Uniform stability of spectral nonlinear Galerkin methods

This article provides a stability analysis for the backward Euler schemes of time discretization applied to the spatially discrete spectral standard and nonlinear Galerkin approximations of the nonstationary Navier‐Stokes equations with some appropriate assumption of the data (λ, u₀, f). If the backward Euler scheme with the semi‐implicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraint Δt ≤ (2/λλ₁). Moreover, if the backward Euler scheme with the explicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraints Δt = O(λ) and Δt = O(λ), respectively, where λ ≤ λ, which shows that the restriction on the time step of the spectral nonlinear Galerkin method is less than that of the spectral standard Galerkin method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

A Crank‐Nicolson Petrov‐Galerkin method with quadrature for semi‐linear parabolic problems

We propose and analyze an application of a fully discrete C² spline quadrature Petrov‐Galerkin method for spatial discretization of semi‐linear parabolic initial‐boundary value problems on rectangular domains. We prove second order in time and optimal order H¹ norm convergence in space for the extrapolated Crank‐Nicolson quadrature Petrov‐Galerkin scheme. We demonstrate numerically both L² and H¹ norm optimal order convergence of the scheme even if the nonlinear source term is not smooth. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

Stability and error analysis for spectral Galerkin method for the Navier–Stokes equations with L2 initial data

In this article we consider the spectral Galerkin method with the implicit/explicit Euler scheme for the two‐dimensional Navier–Stokes equations with the L² initial data. Due to the poor smoothness of the solution on [0,1), we use the the spectral Galerkin method based on high‐dimensional spectral space H_M and small time step Δt² on this interval. While on [1,∞), we use the spectral Galerkin method based on low‐dimensional spectral space H_m(m = O(M^1/2)) and large time step Δt. For the spectral Galerkin method, we provide the standard H²‐stability and the L²‐error analysis. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

A Petrov‐Galerkin method with quadrature for semi‐linear second‐order hyperbolic problems

We propose and analyze a Crank–Nicolson quadrature Petrov–Galerkin (CNQPG) ‐spline method for solving semi‐linear second‐order hyperbolic initial‐boundary value problems. We prove second‐order convergence in time and optimal order H² norm convergence in space for the CNQPG scheme that requires only linear algebraic solvers. We demonstrate numerically optimal order H^k, k = 0,1,2, norm convergence of the scheme for some test problems with smooth and nonsmooth nonlinearities. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A two‐level method in space and time for the Navier‐Stokes equations

A two‐level method in space and time for the time‐dependent Navier‐Stokes equations is considered in this article. The approximate solution u_M∈H_M is decomposed into the large eddy component v∈H_m(m < M) and the small eddy component w∈H. We obtain the large eddy component v by solving a standard Galerkin equation in a coarse‐level subspace H_m with a time step length k, whereas the small eddy component w is derived by solving a linear equation in an orthogonal complement subspace H with a time step length pk, where p is a positive integer. The analysis shows that our two‐level scheme has long‐time stability and can reach the same accuracy as the standard Galerkin method in fine‐level subspace H_M for an appropriate configuration of p and m. Moreover, some numerical examples are provided to complement our theoretical analysis. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Global existence of solutions for 2‐D semilinear wave equations with dissipation localized near infinity in an exterior domain

We shall derive some global existence results to semilinear wave equations with a damping coefficient localized near infinity for very special initial data in H×L². This problem is dealt with in the two‐dimensional exterior domain with a star‐shaped complement. In our result, a power p on the non‐linear term |u|^p is strictly larger than the two‐dimensional Fujita‐exponent. Copyright © 2005 John Wiley & Sons, Ltd.     

A quantitative compactness estimate for scalar conservation laws

In the case of a scalar conservation law with convex flux in space dimension one, P. D. Lax proved [Comm. Pure and Appl. Math. 7 (1954)] that the semigroup defining the entropy solution is compact in L for each positive time. The present note gives an estimate of the ?‐entropy in L of the set of entropy solutions at time t > 0 whose initial data run through a bounded set in L¹. © 2005 Wiley Periodicals, Inc.     

Longtime behavior for a nonlinear wave equation arising in elasto‐plastic flow

The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in elasto‐plastic flow u_tt?div{|?u|^m?1?u}?λΔu_t+Δ²u+g(u)=f(x). It proves that under rather mild conditions, the dynamical system associated with above‐mentioned IBVP possesses a global attractor, which is connected and has finite Hausdorff and fractal dimension in the phase spaces X₁=H(Ω) × L²(Ω) and X=(H³(Ω)∩H(Ω)) × H(Ω), respectively. Copyright © 2008 John Wiley & Sons, Ltd.     

