Showing papers by "Susanne C. Brenner published in 2020"
01 Nov 2020
TL;DR: In this paper, two P 1 finite element methods for an elliptic state-constrained distributed optimal control problem with Neumann boundary conditions on general polygonal domains were investigated.
Abstract: We investigate two P 1 finite element methods for an elliptic state-constrained distributed optimal control problem with Neumann boundary conditions on general polygonal domains.
12 citations
TL;DR: Theoretical and numerical results for two finite element methods for an elliptic distributed optimal control problem on general polygonal/polyhedral domains with pointwise state constraints are presented.
Abstract:
We present theoretical and numerical results for two $P_1$ finite element methods for an elliptic distributed optimal control problem on general polygonal/polyhedral domains with pointwise state constraints.
11 citations
TL;DR: Finite element methods for a model elliptic distributed optimal control problem with pointwise state constraints are considered from the perspective of fourth order boundary value problems in this article, where finite element methods are used to solve the problem.
Abstract: Finite element methods for a model elliptic distributed optimal control problem with pointwise state constraints are considered from the perspective of fourth order boundary value problems.
5 citations
TL;DR: The uniform convergence of the W -cycle algorithm is proved and the performance of V -cycle and W - cycle algorithms are demonstrated through numerical experiments.
5 citations
01 Jan 2020
5 citations
TL;DR: A robust solver for a second order mixed finite element splitting scheme for the Cahn-Hilliard equation with a preconditioned minimal residual algorithm whose performance is independent of the spacial mesh size and the time step size for a given interfacial width parameter.
4 citations
01 Aug 2020
TL;DR: In this paper, a cubic C 0 interior penalty method for linear-quadratic elliptic distributed optimal control problems with pointwise state and control constraints is proposed and its theoretical results corroborate the theoretical error estimates.
Abstract: We design and analyze a cubic C 0 interior penalty method for linear–quadratic elliptic distributed optimal control problems with pointwise state and control constraints. Numerical results that corroborate the theoretical error estimates are also presented.
3 citations
TL;DR: In this paper, a one dimensional elliptic distributed optimal control problem with pointwise constraints on the derivative of the state was considered and the variational inequality satisfied by the derivation was exploited.
Abstract: We consider a one dimensional elliptic distributed optimal control problem with pointwise constraints on the derivative of the state. By exploiting the variational inequality satisfied by the deriv...
2 citations
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TL;DR: A general superapproximation result is derived in this paper which is useful for the local/interior error analysis of finite element methods.
Abstract: A general superapproximation result is derived in this paper which is useful for the local/interior error analysis of finite element methods.
2 citations
TL;DR: This paper investigates C 1 finite element methods for one dimensional elliptic distributed optimal control problems with pointwise constraints on the derivative of the state formulated as fourth order variational inequalities for the state variable and obtains O ( h ) convergence for the approximation of the optimal state in the H 2 norm.
Abstract: We investigate $C^1$ finite element methods for one dimensional elliptic distributed optimal control problems with pointwise constraints on the derivative of the state formulated as fourth order variational inequalities for the state variable. For the problem with Dirichlet boundary conditions, we use an existing $H^{\frac52-\epsilon}$ regularity result for the optimal state to derive $O(h^{\frac12-\epsilon})$ convergence for the approximation of the optimal state in the $H^2$ norm. For the problem with mixed Dirichlet and Neumann boundary conditions, we show that the optimal state belongs to $H^3$ under appropriate assumptions on the data and obtain $O(h)$ convergence for the approximation of the optimal state in the $H^2$ norm.
2 citations
TL;DR: In this article, a general superapproximation result is derived for the local/interior error analysis of finite element methods, which is useful for the analysis of FER methods.
Abstract: A general superapproximation result is derived in this paper which is useful for the local/interior error analysis of finite element methods.
TL;DR: By exploiting the variational inequality satisfied by the derivative of the optimal state, the optimal control problem is solved as a fourth order Variational inequality by a C 1 finite element method, and the error analysis is presented together with numerical results.
Abstract: We consider a one dimensional elliptic distributed optimal control problem with pointwise constraints on the derivative of the state. By exploiting the variational inequality satisfied by the derivative of the optimal state, we obtain higher regularity for the optimal state under appropriate assumptions on the data. We also solve the optimal control problem as a fourth order variational inequality by a $C^1$ finite element method, and present the error analysis together with numerical results.