Document Type : Applied Article

Author

• Maha Mohsin Mohammed Ali

Department of Information and Communication Technology, Polytechnic College of Engineering Specializations, Middle Technical University, Baghdad, Iraq

Abstract

A core challenge in numerically solving partial differential equations (PDEs) is achieving accuracy without sacrificing computational tractability. Globally supported methods provide high-order precision but at a computational cost that grows prohibitively with problem size, while locally supported schemes scale more efficiently, though often with a reduction in solution accuracy. The proposed localized meshless framework resolves this tension through the integration of Radial Basis Function (RBF) approximation with the Partition of Unity (PU) decomposition, with applications to elliptic, convection-diffusion, and Helmholtz boundary value problems and their corresponding PDE-constrained optimal control problems. The domain is partitioned into overlapping subdomains, each accommodating an independent RBF expansion that is subsequently blended by compactly supported PU weight functions. This localization strategy replaces the single ill-conditioned dense system of global RBF collocation with a collection of smaller, well-conditioned local systems, reducing the theoretical computational complexity from O(N³) to O(N log N). Three canonical benchmark problems validate the framework: Poisson’s equation, a singularly perturbed convection-diffusion equation, and the Helmholtz equation on a circular domain, each embedded in a distributed optimal control formulation. The proposed scheme achieves L₂-norm errors of order 10^-4 at N = 1,000 nodes, outperforming both standard finite difference methods and global RBF collocation by factors of 28 to 49 in accuracy. Optimal control problems are handled through a Lagrange multiplier formulation yielding coupled state and adjoint systems, both discretized within the same RBF-PU architecture. The framework is illustrated through a detailed engineering case study — optimal heat-source regulation in a thermally loaded plate — in which the complete optimality system is derived, physically interpreted, and numerically solved. Numerical stability, current limitations, and directions for future investigation are discussed.

Highlights

• RBF-Partition of Unity collocation framework is developed for elliptic PDE-constrained optimal control problems.
• L₂-norm errors of order 10⁻⁴ were achieved at N = 1,000 nodes across three benchmark PDE problems.
• Accuracy surpasses unstabilized FDM by 28–49× and global RBF collocation by 3.5–7.1×.
• Computational complexity is reduced from O(N³) to O(N log N) via localized partition-of-unity decomposition.
• Local patch condition numbers remain below 10³, versus 10¹⁰–10¹² for global RBF at the same N.
• Lagrange multiplier formulation yields coupled state–adjoint systems solved within a unified meshless architecture.
• Fixed-point iteration converges in 14 steps for the thermal plate case study with γ = 10⁻³.
• Optimal heat-source regulation in a thermally loaded plate is demonstrated with full physical interpretation.

Keywords

• Radial basis functions
• Partition of unity
• Meshless methods
• PDE-constrained optimization
• Optimal control
• Convection-diffusion equations
• Lagrange multiplier method

Main Subjects

• Optimal Control