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Volume 10, Issue 3
Convergence of Discontinuous Time-Stepping Schemes for a Robin Boundary Control Problem Under Minimal Regularity Assumptions

K. Chrysafinos

Int. J. Numer. Anal. Mod., 10 (2013), pp. 673-696.

Published online: 2013-10

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  • Abstract

The minimization of the energy functional having states constrained to semi-linear parabolic PDEs is considered. The controls act on the boundary and are of Robin type. The discrete schemes under consideration are discontinuous in time but conforming in space. Stability estimates are presented at the energy norm and at arbitrary times for the state, and adjoint variables. The estimates are derived under minimal regularity assumptions and are applicable for higher order elements. Using these estimates and an appropriate compactness argument (see Walkington [49, Theorem 3.1]) for discontinuous Galerkin schemes, convergence of the discrete solution to the continuous solution is established. In addition, a discrete optimality system is derived and convergence of the corresponding discrete solutions is also demonstrated.

  • Keywords

Discontinuous Time-Stepping Schemes, Finite Element Approximations, Robin Boundary Control, Semi-linear Parabolic PDEs.

  • AMS Subject Headings

65M60,49J20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-10-673, author = {K. Chrysafinos}, title = {Convergence of Discontinuous Time-Stepping Schemes for a Robin Boundary Control Problem Under Minimal Regularity Assumptions}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2013}, volume = {10}, number = {3}, pages = {673--696}, abstract = {

The minimization of the energy functional having states constrained to semi-linear parabolic PDEs is considered. The controls act on the boundary and are of Robin type. The discrete schemes under consideration are discontinuous in time but conforming in space. Stability estimates are presented at the energy norm and at arbitrary times for the state, and adjoint variables. The estimates are derived under minimal regularity assumptions and are applicable for higher order elements. Using these estimates and an appropriate compactness argument (see Walkington [49, Theorem 3.1]) for discontinuous Galerkin schemes, convergence of the discrete solution to the continuous solution is established. In addition, a discrete optimality system is derived and convergence of the corresponding discrete solutions is also demonstrated.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/589.html} }
TY - JOUR T1 - Convergence of Discontinuous Time-Stepping Schemes for a Robin Boundary Control Problem Under Minimal Regularity Assumptions AU - K. Chrysafinos JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 673 EP - 696 PY - 2013 DA - 2013/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/589.html KW - Discontinuous Time-Stepping Schemes, Finite Element Approximations, Robin Boundary Control, Semi-linear Parabolic PDEs. AB -

The minimization of the energy functional having states constrained to semi-linear parabolic PDEs is considered. The controls act on the boundary and are of Robin type. The discrete schemes under consideration are discontinuous in time but conforming in space. Stability estimates are presented at the energy norm and at arbitrary times for the state, and adjoint variables. The estimates are derived under minimal regularity assumptions and are applicable for higher order elements. Using these estimates and an appropriate compactness argument (see Walkington [49, Theorem 3.1]) for discontinuous Galerkin schemes, convergence of the discrete solution to the continuous solution is established. In addition, a discrete optimality system is derived and convergence of the corresponding discrete solutions is also demonstrated.

K. Chrysafinos. (2013). Convergence of Discontinuous Time-Stepping Schemes for a Robin Boundary Control Problem Under Minimal Regularity Assumptions. International Journal of Numerical Analysis and Modeling. 10 (3). 673-696. doi:
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