zbMATH — the first resource for mathematics

Controllability methods for the computation of time-periodic solutions; application to scattering. (English) Zbl 0926.65054
The general framework of the problems considered in this paper is the following: We have to compute solutions of an abstract differential equation \[ {dy\over{dt}}+Ay=f, \] with \(T\)-periodic boundary condition in the \(t\)-variable \[ y(0)=y(T). \] Here \(A\) is a linear operator, in general unbounded, in a Hilbert space \(H\) with norm \(| \cdot| \), and \(f\) is a \(T\)-periodic function of the variable \(t\). The authors suggest to understand the above problem in the least squares sense: We are looking for \(e\in H\), such that \(J(e)\leq J(v)\) for all \(v\in H\), where \(J(v)={1\over 2}| y(T)-v| ^2\), and \({dy\over {dt}}+Ay=f\), with the initial condition \(y(0)=v\). This least squares problem is equivalent to the following: We are looking for a triple \(\{e,y,p\}\) satisfying \(e=y(t)-p(0)\), \({dy\over{dt}}+Ay=f\), \(y(0)=e\), \(-{dp\over{dt}}+A^*p=0\), and \(p(T)=y(T)-e\). To this affine problem, a kind of the conjugate gradient iterative method is applied. A completely analogous procedure is applied to its version discretized with respect to the time variable \(t\). The authors suggest to use, as the discretized version, the explicit Euler method, though it is unconditionally unstable for \(A\) unbounded. The argument for the explicit Euler algorithm is its simplicity; moreover, the authors argue, that the explicit Euler method works well with a time-step small enough if a finite rank approximation appears instead of \(A\).
Reviewer’s remark: This procedure seems to be very uncertain, and surely will fall down if some stiffness is present in the problem.
The above method is applied to the scattering problem, originally formulated as the second order equation in time \(u_{tt}-\Delta u=0,\) for which an external boundary value problem is considered. To assure the domain to be bounded, the authors introduce an additional artificial boundary. Further this problem is transformed into a first order system in time \(t\). A large part of the paper is devoted to the detailed formulation and to the results of numerical computations.

65J10 Numerical solutions to equations with linear operators (do not use 65Fxx)
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
34G10 Linear differential equations in abstract spaces
35L05 Wave equation
35P25 Scattering theory for PDEs
Full Text: DOI
[1] Lasiecka, I.; Triggiani, R., Control problems for systems described by partial differential equations and applications, 97, (1987) · Zbl 0619.00014
[2] Auchmuty, G.; Dean, E.J.; Glowinski, R.; Zhang, S.C., Control methods for the numerical computation of periodic solutions of autonomous differential equations, Control problems for systems described by partial differential equations and applications, 97, 64, (1987)
[3] M. O. Bristeau, R. Glowinski, J. Périaux, Scattering waves using exact controllability methods, 31st AIAA Aerospace Sciences Meeting, Reno, Nevada. · Zbl 0936.76518
[4] M. O. Bristeau, R. Glowinski, J. Périaux, Using exact controll ability to solve the Helmholtz equation at high wave numbers, Mathematical and Numerical Aspects of Wave Propagation, T. KleinmanTh. AngellD. ColtonF. SantosaI. Stakgold, SIAM, Philadelphia, PA, 1993, 113
[5] Glowinski, R.; Lions, J.L., Exact and approximatecontrollability for distributed parameter systems, Acta numer., 159, (1996) · Zbl 0838.93014
[6] M. O. Bristeau, E. J. Dean, R. Glowinski, V. K. Kwok, J. Périaux, Application of exact controllability to the computation of scattering waves, Control Problems in Industry, I. LasieckaB. Morton, Birkhauser, Boston, 1995, 17
