Stability and convergence of fully discrete finite element schemes for the acoustic wave equation

In this paper, we investigate the stability and convergence of some fully discrete finite element schemes for solving the acoustic wave equation where a discontinuous Galerkin discretization in space is used. We first review and compare conventional time-stepping methods for solving the acoustic wave equation. We identify their main properties and investigate their relationship. The study includes the Newmark algorithm which has been used extensively in applications. We present a rigorous stability analysis based on the energy method and derive sharp stability results covering some well-known CFL conditions. A convergence analysis is carried out and optimal a priori error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L ²-norm error over a finite time interval converges optimally as O(h ^p+1+??t ^s ), where p denotes the polynomial degree, s=1 or 2, h the mesh size, and ??t the time step.     

A New Compact Finite Difference Method for Solving the Generalized Long Wave Equation

The aim of this article is to analyze a new compact finite difference method (CFDM) for solving the generalized regularized long wave (GRLW) equation. This method leads to a system of linear equations involving tridiagonal matrices and the rate of convergence of the method is of order O(k ² + h ⁴) where k and h are mesh sizes of time and space variables, respectively. Stability analysis of the method is investigated by the energy method and an error estimate is given. The propagation of single solitons and interaction of two solitary waves are applied to validate the method which is found to be accurate and efficient. Three invariants of the motion are evaluated to determine conservation properties of the method.     

A Priori Error Estimates for Finite Element Methods for H (2,1)-Elliptic Equations

The convergence of finite element methods for linear elliptic boundary value problems of second and forth order is well understood. In this article, we introduce finite element approximations of some linear semi-elliptic boundary value problem of mixed order on a two-dimensional rectangular domain Q. The equation is of second order in one direction and forth order in the other and appears in the optimal control of parabolic partial differential equations if one eliminates the control and the state (or the adjoint state) in the first order optimality conditions. We establish a regularity result and estimate for the finite element error of conforming approximations of this equation. The finite elements in use have a tensor product structure, in one dimension we use linear, quadratic or cubic Lagrange elements in the other dimension cubic Hermite elements. For these elements, we prove the error bound O(h ² + τ ^k ) in the energy norm and O((h ² + τ ^k )(h ² + τ)) in the L ²(Q)-norm.     

Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids

In this paper, we study the convergence of a finite difference scheme on nonuniform grids for the solution of second-order elliptic equations with mixed derivatives and variable coefficients in polygonal domains subjected to Dirichlet boundary conditions. We show that the scheme is equivalent to a fully discrete linear finite element approximation with quadrature. It exhibits the phenomenon of supraconvergence, more precisely, for s ∈ [1,2] order O(h ^s )-convergence of the finite difference solution, and its gradient is shown if the exact solution is in the Sobolev space H ^1+s (Ω). In the case of an equation with mixed derivatives in a domain containing oblique boundary sections, the convergence order is reduced to O(h ^3/2?ε) with ε > 0 if u ∈ H ³(Ω). The second-order accuracy of the finite difference gradient is in the finite element context nothing else than the supercloseness of the gradient. For s ∈ {1,2}, the given error estimates are strictly local.     

On Kirchhoff's Model of Parabolic Type

In this article, the existence of a global strong solution for all finite time is derived for the Kirchhoff's model of parabolic type. Based on exponential weight function, some new regularity results which reflect the exponential decay property are obtained for the exact solution. For the related dynamics, the existence of a global attractor is shown to hold for the problem when the non-homogeneous forcing function is either independent of time or in L^∞(L²). With the finite element Galerkin method applied in spatial direction keeping time variable continuous, a semidiscrete scheme is analyzed, and it is also established that the semidiscrete system has a global discrete attractor. Optimal error estimates in L^∞(H¹) norm are derived which are valid uniformly in time. Further, based on a backward Euler method, a completely discrete scheme is analyzed and error estimates are derived. It is also further, observed that in cases where f = 0 or f = O(e^?γ₀t) with γ₀ > 0, the discrete solutions and error estimates decay exponentially in time. Finally, some numerical experiments are discussed which confirm our theoretical findings.     

Small Data Scattering for the One-Dimensional Nonlinear Dirac Equation with Power Nonlinearity

We study scattering problems for the one-dimensional nonlinear Dirac equation (?_t + α?_x + iβ)Φ = λ|Φ|^p?1Φ. We prove that if p > 3 (resp. p > 3 + 1/6), then the wave operator (resp. the scattering operator) is well-defined on some 0-neighborhood of a weighted Sobolev space. In order to prove these results, we use linear operators D(t)xD(?t) and t?_x + x?_t ? α/2, where {D(t)}_t∈? is the free Dirac evolution group. For the reader's convenience, in an appendix we list and prove fundamental properties of D(t)xD(?t) and t?_x + x?_t ? α/2.     