Global attractors for 2‐D wave equations with displacement‐dependent damping

In this paper the long‐time behaviour of the solutions of 2‐D wave equation with a damping coefficient depending on the displacement is studied. It is shown that the semigroup generated by this equation possesses a global attractor in H(Ω) × L₂(Ω) and H²(Ω)∩H(Ω) × H(Ω). Copyright © 2009 John Wiley & Sons, Ltd.     

On the asymptotic behaviour of the discrete spectrum in buckling problems for thin plates

We consider the buckling problem for a family of thin plates with thickness parameter ε. This involves finding the least positive multiple λ of the load that makes the plate buckle, a value that can be expressed in terms of an eigenvalue problem involving a non‐compact operator. We show that under certain assumptions on the load, we have λ = ??(ε²). This guarantees that provided the plate is thin enough, this minimum value can be numerically approximated without the spectral pollution that is possible due to the presence of the non‐compact operator. We provide numerical computations illustrating some of our theoretical results. Copyright © 2005 John Wiley & Sons, Ltd.     

On the global well‐posedness of discrete Boltzmann systems with chemical reaction

We study an initial‐boundary value problem in one‐space dimension for the discrete Boltzmann equation extended to a diatomic gas undergoing both elastic multiple collisions and chemical reactions. By integration of conservation equations, we prove a global existence result in the half‐space for small initial data N₀∈??∩L¹. Copyright © 2005 John Wiley & Sons, Ltd.     

Dissipativity of the linearly implicit Euler scheme for Navier‐Stokes equations with delay

In this article, we study the dissipativity of the linearly implicit Euler scheme for the 2D Navier‐Stokes equations with time delay volume forces (NSD). This scheme can be viewed as an application of the implicit Euler scheme to linearized NSD. Therefore, only a linear system is needed to solve at each time step. The main results we obtain are that this scheme is L² dissipative for any time step size and H¹ dissipative under a time‐step constraint. As a consequence, the existence of a numerical attractor of the discrete dynamical system is established. A by‐product of the dissipativity analysis of the linearly implicit Euler scheme for NSD is that the dissipativity of an implicit‐explicit scheme for the celebrated Navier‐Stokes equations that treats the volume forces term explicitly is obtained.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2114–2140, 2017     

A lagrangian finite element method for the 2-D euler equations

A grid free Lagrangian finite element method is introduced for the 2-D incompressible Euler equations. The method is derived based on the observation that the product of the Biot-Savart kernel and a polynomial can be integrated analytically over any triangle. This enables us to obtain a numerically stable Lagrangian method without using numerical smoothing. Moreover, we show that the method converges uniformly with second-order accuracy. Actually, we establish a l^∞ stability result which applies to kernels that are more singular than the Biot-Savart kernel, as long as the kernel is L integrable. Another useful result is that we prove convergence of our method when using local regridding, which allows the method to run for longer time even with a fixed mesh. The second-order convergence is also illustrated by our numerical experiments.     

Blowup for nonlinear wave equations describing boson stars

We consider the nonlinear wave equation modeling the dynamics of (pseudorelativistic) boson stars. For spherically symmetric initial data, u₀(x) ∈ C (?³), with negative energy, we prove blowup of u(t, x) in the H^1/2‐norm within a finite time. Physically this phenomenon describes the onset of “gravitational collapse” of a boson star. We also study blowup in external, spherically symmetric potentials, and we consider more general Hartree‐type nonlinearities. As an application, we exhibit instability of ground state solitary waves at rest if m = 0. © 2007 Wiley Periodicals, Inc.     

Non-linear parabolic problem associated with the penetration of a magnetic field into a substance

We study the following initial and boundary value problem: In section 1, with u₀ in L²(Ω), f continuous such that f(u) + ? non-decreasing for ? positive, we prove the existence of a unique solution on (0,T), for each T > 0. In section 2 it is proved that the unique soluition u belongs to L²(0, T; H ∩ H²) ∩ L^∞(0, T; H) if we assume u₀ in H and f in C¹(?,?). Numerical results are given for these two cases.     

