A $C^0$ weak Galerkin method with preconditioning for constrained optimal control problems with general tracking
It provides a numerically efficient and preconditioned solver for challenging fourth-order variational inequality problems in optimal control.
The paper develops a $C^0$ weak Galerkin method with an additive Schwarz preconditioner for optimal control problems governed by PDEs with general tracking and pointwise state constraints, demonstrating effectiveness and robustness through numerical experiments.
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.