A Tree-Based Localized Reduced Basis Method for Parametrized Parabolic PDEs

Mohamed Barakat; Diane Guignard

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arXiv · 2609.36141

A Tree-Based Localized Reduced Basis Method for Parametrized Parabolic PDEs

Mohamed Barakat·Diane Guignard

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Abstract

We propose a localized reduced basis method for parametrized parabolic partial differential equations based on an adaptive partitioning of the parameter domain using hierarchical binary trees. A POD-Greedy procedure is employed to construct reduced spaces associated with each parameter subdomain. A certified \emph{a posteriori} error estimator is derived from an inf-sup stability analysis and shown to be equivalent to the underlying approximation error. This equivalence, together with suitable regularity assumptions on the parameter-to-solution map, enables a rigorous convergence analysis and is used to derive a strategy for selecting the number of POD modes to retain in each local reduced space. Numerical experiments for a parametrized convection--diffusion problem demonstrate the effectiveness of the proposed approach and confirm the predicted convergence behavior.

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Mohamed Barakat, Diane Guignard. 2026-09-28. A Tree-Based Localized Reduced Basis Method for Parametrized Parabolic PDEs. https://arxiv.org/abs/2609.36141

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