Singularities of the resolvent at the thresholds of a stratified operator: a general method

Our problem is about propagation of waves in stratified strips. The operators are quite general, a typical example being a coupled elasto‐acoustic operator H defined in ?² × I where I is a bounded interval of ? with coefficients depending only on z∈I. One applies the ‘conjugate operator method’ to an operator obtained by a spectral decomposition of the partial Fourier transform ? of H. Around each value of the spectrum (except the eigenvalues) including the thresholds, a conjugate operator may be constructed which ensures the ‘good properties’ of regularity for H. A limiting absorption principle is then obtained for a large class of operators at every point of the spectrum (except eigenvalues). If the point is a threshold, the limiting absorption principle is valid in a closed subspace of the usual one (namely L, with s>½) and we are interested by the behaviour of R(z), z close to a threshold, applying in the usual space L, with s>½ when z tends to the threshold. Copyright © 2004 John Wiley & Sons, Ltd.     

On the well‐posedness of the Cauchy problem for an MHD system in Besov spaces

This paper is devoted to the study of the Cauchy problem of incompressible magneto‐hydrodynamics system in the framework of Besov spaces. In the case of spatial dimension n?3, we establish the global well‐posedness of the Cauchy problem of an incompressible magneto‐hydrodynamics system for small data and the local one for large data in the Besov space ? (?ⁿ), 1?p<∞ and 1?r?∞. Meanwhile, we also prove the weak–strong uniqueness of solutions with data in ? (?ⁿ)∩L²(?ⁿ) for n/2p+2/r>1. In the case of n=2, we establish the global well‐posedness of solutions for large initial data in homogeneous Besov space ? (?²) for 2<p<∞ and 1?r<∞. Copyright © 2008 John Wiley & Sons, Ltd.     

Superconvergence of a 3D finite element method for stationary Stokes and Navier‐Stokes problems

For the Poisson equation on rectangular and brick meshes it is well known that the piecewise linear conforming finite element solution approximates the interpolant to a higher order than the solution itself. In this article, this type of supercloseness property is established for a special interpolant of the Q₂ ? P element applied to the 3D stationary Stokes and Navier‐Stokes problem, respectively. Moreover, applying a Q₃ ? P postprocessing technique, we can also state a superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself. Finally, we show that inhomogeneous boundary values can be approximated by the Lagrange Q₂‐interpolation without influencing the superconvergence property. Numerical experiments verify the predicted convergence rates. Moreover, a cost‐benefit analysis between the two third‐order methods, the post‐processed Q₂ ? P discretization, and the Q₃ ? P discretization is carried out. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Monotone numerical schemes for a Dirichlet problem for elliptic operators in divergence form

We consider a second‐order differential operator A( x )=?∑?_ia_ij( x )?_j+ ∑?_j(b_j( x )·)+c( x ) on ?^d, on a bounded domain D with Dirichlet boundary conditions on ?D, under mild assumptions on the coefficients of the diffusion tensor a_ij. The object is to construct monotone numerical schemes to approximate the solution of the problem A( x )u( x )=µ( x ), x ∈D, where µ is a positive Radon measure. We start by briefly mentioning questions of existence and uniqueness introducing function spaces needed to prove convergence results. Then, we define non‐standard stencils on grid‐knots that lead to extended discretization schemes by matrices possessing compartmental structure. We proceed to discretization of elliptic operators, starting with constant diffusion tensor and ending with operators in divergence form. Finally, we discuss W‐convergence in detail, and mention convergence in C and L₁ spaces. We conclude by a numerical example illustrating the schemes and convergence results. Copyright © 2008 John Wiley & Sons, Ltd.     

A multilevel finite element method in space‐time for the Navier‐Stokes problem

A multilevel finite element method in space‐time for the two‐dimensional nonstationary Navier‐Stokes problem is considered. The method is a multi‐scale method in which the fully nonlinear Navier‐Stokes problem is only solved on a single coarsest space‐time mesh; subsequent approximations are generated on a succession of refined space‐time meshes by solving a linearized Navier‐Stokes problem about the solution on the previous level. The a priori estimates and error analysis are also presented for the J‐level finite element method. We demonstrate theoretically that for an appropriate choice of space and time mesh widths: h_j ～ h, k_j ～ k, j = 2, …, J, the J‐level finite element method in space‐time provides the same accuracy as the one‐level method in space‐time in which the fully nonlinear Navier‐Stokes problem is solved on a final finest space‐time mesh. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005