A two-grid method with expanded mixed element for nonlinear reaction-diffusion equations

Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O(△t + h k+1 + H 2k+2 d/2 ) (k ≥ 1), where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.     

The Stability in W s,p (Γ) Spaces of L 2-Projections on Some Convex Sets

ABSTRACT

For a polygonal open bounded subset of ?², of boundary Γ, we study stability estimates for the projection operator from L ¹(Γ) on a convex set K _h of continuous piecewise affine functions satisfying bound constraints. We establish stability estimates in L ^p (Γ) and in W ^s,p (Γ) for 1 ≤ p ≤ ∞ and 0 < s ≤ 1. This kind of result plays a crucial role in error estimates for the numerical approximation of optimal control problems of partial differential equations with bilateral control constraints.     

Modified Lagrange–Galerkin methods of first and second order in time for convection–diffusion problems

We introduce modified Lagrange–Galerkin (MLG) methods of order one and two with respect to time to integrate convection–diffusion equations. As numerical tests show, the new methods are more efficient, but maintaining the same order of convergence, than the conventional Lagrange–Galerkin (LG) methods when they are used with either P ₁ or P ₂ finite elements. The error analysis reveals that: (1) when the problem is diffusion dominated the convergence of the modified LG methods is of the form O(h ^m+1 + h ² + Δt ^q ), q = 1 or 2 and m being the degree of the polynomials of the finite elements; (2) when the problem is convection dominated and the time step Δt is large enough the convergence is of the form O(\frach^m+1Dt+h²+Dt^q){O(\frac{h^{m+1}}{\Delta t}+h^{2}+\Delta t^{q})} ; (3) as in case (2) but with Δt small, then the order of convergence is now O(h ^m  + h ² + Δt ^q ); (4) when the problem is convection dominated the convergence is uniform with respect to the diffusion parameter ν (x, t), so that when ν → 0 and the forcing term is also equal to zero the error tends to that of the pure convection problem. Our error analysis shows that the conventional LG methods exhibit the same error behavior as the MLG methods but without the term h ². Numerical experiments support these theoretical results.     

Finite volume method based on stabilized finite elements for the nonstationary Navier–Stokes problem

A finite volume method based on stabilized finite element for the two‐dimensional nonstationary Navier–Stokes equations is investigated in this work. As in stabilized finite element method, macroelement condition is introduced for constructing the local stabilized formulation of the nonstationary Navier–Stokes equations. Moreover, for P₁ ? P₀ element, the H¹ error estimate of optimal order for finite volume solution (u_h,p_h) is analyzed. And, a uniform H¹ error estimate of optimal order for finite volume solution (u_h, p_h) is also obtained if the uniqueness condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Galerkin discontinuous approximation of the transport equation and viscoelastic fluid flow on quadrilaterals

Numerical simulation of industrial processes involving viscoelastic liquids is often based on finite element methods on quadrilateral meshes. However, numerical analysis of these methods has so far been limited to triangular meshes. In this work, we consider quadrilateral meshes. We first study the approximation of the transport equation by a Galerkin discontinuous method and prove an 𝒪(h^k+1/2) error estimates for the Q_k finite element. Then we study a differential model for viscoelastic flow with unknowns u the velocity, p the pressure, and σ the viscoelastic part of the extra-stress tensor. The approximations are ((Q₁)² transforms of) Q_k+1 continuous for u, Q_k discontinuous for σ, and P_k discontinuous for p, with k ≥ 1. Upwinding for σ is obtained by the Galerkin discontinuous method. We show that an error estimate of order 𝒪(h^k+1/2) is valid in the energy norm for the three unknowns. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 97–114, 1998     

Combination of standard Galerkin and subspace methods for the time‐dependent Navier‐Stokes equations with nonsmooth initial data

We consider a combination of the standard Galerkin method and the subspace decomposition methods for the numerical solution of the two‐dimensional time‐dependent incompressible Navier‐Stokes equations with nonsmooth initial data. Because of the poor smoothness of the solution near t = 0, we use the standard Galerkin method for time interval [0, 1] and the subspace decomposition method time interval [1, ∞). The subspace decomposition method is based on the solution into the sum of a low frequency component integrated using a small time step Δt and a high frequency integrated using a larger time step pΔt with p > 1. From the H¹‐stability and L²‐error analysis, we show that the subspace decomposition method can yield a significant gain in computing time. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009     

