A $C^0$ weak Galerkin method with preconditioning for constrained optimal control problems with general tracking

TL;DR

This paper introduces a $C^0$ weak Galerkin method with an additive Schwarz preconditioner for efficiently solving complex optimal control problems with constraints, ensuring accuracy and robustness.

Contribution

The paper develops a $C^0$-WG method with a tailored preconditioner for constrained OCPs, addressing challenges of reduced regularity and high-order inequalities.

Findings

The $C^0$-WG method achieves optimal accuracy with efficient implementation.

The additive Schwarz preconditioner significantly improves solver performance.

Numerical tests demonstrate robustness for biharmonic and control problems.

Abstract

This paper presents a $C^0$ weak Galerkin ($C^0$-WG) method combined with an additive Schwarz preconditioner for solving optimal control problems (OCPs) governed by partial differential equations with general tracking cost functionals and pointwise state constraints. These problems pose significant analytical and numerical challenges due to the presence of fourth-order variational inequalities and the reduced regularity of solutions. Our first contribution is the design of a $C^0$-WG method based on globally continuous quadratic Lagrange elements, enabling efficient elementwise stiffness matrix assembly and parameter-free implementation while maintaining accuracy, as supported by a rigorous error analysis. As a second contribution, we develop an additive Schwarz preconditioner tailored to the $C^0$-WG method to improve solver performance for the resulting ill-conditioned linear systems. Numerical experiments confirm the effectiveness and robustness of the proposed method and preconditioner for both biharmonic and optimal control problems.