A non-intrusive proper generalized decomposition (PGD) strategy, coupled with an overlapping domain decomposition (DD) method, is proposed to efficiently construct surrogate models of parametric linear elliptic problems. A parametric multi-domain formulation is presented, with local subproblems featuring arbitrary Dirichlet interface conditions represented through the traces of the finite element functions used for spatial discretization at the subdomain level, with no need for additional auxiliary basis functions. The linearity of the operator is exploited to devise low-dimensional problems with only few active boundary parameters. An overlapping Schwarz method is used to glue the local surrogate models, solving a linear system for the nodal values of the parametric solution at the interfaces, without introducing Lagrange multipliers to enforce the continuity in the overlapping region. The proposed DD-PGD methodology relies on a fully algebraic formulation allowing for real-time computation based on the efficient interpolation of the local surrogate models in the parametric space, with no additional problems to be solved during the execution of the Schwarz algorithm. Numerical results for parametric diffusion and convection-diffusion problems are presented to showcase the accuracy of the DD-PGD approach, its robustness in different regimes and its superior performance with respect to standard high-fidelity DD methods.

翻译：本文提出一种非侵入式本征正交分解（PGD）策略，结合重叠区域分解（DD）方法，用于高效构建参数化线性椭圆问题的替代模型。该研究构建了参数化多域公式，其中局部子问题通过子域空间离散的有限元函数迹表示任意狄利克雷界面条件，无需引入额外辅助基函数。利用算子的线性特性，设计了仅包含少量活动边界参数的低维问题。采用重叠Schwarz方法拼接局部替代模型，通过求解界面处参数化解的节点值线性系统实现，无需引入拉格朗日乘子强制重叠区域的连续性。所提出的DD-PGD方法基于完全代数化公式，可在参数空间中对局部替代模型进行高效插值实现实时计算，且在执行Schwarz算法过程中无需求解额外问题。针对参数化扩散问题和对流扩散问题的数值结果，展示了DD-PGD方法的准确性、在不同工况下的鲁棒性，以及相较标准高保真DD方法的优越性能。