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arXiv · 2608.22528

Quasi-optimality of adaptive FEM for optimal control problems involving Dirac measures governed by biharmonic equation

Asha K. Dond·Neela Nataraj·Subham Nayak
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Abstract

This article establishes the quasi-optimality of adaptive nonconforming finite element methods for a class of optimal control problems involving Dirac measures governed by the biharmonic equation. The nonconforming Morley finite elements are employed for discretising both the state and adjoint variables. A modified right-hand side through a companion operator that maps Morley finite elements toaconformingspacehelpstoovercomethechallengeinhandlingpointsourcesontheright-hand side. A priori and a posteriori error estimates for the optimal control problems are derived. Further,optimal convergence rates for adaptive finite element methods are established using an axiomatic framework: by proving key properties such as stability, reduction, discrete reliability, and quasi-orthogonality. Numerical experiments for three types of optimal control problems are discussed extensively and they validate the theoretical results.

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Asha K. Dond, Neela Nataraj, Subham Nayak. 2026-08-23. Quasi-optimality of adaptive FEM for optimal control problems involving Dirac measures governed by biharmonic equation. https://arxiv.org/abs/2608.22528

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