Monotone difference schemes stabilized by discrete mollification for strongly degenerate parabolic equations

The discrete mollification method is a convolution‐based filtering procedure suitable for the regularization of ill‐posed problems and for the stabilization of explicit schemes for the solution of PDEs. This method is applied to the discretization of the diffusive terms of a known first‐order monotone finite difference scheme [Evje and Karlsen, SIAM J Numer Anal 37 (2000) 1838–1860] for initial value problems of strongly degenerate parabolic equations in one space dimension. It is proved that the mollified scheme is monotone and converges to the unique entropy solution of the initial value problem, under a CFL stability condition which permits to use time steps that are larger than with the unmollified (basic) scheme. Several numerical experiments illustrate the performance and gains in CPU time for the mollified scheme. Applications to initial‐boundary value problems are included. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 38–62, 2012     

Analysis of the grad-div stabilization for the time-dependent Navier–Stokes equations with inf-sup stable finite elements

This paper studies inf-sup stable finite element discretizations of the evolutionary Navier–Stokes equations with a grad-div type stabilization. The analysis covers both the case in which the solution is assumed to be smooth and consequently has to satisfy nonlocal compatibility conditions as well as the practically relevant situation in which the nonlocal compatibility conditions are not satisfied. The constants in the error bounds obtained do not depend on negative powers of the viscosity. Taking into account the loss of regularity suffered by the solution of the Navier–Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data, error bounds of order \(\mathcal O(h^{2})\) in space are proved. The analysis is optimal for quadratic/linear inf-sup stable pairs of finite elements. Both the continuous-in-time case and the fully discrete scheme with the backward Euler method as time integrator are analyzed.     

Degenerate two-phase incompressible flow II: regularity, stability and stabilization

In this paper, we analyze a coupled system of highly degenerate elliptic-parabolic partial differential equations for two-phase incompressible flow in porous media. This system involves a saturation and a global pressure (or a total flow velocity). First, we show that the saturation is Hölder continuous both in space and time and the total velocity is Hölder continuous in space (uniformly in time). Applying this regularity result, we then establish the stability of the saturation and pressure with respect to initial and boundary data, from which uniqueness of the solution to the system follows. Finally, we establish a stabilization result on the asymptotic behavior of the saturation and pressure; we prove that the solution to the present system converges (in appropriate norms) to the solution of a stationary system as time goes to infinity. An example is given to show typical regularity of the saturation.     

Multilevel preconditioning and low‐rank tensor iteration for space–time simultaneous discretizations of parabolic PDEs

This paper addresses the solution of parabolic evolution equations simultaneously in space and time as may be of interest in, for example, optimal control problems constrained by such equations. As a model problem, we consider the heat equation posed on the unit cube in Euclidean space of moderately high dimension. An a priori stable minimal residual Petrov–Galerkin variational formulation of the heat equation in space–time results in a generalized least squares problem. This formulation admits a unique, quasi‐optimal solution in the natural space–time Hilbert space and serves as a basis for the development of space–time compressive solution algorithms. The solution of the heat equation is obtained by applying the conjugate gradient method to the normal equations of the generalized least squares problem. Starting from stable subspace splittings in space and in time, multilevel space–time preconditioners for the normal equations are derived. In order to reduce the complexity of the full space–time problem, all computations are performed in a compressed or sparse format called the hierarchical Tucker format, supposing that the input data are available in this format. In order to maintain sparsity, compression of the iterates within the hierarchical Tucker format is performed in each conjugate gradient iteration. Its application to vectors in the hierarchical Tucker format is detailed. Finally, numerical results in up to five spatial dimensions based on the recently developed htucker toolbox for MATLAB are presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Space–time DG methods for the coupled electro–mechanical activation of the human heart

We consider the coupled system of time–dependent nonlinear partial differential equations modeling the electromechanical response of human heart tissue. Instead of time–stepping schemes we use a discontinuous Galerkin finite element method in the space–time domain to be able to resolve the solution in space and time simultaneously. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A connection between filter stabilization and eddy viscosity models

Recently, a new approach for the stabilization of the incompressible Navier–Stokes equations for high Reynolds numbers was introduced based on the nonlinear differential filtering of solutions on every time step of a discrete scheme. In this article, the stabilization is shown to be equivalent to a certain eddy‐viscosity model in Large Eddy Simulation. This allows a refined analysis and further understanding of desired filter properties. We also consider the application of the filtering in a projection (pressure correction) method, the standard splitting algorithm for time integration of the incompressible fluid equations. The article proves an estimate on the convergence of the filtered numerical solution to the corresponding Navier‐Stokes solution. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Internal Optimal Controller Synthesis for Navier–Stokes Equations

