Energy Norm Research Articles

Global well-posedness and exponential decay estimates for semilinear Newell–Whitehead–Segel equation

Abstract This article presents the application of the Faedo–Galerkin compactness method to establish the local well-posedness of the Newell–Whitehead–Segel equation. By analyzing a finite-dimensional approximate problem, the existence and uniqueness of a local solution were demonstrated. A priori estimates were derived, enabling the transition to the limit and the recovery of the original problem’s local solution. The study further proves the uniqueness and continuous dependence of the solution on initial data. Additionally, under certain conditions, it is shown that the energy norm of the solution decays exponentially over time, and the L 2 {L}^{2} -norm of the time derivative of the solution approaches zero asymptotically.

The pseudospectrum and transient of Kaluza–Klein black holes in Einstein–Gauss–Bonnet gravity

Abstract The spectrum and dynamical instability, as well as the transient effect of the tensor perturbation for the so-called Maeda–Dadhich black hole, a type of Kaluza–Klein black hole, in Einstein–Gauss–Bonnet gravity have been investigated in framework of pseudospectrum. We cast the problem of solving quasinormal modes (QNMs) in AdS-like spacetime as the linear evolution problem of the non-normal operator in null slicing by using ingoing Eddington–Finkelstein coordinates. In terms of spectrum instability, based on the generalized eigenvalue problem, the QNM spectrum and ε-pseudospectrum has been studied, while the open structure of ε-pseudospectrum caused by the non-normality of operator indicates the spectrum instability. In terms of dynamical instability, we introduce the concept of the distance to dynamical instability, which plays a crucial role in bridging the spectrum instability and the dynamical instability. We calculate such distance, named the complex stability radius, as parameters vary. Finally, we show the behavior of the energy norm of the evolution operator, which can be roughly reflected by the three kinds of abscissas in context of pseudospectrum, and find the transient growth of the energy norm of the evolution operator.

Existence and stability of almost finite energy weak solutions to the quantum Euler-Maxwell system

In this paper we prove the existence of global in time, finite energy weak solutions to the quantum magnetohydrodynamics (QMHD) system, for a large class of initial data that are slightly more regular than being of finite energy. No restriction is given on their size or the uniform positivity of the mass density. The QMHD system is a well-established model in superconductivity due to Feynman, and is intimately related to a nonlinear Maxwell-Schrödinger (NLMS) system via the Madelung transformations. The lack of global well-posedness (GWP) results for the NLMS system in the energy norm provides an obstruction to stability of its solutions, and consequently to the use of approximation arguments to make rigorous the Madelung approach. We implement here a new strategy, that derives weak solutions to QMHD starting from weak solutions to NLMS, by exploiting its Duhamel formulation. This derivation is rigorously justified by means of suitable dispersive/smoothing estimates for almost finite energy weak solutions to NLMS. The corresponding weak solutions to QMHD satisfy a generalized irrotationality condition, that allows for the presence of non-trivial vorticity (e.g., vortex filaments) concentrated in the vacuum region. This is in sharp contrast with the Euler-Maxwell system where the dispersion is not able to deal with the vorticity transport, therefore almost GWP (for regular solutions that are small perturbations of constant states) holds only in a lifespan, reciprocal of the size of the initial vorticity. Moreover, we also prove a stability property in the energy norm of the weak solutions constructed by our approach. This result follows from new local smoothing estimates for NLMS that are also of independent interest. As a byproduct, we also show the stability for the electromagnetic Lorentz force.

Unconditionally superconvergence error analysis of an energy-stable finite element method for Schrödinger equation with cubic nonlinearity

In this paper, the unconditionally superconvergence analysis is studied for the cubic Schrödinger equation with an energy-stable finite element method. A different approach is proposed to obtain the unconditionally superclose error estimate in H1-norm firstly without using the time splitting technique required in the previous literature. The key to the analysis is to use a priori boundedness of the numerical solution in energy norm and control the nonlinear terms rigorously by two cases, i.e., τ≤h2 and τ≥h2, where τ denotes the temporal size and h is the spatial size. Subsequently, the global superconvergence error estimate in H1-norm is derived by an effective interpolation post-processing approach. Finally, some numerical experiments are carried out to confirm the theoretical findings.

