Exponential Convergence of $hp$-Finite Element Discretization of Optimal Boundary Control Problems with Elliptic Partial Differential Equations

2016 ◽  
Vol 54 (5) ◽  
pp. 2526-2552 ◽  
Author(s):  
D. Wachsmuth ◽  
J.-E. Wurst
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Control ◽  
Exponential Convergence ◽  
Elliptic Partial Differential Equations ◽  
Control Problems ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Boundary Control Problems

Improved error estimates for semidiscrete finite element solutions of parabolic Dirichlet boundary control problems

2019 ◽  
Vol 40 (4) ◽  
pp. 2898-2939 ◽  
Author(s):  
Wei Gong ◽  
Buyang Li
Keyword(s):  
Finite Element ◽  
Boundary Control ◽  
Parabolic Equations ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Finite Element Discretization ◽  
Dirichlet Boundary Control ◽  
Element Discretization ◽  
Boundary Control Problem ◽  
Boundary Control Problems

Abstract The parabolic Dirichlet boundary control problem and its finite element discretization are considered in convex polygonal and polyhedral domains. We improve the existing results on the regularity of the solutions by establishing and utilizing the maximal $L^p$-regularity of parabolic equations under inhomogeneous Dirichlet boundary conditions. Based on the proved regularity of the solutions, we prove ${\mathcal O}(h^{1-1/q_0-\epsilon })$ convergence for the semidiscrete finite element solutions for some $q_0>2$, with $q_0$ depending on the maximal interior angle at the corners and edges of the domain and $\epsilon$ being a positive number that can be arbitrarily small.

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