Barbu and Triggiani (Indiana Univ. Math. J. 2004; 53:1443–1494) have proposed a solution of the internal feedback stabilization problem of Navier–Stokes equations with no-slip boundary conditions. They have shown that any unstable steady-state solution can be exponentially stabilized by a finite-dimensional feedback controller with support in an arbitrary open subset of positive measure. The finite dimension of the feedback controller is minimal and is related to the largest algebraic multiplicity of the unstable eigenvalues of the linearized equation. The feedback law is obtained as a solution of a linear-quadratic control problem. In this paper, we formulate a practical algorithm implementation of the proposed stabilization approach, based on the finite element method, and demonstrate its applicability and effectiveness using an example involving the stabilization of two-dimensional Navier–Stokes equations.     

Strong convergence for explicit space–time discrete numerical approximation methods for stochastic Burgers equations

In this paper we propose and analyze explicit space–time discrete numerical approximations for additive space–time white noise driven stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities such as the stochastic Burgers equation with space–time white noise. The main result of this paper proves that the proposed explicit space–time discrete approximation method converges strongly to the solution process of the stochastic Burgers equation with space–time white noise. To the best of our knowledge, the main result of this work is the first result in the literature which establishes strong convergence for a space–time discrete approximation method in the case of the stochastic Burgers equations with space–time white noise.     

Can Galileons support Lorentzian wormholes?

We discuss the possibility of constructing stable, static, spherically symmetric, asymptotically flat Lorentzian wormhole solutions in general relativity coupled to a generalized Galileon field π. Assuming that the Minkowski space–time is obtained at ?_π = 0, we find that there is tension between the properties of the energy–momentum tensor required to support a wormhole (violation of the average null energy conditions) and stability of the Galileon perturbations about the putative solution (absence of ghosts and gradient instabilities). In three-dimensional space–time, this tension is strong enough to rule out wormholes with the above properties. In higher dimensions, including the most physically interesting case of four-dimensional space–time, wormholes, if any, must have fairly contrived shapes.     

对流扩散反应方程的局部投影稳定化连续时空有限元方法

本文将局部投影稳定化（LPS）方法和连续时空有限元方法相结合研究对流扩散反应方程,给出稳定化连续时空有限元离散格式.与传统的时空有限元研究思路不同,时间方向利用Lagrange插值多项式,解耦时间和空间变量,降低时空有限元解的维数,具有减少计算量和简化理论分析的优点.通过引入Legendre多项式给出了有限元解的稳定性分析,进一步引进Lobatto多项式证明了有限元解的全局L^∞（L²）和局部L²（J_n;LPS）范数误差估计.最后给出数值算例验证理论分析的正确性,以及稳定化格式的可行性和有效性.     

Nonlinear analysis of lattice model with consideration of optimal current difference

Based on Li’s lattice model, a new lattice model considering the influence of optimal current difference information is presented. The linear stability criterion of this model is obtained by employing the linear stability theory. The results show that the new consideration contributes to the stabilization of traffic systems. The modified Korteweg-de Vries (mKdV) equation is derived by using the nonlinear analysis method. The occurrence of traffic jamming transitions could be thus described by the kink-antikink soliton solution for the mKdV equation. From the simulation results of space–time evolution of the vehicle density, it is shown that the traffic jam is suppressed efficiently with considering the information of optimal current difference, and the analytical results is in good agreement with the simulation ones.     

Error estimates for space–time discontinuous Galerkin formulation based on proper orthogonal decomposition

In this study, proper orthogonal decomposition (POD) method is applied to diffusion–convection–reaction equation, which is discretized using space–time discontinuous Galerkin (dG) method. We provide estimates for POD truncation error in dG-energy norm, dG-elliptic projection, and space–time projection. Using these new estimates, we analyze the error between the dG and the POD solution, and the error between the exact and the POD solution. Numerical results, which are consistent with theoretical convergence rates, are presented.     

Global in time existence of self-interacting scalar field in de Sitter spacetimes

We prove the existence of a global in time solution of the semilinear Klein–Gordon equation in the de Sitter space–time. The coefficients of the equation depend on spatial variables as well, that make results applicable to the space–time with the time slices being Riemannian manifolds.     