The weak Galerkin finite element method for Stokes interface problems with curved interface

In this paper, we develop a weak Galerkin (WG) finite element scheme for the Stokes interface problems with curved interface. The conventional numerical schemes rely on the use of straight segments to approximate the curved interface and the accuracy is limited by geometric errors. Hence in our method, we directly construct the weak Galerkin finite element space on the curved cells to avoid geometric errors. For the integral calculation on curved cells, we employ non-affine transformations to map curved cells onto the reference element. The optimal error estimates are obtained in both the energy norm and the L2 norm. A series of numerical experiments are provided to validate the efficiency of the proposed WG method.

An adaptive hybridizable discontinuous Galerkin method for Darcy–Forchheimer flow in fractured porous media

In this paper, we consider an adaptive hybridizable discontinuous Galerkin (HDG) method based on the discrete fracture model for approximation of a single-phase flow in fractured porous media. We are interested in the case that the flow rate in fracture is large enough to justify the use of Forchheimer’s law for modeling flow within the fracture, while Darcy’s law is applied to the surrounding matrix. The HDG method could be designed to simulate the flow in porous media with reduced fractures which consist of many straight lines or planes. More specifically, we use piecewise polynomials of degree [Formula: see text] to approximate the velocity and pressure in fracture and surrounding porous media. The existence and uniqueness of discrete solutions are proved by the Brouwer fixed point theorem, and an efficient and reliable a posteriori error estimator is obtained with respect to an energy norm. Moreover, the HDG scheme, the existence and uniqueness of discrete solutions, and the a posteriori error estimates are also extended to the problem with non-planar, embedded, and intersecting fractures. Finally, several numerical examples are provided to validate the performance of the obtained a posteriori error estimator.

Uniform convergence of finite element method on Bakhvalov-type mesh for a 2-D singularly perturbed convection–diffusion problem with exponential layers

Analyzing uniform convergence of finite element method for a 2-D singularly perturbed convection–diffusion problem with exponential layers on Bakhvalov-type mesh remains a complex, unsolved problem. Previous attempts to address this issue have encountered significant obstacles, largely due to the constraints imposed by a specific mesh. These difficulties stem from three primary factors: the width of the mesh subdomain adjacent to the transition point, constraints imposed by the Dirichlet boundary condition, and the structural characteristics of exponential layers. In response to these challenges, this paper introduces a novel analysis technique that leverages the properties of interpolation and the relationship between the smooth function and the layer function on the boundary. By combining this technique with a simplified interpolation, we establish the uniform convergence of optimal order k under an energy norm for finite element method of any order k. Numerical experiments validate our theoretical findings.

An optimally convergent Fictitious Domain method for interface problems

We introduce a novel Fictitious Domain (FD) unfitted method for interface problems associated with a second-order elliptic linear differential operator, that achieves optimal convergence without the need for adaptive mesh refinements nor enrichments of the Finite Element spaces. The key aspect of the proposed method is that it extends the solution into the fictitious domain in a way that ensures high global regularity. Continuity of the solution across the interface is enforced through a boundary Lagrange multiplier. The subdomains coupling, however, is not achieved by means of the duality pairing with the Lagrange multiplier, but through an L2 product with the H1 Riesz representative of the latter, thus avoiding gradient jumps across the interface. Thanks to the enhanced regularity, the proposed method attains an increase, with respect to standard FD methods, of up to one order of convergence in energy norm. The Finite Element formulation of the method is presented, followed by its analysis. Numerical tests on a model problem demonstrate its effectiveness and its superior accuracy compared to standard unfitted methods.

