TY - JOUR
T1 - Convergence of Discontinuous Time-Stepping Schemes for a Robin Boundary Control Problem Under Minimal Regularity Assumptions
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 - <p style="text-align: justify;">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.</p>