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Numerical aspects of one stabilization method. (English) Zbl 1058.65095

The author considers a discrete analogue of the stabilization algorithm proposed by A. V. Fursikov [Math. Sb. 192, No. 4, 115–160 (2001; Zbl 1019.93047)]. The task is to find boundary conditions for (generally non-linear) parabolic equations which drive the solution from a given intial condition to a fixed stationary solution with a prescribed exponential rate of convergence. The basic idea to obtain these boundary conditions is to extend the solution domain and to project the zero extension of the initial value to a suitable subspace which guarantees a high enough decay rate (the projection is done in such a way that the initial values are not altered on the original domain). Evolving the problem on the enlarged domain then allows to read off the values at the original boundary which leads to the desired solution behavior.
For the one-dimensional diffusion equation as a model problem, the author considers a discrete version of this approach. In this case, the eigenfunctions of the discretized second derivative (the elliptic operator) are explicitly known and the projection matrix needed to define the proper initial value on the extended domain can be studied in detail. The author reports that the condition number behaves very similar to the continuous case, i.e. it grows quickly with the dimension (similar to the condition number of the Hilbert matrix). While a higher decay rate leads to a higher dimension and thus to a higher condition number of the projection matrix, increasing the size of the enlarged domain has the opposite effect on the condition number. The author discusses the choice of optimal parameters (size of domain and size of the projection matrix) to achieve a given decay rate.
Finally, the approach is applied to the Chafee-Infante equation [D. Henry, Geometric theory of semilinear parabolic equations. (1981; Zbl 0456.35001)] which is close to the model example but includes both unstable and non-linear behavior with a suitable choice of parameters. It turns out that the projection which is necessary only initially in the stable linear case has to be repeated periodically during the evolution if the process is non-linear or unstable (in the latter case because of amplified round-off errors).

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
35B37 PDE in connection with control problems (MSC2000)
65K10 Numerical optimization and variational techniques
49M25 Discrete approximations in optimal control
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
49J20 Existence theories for optimal control problems involving partial differential equations

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References:

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[2] DOI: 10.1016/S0024-3795(98)00015-9 · Zbl 0934.15005
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[5] DOI: 10.1007/PL00000972 · Zbl 0983.93021
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