We propose a new method, the continuous Galerkin method with globally and locally supported basis functions (CG-GL), to address the parametric robustness issues of reduced-order models (ROMs) by incorporating solution-based adaptivity with locally supported finite element basis functions. The CG-GL method combines the accuracy of locally supported basis functions with the efficiency of globally supported data-driven basis functions. Efficient output-based dual-weighted residual error estimates are derived and implemented for the CG-GL method and used to drive efficient online trial space adaptation. An empirical quadrature procedure is introduced for rapid evaluation of nonlinear terms that does not require retraining throughout the adaptation process. Two numerical experiments demonstrate the potential of the CG-GL method to produce accurate approximations with limited training and its tunable tradeoff between accuracy and computational cost.

翻译：我们提出了一种新方法——连续伽辽金法与全局和局部支撑基函数（CG-GL），通过将基于解的适应性与局部支撑有限元基函数相结合，来解决降阶模型（ROMs）的参数鲁棒性问题。CG-GL方法将局部支撑基函数的精确性与全局支撑数据驱动基函数的效率相结合。该方法推导并实现了基于输出的高效对偶加权残差误差估计，并用于驱动高效的在线试验空间自适应。引入了一种经验求积过程，用于非线性项的快速评估，且在整个自适应过程中无需重新训练。两个数值实验证明了CG-GL方法在有限训练下生成精确近似解的潜力，以及其在精度与计算成本之间的可调权衡。