[7] G. Auchmuty, Control theoretic methods for the computation of special solutions of differential equations, Computational Sciences for the 21st Century, M. O. BristeauG. EtgenW. FitzgibbonJ. L. LionsJ. PériauxM. F. Wheeler, Wiley, Chichester, 1997, 655 · Zbl 0911.65061
[8] Bardos, C.; Rauch, J., Variational algorithms for the helmholtzequation using time evolution and artificial boundaries, Asymptotic anal., 9, 101, (1994) · Zbl 0818.65100
[9] Morgan, J.; Hollis, S., The existence of periodic solutions toreaction-diffusion equations with periodic data, SIAM J. math. anal., 26, 1225, (1995) · Zbl 0849.35052
[10] M. O. Bristeau, R. Glowinski, J. Périaux, Exact controllability to solve the Helmholtz equation with absorbing boundary conditions, Finite Element Methods, D. KrizekP. NeittaanmakiR. Stenberg, Dekker, New York, 1994, 79 · Zbl 0809.65099
[11] Kleinman, R.E.; Roach, G.F., Boundary integral equations for the three-dimensional Helmholtz equation, SIAM rev., 16, 214, (1974) · Zbl 0253.35023
[12] Harari, I.; Hughes, T.J.R., A cost comparison of boundary element and finite element methods for problems of time-harmonic acoustics, Comput. methods appl. mech. engrg., 97, 77, (1991)
[13] Ammari, H.; Nedelec, J.C., Couplage éléments finis/équations intégrales pour la résolution des équations de Maxwell en milieu hétérogène, Equations aux Dérivées partielles et applications, 19, (1998) · Zbl 0940.78010
[14] E. Heikkola, Y. A. Kuznetsov, P. Neittaanmaki, J. Toivanen, Different approximations for the exterior Helmholtz equation: A numerical study, J. Comput. Phys. · Zbl 0909.65119
[15] Lions, J.L., Exact controllability, stabilization and perturbation for distributed systems, SIAM rev., 30, 1, (1988) · Zbl 0644.49028
[16] Lions, J.L., Contrôlabilité exacte, perturbation et stabilisation des systèmes distribués, 1, (1988) · Zbl 0653.93002
[17] P. L. George, E. Seveno, Génération de Maillages par une Méthode de Type Frontal, 1992
[18] H. Bourouchaki, F. Hecht, E. Saltel, P. L. George, Reasonably efficient Delaunay-based mesh generator in 3 dimensions, Proc. of the 4th Int. Meshing Roundtable, Albuquerque, NM, 1995
[19] J. Erhel, F. Guyomarch, An augmented subspace conjugate gradient, 1997
[20] Mur, G., The finite element modeling of three-dimensional electromagnetic fields using edge and nodal elements, IEEE trans. antennas propag., 41, 948, (1993)
[21] Bowman, J.J.; Senior, T.B.A.; Uslenghi, P.L.E., Electromagnetic and acoustic scattering by simple shapes, (1987)
[22] F. El, Dabaghi, K. Parrott, H. Steve, Proceedings of the3rd Workshop on Approximations and Numerical Methods for the Solution of Maxwell Equations, Wiley, Chichester
[23] Y. Kuznetsov, K. Lipnikov, On the application of fictitious domain and domain decomposition methods for scattering problems on Cray Y-MP C98, Department of Mathematics, University of Nijmegen, December 1995
[24] M. O. Bristeau, V. Girault, R. Glowinski, T. W. Pan, J. Périaux, Y. Xiang, On a fictitious domain method for flow and wave problems, Domain Decomposition Methods in Sciences and Engineering, R. GlowinskiJ. PériauxZ-C. ShiO. Widlund, Wiley, Chichester, 1997, 361
[25] M. O. Bristeau, Exact controllability methods for calculation of 3D time-periodic Maxwell solutions, Computational Science for the 21st Century, M. O. BristeauG. EtgenW. FitzgibbonJ. L. LionsJ. PériauxM. F. Wheeler, Wiley, Chichester, 1997, 492 · Zbl 0911.65126
[26] Berggren, M.; Glowinski, R.; Lions, J.L., A computationalapproach to controllability issues for flow-related models (I): pointwise control of the viscous Burgers equation, Internat. J. comput. fluid dynamics, 7, 237, (1996) · Zbl 0894.76056
[27] Berggren, M., Numerical solution of a flow control problem: vorticity reduction by dynamic boundary action, SIAM J. sci. comput., 19, 1024, (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.