This paper formulates, analyzes, and demonstrates numerically a method for the partitioned solution of coupled interface problems involving combinations of projection-based reduced order models (ROM) and/or full order methods (FOMs). The method builds on the partitioned scheme developed in [1], which starts from a well-posed formulation of the coupled interface problem and uses its dual Schur complement to obtain an approximation of the interface flux. Explicit time integration of this problem decouples its subdomain equations and enables their independent solution on each subdomain. Extension of this partitioned scheme to coupled ROM-ROM or ROM-FOM problems required formulations with non-singular Schur complements. To obtain these problems, we project a well-posed coupled FOM-FOM problem onto a composite reduced basis comprising separate sets of basis vectors for the interface and interior variables, and use the interface reduced basis as a Lagrange multiplier. Our analysis confirms that the resulting coupled ROM-ROM and ROM-FOM problems have provably non-singular Schur complements, independent of the mesh size and the reduced basis size. In the ROM-FOM case, analysis shows that one can also use the interface FOM space as a Lagrange multiplier. We illustrate the theoretical and computational properties of the partitioned scheme through reproductive and predictive tests for a model advection-diffusion transmission problem.

翻译：本文针对涉及基于投影的降阶模型（ROM）和/或全阶方法（FOM）组合的耦合界面问题，提出、分析并数值验证了一种分区求解方法。该方法建立在文献[1]发展的分区方案基础上，该方案从耦合界面问题的适定表述出发，利用其对偶Schur补获得界面通量的近似。对该问题进行显式时间积分可解耦子域方程，并允许在各子域独立求解。将这种分区方案推广至耦合ROM-ROM或ROM-FOM问题，需要具有非奇异Schur补的表述形式。为获得此类问题，我们将适定的耦合FOM-FOM问题投影到组合简化基上（该基包含界面变量和内部变量的独立基向量集），并采用界面简化基作为拉格朗日乘子。我们的分析证实，所得耦合ROM-ROM和ROM-FOM问题的Schur补具有可证明的非奇异性，且与网格尺寸和简化基尺寸无关。在ROM-FOM情形中，分析表明也可将界面FOM空间作为拉格朗日乘子。我们通过对模型对流-扩散传输问题开展复现性预测与有效性预测测试，阐明了该分区方案的理论与计算特性。