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A non-intrusive proper generalized decomposition (PGD) strategy, coupled withan overlapping domain decomposition (DD) method, is proposed to efficientlyconstruct surrogate models of parametric linear elliptic problems. A parametricmulti-domain formulation is presented, with local subproblems featuringarbitrary Dirichlet interface conditions represented through the traces of thefinite element functions used for spatial discretization at the subdomainlevel, with no need for additional auxiliary basis functions. The linearity ofthe operator is exploited to devise low-dimensional problems with only fewactive boundary parameters. An overlapping Schwarz method is used to glue thelocal surrogate models, solving a linear system for the nodal values of theparametric solution at the interfaces, without introducing Lagrange multipliersto enforce the continuity in the overlapping region. The proposed DD-PGDmethodology relies on a fully algebraic formulation allowing for real-timecomputation based on the efficient interpolation of the local surrogate modelsin the parametric space, with no additional problems to be solved during theexecution of the Schwarz algorithm. Numerical results for parametric diffusionand convection-diffusion problems are presented to showcase the accuracy of theDD-PGD approach, its robustness in different regimes and its superiorperformance with respect to standard high-fidelity DD methods.

An overlapping domain decomposition method for the solution of parametric elliptic problems via proper generalized decomposition