An invitation to resolvent analysis

AbstractResolvent analysis is a powerful tool that can reveal the linear amplification mechanisms between the forcing inputs and the response outputs about a base flow. These mechanisms can be revealed in terms of a pair of forcing and response modes and the associated energy gains (amplification magnitude) at a given frequency. The linear relationship that ties the forcing and the response is represented through the resolvent operator (transfer function), which is constructed through spatially discretizing the linearized Navier–Stokes operator. One of the unique strengths of resolvent analysis is its ability to analyze statistically stationary turbulent flows. In light of the increasing interest in using resolvent analysis to study a variety of flows, we offer this guide in hopes of removing the hurdle for students and researchers to initiate the development of a resolvent analysis code and its applications to their problems of interest. To achieve this goal, we discuss various aspects of resolvent analysis and its role in identifying dominant flow structures about the base flow. The discussion in this paper revolves around the compressible Navier–Stokes equations in the most general manner. We cover essential considerations ranging from selecting the base flow and appropriate energy norms to the intricacies of constructing the linear operator and performing eigenvalue and singular value decompositions. Throughout the paper, we offer details and know-how that may not be available to readers in a collective manner elsewhere. Towards the end of this paper, examples are offered to demonstrate the practical applicability of resolvent analysis, aiming to guide readers through its implementation and inspire further extensions. We invite readers to consider resolvent analysis as a companion for their research endeavors.

Orbital stability of multi-peakons for a generalized Dullin–Gottwald–Holm equation

In this paper, we consider a generalized Dullin–Gottwald–Holm equation. The equation admits single peakons and multi-peakons. Using energy argument and combining the method of the orbital stability of a single peakon with monotonicity of the local energy norm, we prove that the sum of N sufficiently decoupled peakons is orbitally stable in the energy space.

The stabilized nonconforming virtual element method for the Darcy–Stokes problem

A stabilized nonconforming virtual element method is designed in order to solve the Darcy–Stokes problem, which preserves a divergence-free approximation to the velocity. The same degrees of freedom as the usual Crouzeix–Raviart-type virtual element is used, but a different virtual element space is obtained by modifying the conforming Stokes virtual element with the H1-projection operator. The proposed stabilized scheme contains two jump penalty terms over edges. One is the penalty for jumps of velocity approximation and the other one is the penalty for jumps of its normal component. We analyze this method’s well-posedness and prove its uniform convergence in a discrete energy norm. Finally, we verify the validity of this stabilized scheme by some numerical experiments.

A discontinuous Galerkin method for the Brinkman–Darcy-transport problem

In this paper, a weighted discontinuous Galerkin finite element method and an upwind format are presented to solve the coupled Brinkman–Darcy flow and transport model. The flow in a highly permeable region of the model is governed by the Brinkman equations, and the percolation in the porous media is controlled by the Darcy equations, which are coupled by three interface conditions. The permeability coefficients in this model are strongly nonhomogeneous, anisotropic, and discontinuous; the transport equation is a convection-dominated problem; and the velocity field in the porous media domain does not satisfy the divergence-free condition. A weighted discontinuous Galerkin finite element method is used to solve the complex permeability coefficient problem and an upwind scheme is used to solve the convection dominated problem. The interface conditions can be naturally incorporated into the discrete formulation without introducing additional variables. Optimal error estimates are obtained for the semi-discretization and the full discretization with the backward Euler scheme in suitable energy norm. A series of numerical experiments are provided to illustrate the proposed method, including testing the convergence and accuracy of various types of meshes; simulating fluid flow in complex porous media such as obstacles, layered media, and curved interfaces; and studying the coupled flow behavior of surface-subsurface flows with contaminant transport.

Superconvergence analysis of the conforming discontinuous Galerkin method on a Bakhvalov-type mesh for singularly perturbed reaction–diffusion equation

The conforming discontinuous Galerkin (CDG) method maximizes the utilization of all degrees of freedom of the discontinuous Pk polynomial to achieve a convergence rate two orders higher than its counterpart conforming finite element method employing continuous Pk element. Despite this superiority, there is little theory of the CDG methods for singular perturbation problems. In this paper, superconvergence of the CDG method is studied on a Bakhvalov-type mesh for a singularly perturbed reaction–diffusion problem. For this goal, a pre-existing least squares method has been utilized to ensure better approximation properties of the projection. On the basis of that, we derive superconvergence results for the CDG finite element solution in the energy norm and L2-norm and obtain uniform convergence of the CDG method for the first time.

