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<p>In this article, we utilize a unified discontinuous Galerkin approach to approximate the control constrained Dirichlet boundary optimal control problem using finite element method over simplicial triangulation. The continuous optimality system obtained from this method simplifies the control constraints into a simplified Signorini type problem, which is then coupled with boundary value problems for the state and adjoint state variables. The symmetric property of the discrete bilinear forms is required in order to derive the discrete optimality system. The main focus of this article is to derive a posteriori error estimates in the energy norm, where we address the reliability and efficiency of the proposed a posteriori error estimator. The suitable construction of auxiliary problems, continuous and discrete Lagrange multipliers, and intermediate operators are crucial in developing a posteriori error analysis. We have also established optimal a priori error estimates in the energy norm for all the optimal variables (state, adjoint state, and control) under the appropriate regularity assumptions. Theoretical findings are confirmed and illustrated through numerical results on both uniform and adaptive meshes.</p>

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Discontinuous Galerkin Finite Element Methods for the Control Constrained Dirichlet Control Problem Governed by the Diffusion Equation

  • 期刊论文

  • 发表日期：

    2025年12月2日

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  • Divay Garg,
  • Kamana Porwal

摘要

In this article, we utilize a unified discontinuous Galerkin approach to approximate the control constrained Dirichlet boundary optimal control problem using finite element method over simplicial triangulation. The continuous optimality system obtained from this method simplifies the control constraints into a simplified Signorini type problem, which is then coupled with boundary value problems for the state and adjoint state variables. The symmetric property of the discrete bilinear forms is required in order to derive the discrete optimality system. The main focus of this article is to derive a posteriori error estimates in the energy norm, where we address the reliability and efficiency of the proposed a posteriori error estimator. The suitable construction of auxiliary problems, continuous and discrete Lagrange multipliers, and intermediate operators are crucial in developing a posteriori error analysis. We have also established optimal a priori error estimates in the energy norm for all the optimal variables (state, adjoint state, and control) under the appropriate regularity assumptions. Theoretical findings are confirmed and illustrated through numerical results on both uniform and adaptive meshes.