Research Article
A Priori and a Posteriori Error Estimates of Finite Element Method for Source Control Problems Governed by a System of Quasi-linear Elliptic Equations
Issue:
Volume 11, Issue 1, February 2025
Pages:
1-17
Received:
4 March 2025
Accepted:
6 May 2025
Published:
21 June 2025
Abstract: In this paper, we consider the optimal source control of a 2-dimensional steady-state thermistor. The problem is described by a system of two nonlinear elliptic partial differential equations with appropriate boundary conditions which model the coupling of the thermistor to its surroundings. The heat source is Joule heat due to variable resistance. The problem is a source optimal control problem that controls the source term necessary to approximate the temperature to a proper target function. First, we derive the optimality condition of the problem. Based on setting the approximation problem of a given control problem in a first order polynomial finite element function space and deriving the optimality condition of the approximation problem, we evaluated a priori error between the optimal control, the optimal state, the conjugate state and its finite element approximation functions. Then, we evaluate the upper bound of a posteriori error estimates that are currently available for error estimation. For a posteriori error estimates, it is necessary to find the convergence of the error indicator. In this paper, we prove the convergence of a posteriori error indicator by obtaining a lower bound estimate of a posteriori error and finding that the total variance error goes to zero. And, we propose a gradient algorithm to find the optimal control and provide a condition for this algorithm to converge. The validity is also demonstrated by adaptive numerical simulations with a detailed problem. The computational results are obtained on three adaptive meshes and the graphs of the finite element solutions are presented.
Abstract: In this paper, we consider the optimal source control of a 2-dimensional steady-state thermistor. The problem is described by a system of two nonlinear elliptic partial differential equations with appropriate boundary conditions which model the coupling of the thermistor to its surroundings. The heat source is Joule heat due to variable resistance....
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,
Ali Bulama Mamman
, at some finite time t. We derive an explicit formula for the guaranteed time of pursuit, which holds for all admissible strategies of the evader. Additionally, we show that an optimal escape strategy for the evader, known as l-escape, can also be realized at this same optimal time. This simultaneous realization of pursuit and escape conditions indicates a balance in the strategies of both players, thus leading to an optimal pursuit time. Our results contribute to the broader understanding of differential games and optimal control theory.