基于树的参数化抛物型偏微分方程局部化降基方法

中文摘要 AI 辅助

我们提出了一种针对参数化抛物型偏微分方程的局部化降基方法，该方法基于使用层次二叉树的参数域自适应划分。采用POD-Greedy过程来构造与每个参数子域相关联的降基空间。从inf-sup稳定性分析中推导出一个经过认证的后验误差估计器，并证明其与底层逼近误差等价。这种等价性，连同对参数到解映射的适当正则性假设，使得严格的收敛性分析成为可能，并用于推导在每个局部降基空间中保留的POD模态数量的选择策略。针对参数化对流-扩散问题的数值实验证明了所提出方法的有效性，并确认了预测的收敛行为。

英文摘要

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.

发表机构

• University of Ottawa（渥太华大学）

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