Analysis of the immersed boundary method for a finite element Stokes problem

Convergence results are presented for the immersed boundary (IB) method applied to a model Stokes problem. As a discretization method, we use the finite element method. First, the immersed force field is approximated using a regularized delta function. Its error in the W^{?1, p} norm is examined for 1 ≤ p < n/(n ? 1), with n representing the space dimension. Subsequently, we consider IB discretization of the Stokes problem and examine the regularization and discretization errors separately. Consequently, error estimate of order h^{1 ? α} in the W^{1, 1} × L¹ norm for the velocity and pressure is derived, where α is an arbitrary small positive number. The validity of those theoretical results is confirmed from numerical examples.     

Finite element method for Grwünwald–Letnikov time-fractional partial differential equation

In this article, we consider the finite element methods (FEM) for Grwünwald–Letnikov time-fractional diffusion equation, which is obtained from the standard two-dimensional diffusion equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0?h ^r+1?+?τ^2-α), where h, τ and r are the space step size, time step size and polynomial degree, respectively. A numerical example is presented to verify the order of convergence.     

Error Estimation for Discontinuous Galerkin Solutions of Two-Dimensional Hyperbolic Problems

We analyze the discretization errors of discontinuous Galerkin solutions of steady two-dimensional hyperbolic conservation laws on unstructured meshes. We show that the leading term of the error on each element is a linear combination of orthogonal polynomials of degrees p and p+1. We further show that there is a strong superconvergence property at the outflow edge(s) of each element where the average discretization error converges as O(h ^2p+1) compared to a global rate of O(h ^p+1). Our analyses apply to both linear and nonlinear conservation laws with smooth solutions. We show how to use our theory to construct efficient and asymptotically exact a posteriori discretization error estimates and we apply these to some examples.     

Superconvergence of H 1-Galerkin mixed finite element methods for parabolic problems

In this article, we study the semidiscrete H ¹-Galerkin mixed finite element method for parabolic problems over rectangular partitions. The well-known optimal order error estimate in the L ²-norm for the flux is of order 𝒪(h ^k+1) (SIAM J. Numer. Anal. 35 (2), (1998), pp. 712–727), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas element. We derive a superconvergence estimate of order 𝒪(h ^k+3) between the H ¹-Galerkin mixed finite element approximation and an appropriately defined local projection of the flux variable when k ≥ 1. A the new approximate solution for the flux with superconvergence of order 𝒪(h ^k+3) is realized via a postprocessing technique using local projection methods.     

An Explicit Computation of a Family of Trivializing Étale Covers

We explicitly compute étale covers of the smooth Fermat curves Y _p+1 = Proj k[u, v, w]/(u ^p+1 + v ^p+1 ? w ^p+1) which trivialize the vector bundles Syz(u ², v ², w ²)(3), where k is a field of characteristic p ≥ 3.     

Asymptotic Behavior of Poisson Kernels on NA Groups

On a Lie group S = NA, that is a split extension of a nilpotent Lie group N by a one-parameter group of automorphisms A, a probability measure μ is considered and treated as a distribution according to which transformations s ∈ S acting on N = S/A are sampled. Under natural conditions, formulated some over thirty years ago, there is a μ-invariant measure m on N. Properties of m have been intensively studied by a number of authors. The present article deals with the situation when μ(A) = ?(s _t  ∈ A), where ?⁺ ? t → s _t  ∈ S is the diffusion on S generated by a second order subelliptic, hypoelliptic, left-invariant operator on S. In this article the most general operators of this kind are considered. Precise asymptotic for m at infinity and for the Green function of the operator are given. To achieve this goal a pseudodifferential calculus for operators with coefficients of finite smoothness is formulated and applied.     

Unconditional superconvergent analysis of a new mixed finite element method for Ginzburg–Landau equation

In this article, unconditional superconvergent analysis of a linearized fully discrete mixed finite element method is presented for a class of Ginzburg–Landau equation based on the bilinear element and zero‐order Nédélec's element pair (Q₁₁/Q₀₁ × Q₁₀). First, a time‐discrete system is introduced to split the error into temporal error and spatial error, and the corresponding error estimates are deduced rigorously. Second, the unconditional superclose and optimal estimate of order O(h² + τ) for u in H¹‐norm and p = ?u in L²‐norm are derived respectively without the restrictions on the ratio between h and τ, where h is the subdivision parameter and τ, the time step. Third, the global superconvergent results are obtained by interpolated postprocessing technique. Finally, some numerical results are carried out to confirm the theoretical analysis.