Estimates of the Nash–Aronson type for degenerating parabolic equations

We consider second-order parabolic equations describing diffusion with degeneration and diffusion on singular and combined structures. We give a united definition of a solution of the Cauchy problem for such equations by means of semigroup theory in the space L ₂ with a suitable measure. We establish some weight estimates for solutions of Cauchy problems. Estimates of Nash–Aronson type for the fundamental solution follow from them. We plan to apply these estimates to known asymptotic diffusion problems, namely, to the stabilization of solutions and to the “central limit theorem.”     

Optimal State Feedback Input-Output Stabilization of Infinite-Dimensional Discrete Time-Invariant Linear Systems

We study the optimal input-output stabilization of discrete time-invariant linear systems in Hilbert spaces by state feedback. We show that a necessary and sufficient condition for this problem to be solvable is that the transfer function has a right factorization over H-infinity. A necessary and sufficient condition in terms of an (arbitrary) realization is that each state which can be reached in a finite time from the zero initial state has a finite cost. Another equivalent condition is that the control Riccati equation has a solution (in general unbounded and even non densely defined). The optimal state feedback input-output stabilization problem can then be solved explicitly in terms of the smallest solution of this control Riccati equation. We further show that after renorming the state space in terms of the solution of the control Riccati equation, the closed-loop system is not only input-output stable, but also strongly internally stable. Received: July 4, 2007. Revised: October 17, 2007.     

Numerical analysis of a continuous Galerkin method for damped sine-Gordon equation

In this article, we discuss the numerical solution for the two-dimensional (2-D) damped sine-Gordon equation by using a space–time continuous Galerkin method. This method allows variable time steps and space mesh structures and its discrete scheme has good stability which are necessary for adaptive computations on unstructured grids. Meanwhile, it can easily get the higher-order accuracy in both space and time directions. The existence and uniqueness to the numerical solution are strictly proved and a priori error estimate in maximum-norm is given without any space–time grid conditions attached. Also, we prove that if the mesh in each time level is generated in a reasonable way, we can get the optimal order of convergence in both temporal and spatial variables. Finally, the convergence rates are presented and analyzed by some numerical experiments to illustrate the validity of the scheme.     

Interior point stabilization for column generation

Interior point stabilization is an acceleration method for column generation algorithms. It addresses degeneracy and convergence difficulties by selecting a dual solution inside the optimal space rather than retrieving an extreme point. The method is applied to the case of the vehicle routing problem with time windows.     

Stabilized nodal‐based finite element methods for the magnetic induction problem

We consider the time‐dependent magnetic induction model as a step towards the resistive magnetohydrodynamics model in incompressible media. Conforming nodal‐based finite element approximations of the induction model with inf‐sup stable finite elements for the magnetic field and the magnetic pseudo‐pressure are investigated. Based on a residual‐based stabilization technique proposed by Badia and Codina, SIAM J. Numer. Anal. 50 (2012), pp. 398–417, we consider a stabilized nodal‐based finite element method for the numerical solution. Error estimates are given for the semi‐discrete model in space. Finally, we present some examples, for example, for the magnetic flux expulsion problem, Shercliff's test case and singular solutions of the Maxwell problem. Copyright © 2015 John Wiley & Sons, Ltd.     

Analysis of the Yosida method for the incompressible Navier–Stokes equations

The Yosida method was introduced in (Quarteroni et al., to appear) for the numerical approximation of the incompressible unsteady Navier–Stokes equations. From the algebraic viewpoint, it can be regarded as an inexact factorization of the matrix arising from the space and time discretization of the problem. However, its differential interpretation resides on an elliptic stabilization of the continuity equation through the Yosida regularization of the Laplacian (see (Brezis, 1983, Ciarlet and Lions, 1991)). The motivation of this method as well as an extensive numerical validation were given in (Quarteroni et al., to appear).In this paper we carry out the analysis of this scheme. In particular, we consider a first-order time advancing unsplit method. In the case of the Stokes problem, we prove unconditional stability and moreover that the splitting error introduced by the Yosida scheme does not affect the overall accuracy of the solution, which remains linear with respect to the time step. Some numerical experiments, for both the Stokes and Navier–Stokes equations, are presented in order to substantiate our theoretical results.     

Amplitude equation for the stochastic reaction‐diffusion equations with random Neumann boundary conditions

In this paper, we consider a quite general class of reaction‐diffusion equations with cubic nonlinearities and with random Neumann boundary conditions. We derive rigorously amplitude equations, using the natural separation of time‐scales near a change of stability and investigate whether additive degenerate noise and random boundary conditions can lead to stabilization of the solution of the stochastic partial differential equation or not. The nonlinear heat equation (Ginzburg–Landau equation) is used to illustrate our result. Copyright © 2015 John Wiley & Sons, Ltd.