Uniform boundary controllability of a semi-discrete 1-D wave equation

Summary. A numerical scheme for the controlled semi-discrete 1-D wave equation is considered. We analyze the convergence of the boundary controls of the semi-discrete equations to a control of the continuous wave equation when the mesh size tends to zero. We prove that, if the high modes of the discrete initial data have been filtered out, there exists a sequence of uniformly bounded controls and any weak limit of this sequence is a control for the continuous problem. The number of the eliminated frequencies depends on the mesh size and the regularity of the continuous initial data. The case of the HUM controls is also discussed. Received March 3, 2001 / Published online October 17, 2001     

Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations

In arbitrary dimension, in the discrete setting of finite-differences we prove a Carleman estimate for a semi-discrete parabolic operator, in which the large parameter is connected to the mesh size. This estimate is applied for the derivation of a (relaxed) observability estimate, that yield some controlability results for semi-linear semi-discrete parabolic equations. Sub-linear and super-linear cases are considered.     

Numerical solution of the acoustic wave equation using Raviart–Thomas elements

In this paper we discuss the numerical approximation of the displacement form of the acoustic wave equation using mixed finite elements. The mixed formulation allows for approximation of both displacement and pressure at each time step, without the need for post-processing. Lowest-order and next-to-lowest-order Raviart–Thomas elements are used for the spatial discretization, and centered finite differences are used to advance in time. Use of these Raviart–Thomas elements results in a diagonal mass matrix for resolution of pressure, and a mass matrix for the displacement variable that is sparse with a simple structure. Convergence results for a model problem are provided, as are numerical results for a two-dimensional problem with a point source.     

Time-periodic solutions of a nonlinear wave equation

The existence of a time-periodic solution of an n-dimensional nonlinear wave equation is established with n=2 and 3.     

Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation

We are concerned with the boundary controllability to the trajectories of the Kuramoto-Sivashinsky equation. By using a Carleman estimate, we obtain the null controllability of the linearized equation around a given solution. From a local inversion theorem we get the local controllability to the trajectories of the nonlinear system.     

Space–time discontinuous Galerkin finite element method for shallow water flows

A space–time discontinuous Galerkin (DG) finite element method is presented for the shallow water equations over varying bottom topography. The method results in nonlinear equations per element, which are solved locally by establishing the element communication with a numerical HLLC flux. To deal with spurious oscillations around discontinuities, we employ a dissipation operator only around discontinuities using Krivodonova's discontinuity detector. The numerical scheme is verified by comparing numerical and exact solutions, and validated against a laboratory experiment involving flow through a contraction. We conclude that the method is second order accurate in both space and time for linear polynomials.     

On mixed finite element methods for linear elastodynamics

Summary We construct and analyze finite element methods for approximating the equations of linear elastodynamics, using mixed elements for the discretization of the spatial variables. We consider two different mixed formulations for the problem and analyze semidiscrete and up to fourth-order in time fully discrete approximations.L ² optimal-order error estimates are proved for the approximations of displacement and stress.Work supported in part by the Hellenic State Scholarship Foundation     

Error estimates for a mixed finite element discretization of some degenerate parabolic equations

We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases. A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based on the mixed finite element method (MFEM) in space, and is of one step implicit in time. The lowest order Raviart–Thomas elements are used. Here we extend the results in Radu et al. (SIAM J Numer Anal 42:1452–1478, 2004), Schneid et al. (Numer Math 98:353–370, 2004) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart–Thomas elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples.     

On the influence of numerical integration on mixed finite element approximations of a Maxwell eigenvalue problem

The behaviour of electromagnetic resonances in cavities is modelled by a Maxwell eigenvalue problem (EVP). In the present work, we rewrite the corresponding variational problem, as it arises with a view to the application of a finite element method, in a mixed formulation. For the modelling of realistic problems the integrals occurring in this mixed formulation often cannot be evaluated exactly. We take into account the error arising from numerical quadrature and show convergence to the approximations using exact integration. Finally, some numerical results are presented.     

An identification problem for a time periodic nonlinear wave equation in non cylindrical domains

The existence of a time-periodic solution of a free boundary nonlinear wave equation in non cylindrical domains is established. The problem arises in the study of the identification of the coefficient of the wave equation and of the boundary of the region from the observed values of the solution in a fixed subregion.     

Polynomial decay rate for the dissipative wave equation

Using Fourier integral operators with special amplitude functions, we analyze the stabilization of the wave equation in a three-dimensional bounded domain on which exists a trapped ray bouncing up and down infinitely between two parallel parts of the boundary.     

Approximate controllability of a class of semilinear degenerate systems with boundary control

This paper concerns a class of control systems governed by semilinear degenerate equations with boundary control in one-dimensional space. The control is proposed on the ‘degenerate’ part of the boundary. The control systems are shown to be approximately controllable by Kakutani's fixed point theorem.     

