A NOVEL OUTLOOK ON FINITE ELEMENT METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS - Nigam, Deependra; Chauhan, Amit (ShodhKosh: Journal of Visual and Performing Arts, 2024)
We present a novel method for error control and adaptive strategies in finite element discretizations for optimization problems governed by partial differential equations. Utilizing the Lagrangian formalism, the objective is to identify stationary points of the first-order necessary optimality conditions. Mesh adaptation is guided by residual-based a posteriori error estimates derived through duality principles, enabling error control for any specified physical quantity of interest. A distinctive aspect of this method is the natural alignment of the error-control functional with the optimization problem cost functional. This alignment allows the Lagrange multiplier to directly weight the cell residuals in the error estimator, resulting in a straightforward and computationally efficient algorithm tailored to the specific requirements of the optimization problem. The proposed approach is developed and validated on simple model problems related to optimal control in semiconductor applications.