbitrarily high-order schemes under conditions accessible by a PP limiter. In the 2D case, we show that the DDF condition is necessary and crucial for achieving the PP property. It is observed that even slightly violating the proposed DDF condition may cause failure to preserve the positivity of pressure. We prove that the 2D LF type scheme with proper numerical viscosity preserves both the positivity and the DDF condition. Sufficient conditions are derived for 2D PP high-order schemes, and extension to 3D is discussed. Numerical examples provided in the supplementary material further confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

The ensemble Kalman filter (EnKF) is a widely used methodology for state estimation in partially, noisily observed dynamical systems and for parameter estimation in inverse problems. Despite its widespread use in the geophysical sciences, and its gradual adoption in many other areas of application, analysis of the method is in its infancy. Furthermore, much of the existing analysis deals with the large ensemble limit, far from the regime in which the method is typically used. The goal of this paper is to analyze the method when applied to inverse problems with fixed ensemble size. A continuous time limit is derived and the long-time behavior of the resulting dynamical system is studied. Most of the rigorous analysis is confined to the linear forward problem, where we demonstrate that the continuous time limit of the EnKF corresponds to a set of gradient ows for the data misfit in each ensemble member, coupled through a common preconditioner which is the empirical covariance matrix of the ensemble. Numerical results demonstrate that the conclusions of the analysis extend beyond the linear inverse problem setting. Numerical experiments are also given which demonstrate the benefits of various extensions of the basic methodology. [ABSTRACT FROM AUTHOR]

Radiation transport problems are posed in a high-dimensional phase space, limiting the use of finely resolved numerical simulations. An emerging tool to efficiently reduce computational costs and memory footprint in such settings is dynamical low-rank approximation (DLRA). Despite its efficiency, numerical methods for DLRA need to be carefully constructed to guarantee stability while preserving crucial properties of the original problem. Important physical effects that one likes to preserve with DLRA include capturing the diffusion limit in the high-scattering regimes as well as dissipating energy. In this work we propose and analyze a dynamical low-rank method based on the the "unconventional" basis update & Galerkin step integrator. We show that this method is asymptotic preserving, i.e., it captures the diffusion limit, and energy stable under a CFL condition. The derived CFL condition captures the transition from the hyperbolic to the parabolic regime when approaching the diffusion limit. [ABSTRACT FROM AUTHOR]

In this paper we establish the explicit expression for two-dimensional (2D) electro-magnetic cloaks of arbitrary star shapes without explicit contour expressions of the objects. Furthermore, 2D arbitrary star-shaped time domain cloak models are developed. A new finite element time domain (FETD) scheme is developed to solve the governing equations, and its stability is also proved. Numerical results are presented to confirm our theoretical analysis of the cloak models and the effectiveness of our FETD method. To our best knowledge, this is the first time domain finite element simulation of arbitrary star-shaped cloaks. [ABSTRACT FROM AUTHOR]

We propose an adaptive finite element algorithm to approximate solutions of elliptic problems whose forcing data is locally defined and is approximated by regularization (or mollification). We show that the energy error decay is quasi-optimal in two-dimensional space and suboptimal in three-dimensional space. Numerical simulations are provided to confirm our findings. [ABSTRACT FROM AUTHOR]

We derive a posteriori error estimates for a fully discrete time-implicit finite element approximation of the stochastic total variaton flow (STVF) with additive space time noise. The estimates are first derived for an implementable fully discrete approximation of a regularized STVF. We then show that the derived a posteriori estimates remain valid for the unregularized flow up to a perturbation term that can be controlled by the regularization parameter. Based on the derived a posteriori estimates we propose a pathwise algorithm for the adaptive space-time refinement and perform numerical simulation for the regularized STVF to demonstrate the behavior of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Using dimensionally reduced models for the numerical simulation of thin objects is highly attractive as this reduces the computational work substantially. The case of narrow thin elastic bands is considered, and a convergent finite element discretization for the one-dimensional energy functional and a fully practical, energy-monotone iterative method for computing stationary configurations are devised. Numerical experiments confirm the theoretical findings and illustrate the qualitative behavior of elastic narrow bands. [ABSTRACT FROM AUTHOR]

In the analysis of highly oscillatory evolution problems, it is commonly assumed that a single frequency is present and that it is either constant or, at least, bounded from below by a strictly positive constant uniformly in time. Allowing for the possibility that the frequency depends on time and vanishes at some instance introduces additional difficulties from both the asymptotic analysis and numerical simulation points of view. This work is a first step towards the resolution of these difficulties. In particular, we show that it is still possible in this situation to infer the asymptotic behavior of the solution at the price of more intricate computations, and we derive a second order uniformly accurate numerical method. [ABSTRACT FROM AUTHOR]

We study the unphysical recurrence phenomenon arising in the numerical simulation of the transport equations using the Hermite-spectral method. From a mathematical point of view, the suppression of this numerical artifact with filters is theoretically analyzed for two types of transport equations. It is rigorously proven that all the nonconstant modes are damped exponentially by the filters in both models and formally shown that the filter does not affect the damping rate of the electric energy in the linear Landau damping problem. Numerical tests are performed to show the effect of the filters. [ABSTRACT FROM AUTHOR]

In this work, we propose a class of explicit θ-schemes for solving mean-field backward stochastic differential equations. We first prove a rigorous stability result, based on which sharp error estimates are presented, showing that the proposed θ-schemes yield a second order rate of convergence. Several numerical experiments are carried out to verify the theoretical results. It seems that this is the first attempt to design high order numerical schemes for mean-field backward stochastic differential equations. [ABSTRACT FROM AUTHOR]

We analyze the use of nonfitting meshes for simulating the propagation of electromagnetic waves inside the earth with applications to borehole logging. We avoid the use of parameter homogenization and employ standard edge finite element basis functions. For our geophysical applications, we consider a 3D Maxwell's system with piecewise constant conductivity and globally constant permittivity and permeability. The model is analyzed and discretized using both the E-and H-formulations. Our main contribution is to develop a sharp error estimate for both the electric and magnetic fields. In the presence of singularities, our estimate shows that the magnetic field approximation is converging faster than the electric field approximation. As a result, we conclude that error estimates available in the literature are sharp with respect to the electric field error but provide pessimistic convergence rates for the magnetic field in our geophysical applications. Another surprising consequence of our analysis is that nonfitting meshes deliver the same convergence rate as fitting meshes to approximate the magnetic field. Our theoretical results are numerically illustrated via 2D experiments. For the analyzed cases, the accuracy loss due to the use of nonfitting meshes is limited, even for high conductivity contrasts. [ABSTRACT FROM AUTHOR]

We investigate the growth of tumors using a nonlinear model of partial differential equations which incorporates mechanical laws for tissue compression combined with rules for nutrients availability and drug application. Rigorous analysis and simulations are presented which show the effect of nutrient and drug applications on the progression of the tumor. We construct a convergent finite difference scheme to approximate solutions of the nonlinear system of partial differential equations. Extensive numerical tests show that solutions exhibit a necrotic core when the nutrient level falls below a critical level in accordance with medical observations. The same numerical experiment is performed in the case of drug application for the purpose of comparison. Depending on the balance between nutrient and drug both shrinkage and growth of tumors can occur. The role of inhomogeneous boundary conditions, vascularization, and anisotropies in the development of tumor shape irregularities are discussed. [ABSTRACT FROM AUTHOR]