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：提出了一种非侵入式本征正交分解（PGD）策略，耦合重叠区域分解（DD）方法，用于高效构建参数化线性椭圆问题的代理模型。本文给出了一种参数化多区域公式化方法，其中局部子问题通过有限元函数迹表示任意狄利克雷界面条件，这些有限元函数用于子域级别的空间离散化，无需引入额外辅助基函数。利用算子的线性性质，设计出仅含少量活性边界参数的低维问题。采用重叠施瓦兹方法耦合局部代理模型，通过求解界面处参数化解节点值的线性系统，避免在重叠区域引入拉格朗日乘子强制连续性。所提DD-PGD方法基于全代数公式化，允许通过参数空间内局部代理模型的高效插值实现实时计算，在施瓦兹算法执行过程中无需求解额外问题。针对参数化扩散和对流扩散问题的数值结果表明，DD-PGD方法具有高精度、不同工况下的鲁棒性，以及相对于标准高保真度区域分解方法的优越性能。