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Two fast finite difference schemes for elliptic Dirichlet boundary control problems

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Abstract

In this paper, we propose and analyze two standard finite difference schemes (called Scheme I and Scheme II) for discretizing the first-order necessary optimality systems, which characterize the optimal solutions of Dirichlet boundary control problems governed by elliptic equations. We proved that the proposed schemes are uniformly stable on a uniform mesh, which implies a second-order and first-order convergence of the Scheme I and Scheme II, respectively, provided the optimal solutions have the required regularity. The resulting symmetric indefinite sparse linear systems are solved by the preconditioned GMRES iterative solver with a fast (FFT-based) constraint preconditioner, which numerically shows a mesh-independent convergence rate. Numerical examples, including the case with less regular solutions, are presented to validate our theoretical analysis and demonstrate the promising approximation accuracy and computational efficiency of our proposed schemes and preconditioned iterative solver, respectively. Our developed fast finite difference schemes achieve a comparable order of convergence as the other available schemes in the literature.

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Notes

  1. For our considered rectangle domain, the high regularity assumptions of y and z are rather restrictive, since both f and g are required to be sufficiently smooth and also satisfy suitable compatibility conditions at the four corners. With less regular solutions, we expect to obtain lower order of accuracy (see Example 3 in Sect. 4). Our current paper focuses on the case with smooth solutions, which serves as the first step toward more practical cases with nonsmooth solutions. It is likely to enforce only minimal regularity assumptions [5] by analyzing our proposed finite difference schemes within the framework of generalized (weak) solutions [25, Ch. 2], but how to obtain such (optimal) error estimates in various norms is still open for further study. The requirement of high regularity of solutions is an obvious drawback of any standard finite difference schemes. However, their simplicity and flexibility are very attractive to those applications in demand of efficient numerical implementation.

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Acknowledgements

The author would like to thank the editor and two anonymous referees for their valuable comments and detailed suggestions that have significantly contributed to improving the presentation of this paper.

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  1. Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL, 62026, USA

    Jun Liu

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Liu, J. Two fast finite difference schemes for elliptic Dirichlet boundary control problems. J. Appl. Math. Comput. 61, 481–503 (2019). https://doi.org/10.1007/s12190-019-01261-2

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