Controllability methods for the computation of time-periodic solutions; application to scattering.

*(English)*Zbl 0926.65054The 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.

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.

Reviewer: Krzysztof Moszyński (Warszawa)

##### MSC:

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 |

##### Keywords:

time-periodic solutions; conjugate gradients; controllability methods; abstract differential equation; Hilbert space; least squares problem; explicit Euler method; scattering; artificial boundary; numerical examples; wave equation
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\textit{M. O. Bristeau} et al., J. Comput. Phys. 147, No. 2, 265--292 (1998; Zbl 0926.65054)

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