An overlapping domain decomposition method for the solution of parametric elliptic problems via proper generalized decomposition
An overlapping domain decomposition method for the solution of parametric elliptic problems via proper generalized decomposition
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DOI：
10.1016/j.cma.2023.116484
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发表时间：
2023-07
期刊：
影响因子：
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通讯作者：
M. Discacciati;B. Evans;M. Giacomini
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文献类型：
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作者：
M. Discacciati;B. Evans;M. Giacomini
提出了一种非侵入的恰当广义分解（PGD）策略，结合重叠区域分解（DD）方法，有效地构造参数线性椭圆问题的代理模型.提出了一种参数化的多域公式，局部子问题具有任意Dirichlet界面条件，通过用于子域级空间离散化的有限元函数的迹表示，不需要额外的辅助基函数。利用算子的线性性，设计出只有很少的活动边界参数的低维问题。一个重叠的施瓦茨方法被用来粘接本地代理模型，解决了一个线性系统的节点值的参数解决方案的接口，而不引入拉格朗日乘子，以加强在重叠区域的连续性。所提出的DD-PGD方法依赖于一个完全的代数公式，允许实时计算的基础上，在参数空间中的本地代理模型的有效插值，没有额外的问题要解决的施瓦茨算法的执行过程中。参数扩散和对流扩散问题的数值结果展示的DD-PGD方法的准确性，其在不同的制度和其上级性能相对于标准的高保真DD方法的鲁棒性。
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.