The Cauchy problem for the 1-D Dirac–Klein–Gordon equation

The Cauchy problem for the Dirac–Klein–Gordon equation are discussed in one space dimension. Time local and global existence for solutions with rough data, especially the solutions for Klein–Gordon equation in the critical and super critical Sobolev norm of [4] are considered. The solutions with general propagation speeds are dealt with.      

A convergent finite element formulation for transonic flow

Summary A finite element formulation for the full potential equation in the case of two-dimensional transonic flow is presented. The formulation is based on an optimal control approach developed by Glowinski and Pironneau. The solution of the full potential equation is obtained by a minimization problem. Using a new compactness result it is possible to prove convergence for the solutions of the minimization problem. The a priori assumption of existence and uniqueness of a weak solution of the full potential equation satisfying an entropy condition implies that the limit function must be the solution. It is possible to extend the convergence result to the case of three-dimensional transonic potential flow.The research reported here was supported by a grant from the Stiftung Volkswagenwerk, Federal Republic of Germany. It is a part of the doctoral thesis of the above author, Universität Stuttgart 1989     

Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes

In this article, we derive uniform admissibility and observability properties for the finite element space semi-discretizations of , where A ₀ is an unbounded self-adjoint positive definite operator with compact resolvent. To address this problem, we present a new spectral approach based on several spectral criteria for admissibility and observability of such systems. Our approach provides very general admissibility and observability results for finite element approximation schemes of , which stand in any dimension and for any regular mesh (in the sense of finite elements). Our results can be combined with previous works to derive admissibility and observability properties for full discretizations of . We also present applications of our results to controllability and stabilization problems. The author was partially supported by the “Agence Nationale de la Recherche” (ANR), Project C-QUID, number BLAN-3-139579.     

Carleman estimates for degenerate parabolic operators with applications to null controllability

We prove an estimate of Carleman type for the one dimensional heat equation $$ u_t - \left( {a\left( x \right)u_x } \right)_x + c\left( {t,x} \right)u = h\left( {t,x} \right),\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where a(·) is degenerate at 0. Such an estimate is derived for a special pseudo-convex weight function related to the degeneracy rate of a(·). Then, we study the null controllability on [0, 1] of the semilinear degenerate parabolic equation $$ u_t - \left( {a\left( x \right)u_x } \right)_x + f\left( {t,x,u} \right) = h\left( {t,x} \right)\chi _\omega \left( x \right), $$ where (t, x) ∈(0, T) × (0, 1), ω=(α, β) ⊂⊂ [0, 1], and f is locally Lipschitz with respect to u. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

An augmented mixed finite element method for 3D linear elasticity problems

In this paper we introduce and analyze a new augmented mixed finite element method for linear elasticity problems in 3D. Our approach is an extension of a technique developed recently for plane elasticity, which is based on the introduction of consistent terms of Galerkin least-squares type. We consider non-homogeneous and homogeneous Dirichlet boundary conditions and prove that the resulting augmented variational formulations lead to strongly coercive bilinear forms. In this way, the associated Galerkin schemes become well posed for arbitrary choices of the corresponding finite element subspaces. In particular, Raviart-Thomas spaces of order 0 for the stress tensor, continuous piecewise linear elements for the displacement, and piecewise constants for the rotation can be utilized. Moreover, we show that in this case the number of unknowns behaves approximately as 9.5 times the number of elements (tetrahedrons) of the triangulation, which is cheaper, by a factor of 3, than the classical PEERS in 3D. Several numerical results illustrating the good performance of the augmented schemes are provided.     

Superconvergence of the continuous Galerkin finite element method for delay-differential equation with several terms

The continuous Galerkin finite element method for linear delay-differential equation with several terms is studied. Adding some lower terms in the remainder of orthogonal expansion in an element so that the remainder satisfies more orthogonal condition in the element, and obtain a desired superclose function to finite element solution, thus the superconvergence of p  -degree finite element approximate solution on (p+1)$(p+1)$-order Lobatto points is derived.     

A second-order accurate numerical method for a fractional wave equation

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k ² + h ², where k and h are the parameters for the time and space meshes, respectively.     

Comparison between the shifted-Laplacian preconditioning and the controllability methods for computational acoustics

Processes that can be modelled with numerical calculations of acoustic pressure fields include medical and industrial ultrasound, echo sounding, and environmental noise. We present two methods for making these calculations based on Helmholtz equation. The first method is based directly on the complex-valued Helmholtz equation and an algebraic multigrid approximation of the discretized shifted-Laplacian operator; i.e. the damped Helmholtz operator as a preconditioner. The second approach returns to a transient wave equation, and finds the time-periodic solution using a controllability technique. We concentrate on acoustic problems, but our methods can be used for other types of Helmholtz problems as well. Numerical experiments show that the control method takes more CPU time, whereas the shifted-Laplacian method has larger memory requirement.