scholarly journals $L^\infty$ Norm Error Estimates for HDG Methods Applied to the Poisson Equation with an Application to the Dirichlet Boundary Control Problem

2021 ◽  
Vol 59 (2) ◽  
pp. 720-745
Author(s):  
Gang Chen ◽  
Peter B. Monk ◽  
Yangwen Zhang
Keyword(s):  
Control Problem ◽  
Error Estimates ◽  
Poisson Equation ◽  
Boundary Control ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Control ◽  
The Poisson Equation ◽  
Boundary Control Problem ◽  
Norm Error

scholarly journals Crank-Nicolson finite difference schemes for parabolic optimal Dirichlet boundary control problem

2021 ◽  
Author(s):  
Caijie Yang ◽  
Tongjun Sun
Keyword(s):  
Finite Difference ◽  
Control Problem ◽  
Boundary Control ◽  
Dirichlet Boundary ◽  
Second Order ◽  
Difference Schemes ◽  
Optimality System ◽  
Finite Difference Schemes ◽  
Dirichlet Boundary Control ◽  
Boundary Control Problem

In this paper, we adopt the optimize-then-discretize approach to solve parabolic optimal Dirichlet boundary control problem. First, we derive the first-order necessary optimality system, which includes the state, co-state equations and the optimality condition. Then, we propose Crank-Nicolson finite difference schemes to discretize the optimality system in 1D and 2D cases, respectively. In order to build the second order spatial approximation, we use the ghost points on the boundary in the schemes. We prove that the proposed schemes are unconditionally stable, compatible and second-order convergent in both time and space. To avoid solving the large coupled schemes directly, we use the iterative method. Finally, we present a numerical example to validate our theoretical analysis.

scholarly journals Error bounds for a Dirichlet boundary control problem based on energy spaces

2016 ◽  
Vol 86 (305) ◽  
pp. 1103-1126 ◽  
Author(s):  
Sudipto Chowdhury ◽  
Thirupathi Gudi ◽  
A. K. Nandakumaran
Keyword(s):  
Control Problem ◽  
Boundary Control ◽  
Error Bounds ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Control ◽  
Boundary Control Problem

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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