On Error Estimates of a discontinuous Galerkin Method of the Boussinesq System of Equations

Abstract In this paper, we propose and analyze a discontinuous Galerkin finite element method for solving the transient Boussinesq incompressible heat conducting fluid flow equations. This method utilizes an upwind approach to handle the nonlinear convective terms effectively. We discuss new a priori bounds for the semidiscrete discontinuous Galerkin approximations. Furthermore, we establish optimal a priori error estimates for the semidiscrete discontinuous Galerkin velocity approximation in L 2 \mathbf{L}^{2} and energy norms, the temperature approximation in L 2 L^{2} and energy norms and pressure approximation in L 2 L^{2} -norm for t > 0 t>0 . Additionally, under the smallness assumption on the data, we prove uniform in time error estimates. We also consider a backward Euler scheme for full discretization and derive fully discrete error estimates. Finally, we provide numerical examples to support the theoretical conclusions.

Improved Homogenization Estimates for Higher-order Elliptic Operators in Energy Norms

Improved Homogenization Estimates for Higher-order Elliptic Operators in Energy Norms

A Priori error estimates of Runge–Kutta discontinuous Galerkin schemes to smooth solutions of fractional

We give a priori error estimates of second order in time fully explicit Runge–Kutta discontinuous Galerkin schemes using upwind fluxes to smooth solutions of scalar fractional conservation laws in one space dimension. Under the time step restrictions τ ≤ ch for piecewise linear and τ ≲ h4/3 for higher order finite elements, we prove a convergence rate for the energy norm ‖⋅‖Lt∞Lx2+|⋅|Lx2Hxλ/2 that is optimal for solutions and flux functions that are smooth enough. Our proof relies on a novel upwind projection of the exact solution.

Cross-system interactions for positive tipping cascades

Abstract. Positive tipping points are promising leverage points in social systems for accelerated progress towards climate and sustainability targets. Besides their impact in specific social systems such as energy, food, or social norms and values, positive tipping dynamics may in some cases spread across different systems, amplifying the impact of tipping interventions. However, the cross-system interactions that can create such tipping cascades are sparsely examined. Here, we review interactions across sociotechnical, socioecological, socioeconomic, and sociopolitical systems that can lead to tipping cascades based on the emerging and relevant past evidence. We show that there are several feedback mechanisms where a strategic input can trigger secondary impacts for a disproportionately large positive response, and various agents that can trigger such cascades. This review of cross-system interactions facilitates the quantification and analysis of positive tipping cascades in future studies.

An efficient weak Galerkin FEM for third-order singularly perturbed convection-diffusion differential equations on layer-adapted meshes

In this article, we study the weak Galerkin finite element method to solve a class of a third order singularly perturbed convection-diffusion differential equations. Using some knowledge on the exact solution, we prove a robust uniform convergence of order O(N−(k−1/2)) on the layer-adapted meshes including Bakhvalov-Shishkin type, and Bakhvalov-type and almost optimal uniform error estimates of order O((N−1ln⁡N)(k−1/2)) on Shishkin-type mesh with respect to the perturbation parameter in the energy norm using high-order piecewise discontinuous polynomials of degree k. Here N is the number mesh intervals. We conduct numerical examples to support our theoretical results.

Residual-based a posteriori error estimators for algebraic stabilizations

In this note, we extend the analysis for the residual-based a posteriori error estimators in the energy norm defined for the algebraic flux correction (AFC) schemes (Jha, 2021) to the newly proposed algebraic stabilization schemes (John and Knobloch, 2022; Knobloch, 2023). Numerical simulations on adaptively refined grids are performed in two dimensions showing the higher efficiency of an algebraic stabilization with similar accuracy compared with an AFC scheme.

Energy norm error estimates and convergence analysis for a stabilized Maxwell’s equations in conductive media

The aim of this article is to investigate the well-posedness, stability of solutions to the time-dependent Maxwell’s equations for electric field in conductive media in continuous and discrete settings, and study convergence analysis of the employed numerical scheme. The situation we consider would represent a physical problem where a subdomain is emerged in a homogeneous medium, characterized by constant dielectric permittivity and conductivity functions. It is well known that in these homogeneous regions the solution to the Maxwell’s equations also solves the wave equation, which makes computations very efficient. In this way our problem can be considered as a coupling problem, for which we derive stability and convergence analysis. A number of numerical examples validate theoretical convergence rates of the proposed stabilized explicit finite element scheme.