Development of error criteria for adaptive multi-element polynomial chaos approaches

•An extension to the multi-element polynomial chaos methodology is offered.•Two different error criteria for partitioning are proposed.•These criteria are applied to differential equations and mechanical oscillators.•Results are compared with original partitioning methodology.•The proposed criteria are more robust and faster to converge.

This paper presents and compares different methodologies to create an adaptive stochastic space partitioning in polynomial chaos applications which use a multi-element approach. To implement adaptive partitioning, Wan and Karniadakis first developed a criterion based on the relative error in local variance. We propose here two different error criteria: one based on the residual error and the other on the local variance discontinuity created by partitioning. The methods are applied to classical differential equations with long-term integration difficulties, including the Kraichnan–Orszag three-mode problem, and to simple linear and nonlinear mechanical systems whose stochastic dynamic responses are investigated. The efficiency and robustness of the approaches are investigated by comparison with Monte-Carlo simulations. For the different examples considered, they show significantly better convergence characteristics than the original error criterion used.