A recursive multilevel hierarchical domain decomposition approach for efficient spectral stochastic finite element analysis

Spectral stochastic finite element methods (SSFEM) have emerged powerful tools for uncertainty quantification in engineering. Although domain decomposition methods (DDM) have been integrated into SSFEM to enhance computational efficiency, the global extended Schur complement (e-SC) system expands rapidly with increasing stochastic dimensions and polynomial chaos orders, leading to prohibitively large interface systems. Further, Gaussian elimination may become computationally expensive under refined stochastic discretizations. These issues severely limit the applicability of conventional DD-based SSFEM for large-scale engineering systems. In this paper, a recursive multilevel hierarchical domain decomposition method for efficient SSFE analysis is developed. The method firstly reformulates the conventional e-SC system into a global Kronecker e-SC system via tensor-product factorization. The resulting system is then partitioned as a multilevel block form by exploiting its hierarchical structure. Based on this multilevel representation, a recursive multilevel hierarchical strategy is developed to solve the resulting Kronecker system. In the developed method, since the constructed Kronecker system can be assembled from low-dimensional Kronecker factors, Gaussian elimination can be performed through inversion of a set of small-scale matrices, and the prohibitive large-matrix operations inherent in conventional DDM are thus bypassed. Furthermore, since the multilevel partition enables the global system to be converted to a set of small-scale subsystems with block diagonal dominance, the computational complexity for solving large-scale global Kronecker e-SC system can be substantially reduced. In addition, the recursive refinement strategy iteratively mitigates the hierarchy-induced approximation errors, ensuring robust convergence even under high stochastic variability. The stochastic analysis of a practical large-scale gravity dam demonstrates the effectiveness of the developed method. The results show that the method provides an accurate and highly efficient framework for large-scale spectral stochastic finite element analysis, demonstrating its potential for complex engineering applications.