Achievement

We developed a domain decomposition model reduction method for linear steady- state convection-diffusion equations with random coefficients. Of particular interest to this effort are the diffusion equation with random diffusivity and the convection-dominated transport equation with random velocity. We investigated two types of random fields, i.e., colored noises and discrete white noises, both of which can lead to high-dimensional parametric dependence in practice. The motivation is to exploit domain decomposition to reduce the parametric dimension of local problems in subdomains, such that an entire parametric map can be approximated with a small number of expensive partial differential equation (PDE) simulations. The new method combines domain decomposition with model reduction and sparse polynomial approximation, so as to simultaneously handle the high dimensionality and irregular behavior of the PDEs under consideration. The advantages of our method lie in three aspects: (i) online-offline decomposition, i.e., the online cost is independent of the size of the triangle mesh; (ii) sparse approximation of operators involving nonaffine high-dimensional random fields; (iii) an effective strategy to capture irregular behaviors, e.g., sharp transitions of the PDE solutions. Two numerical examples are provided to demonstrate the advantageous performance of our method. 

Publication:

• L. Mu and G. Zhang, A domain-decomposition model reduction method for linear convection- diffusion equations with random coefficients, SIAM Journal on Scientific Computing, 41 (3), pp. A1984-A2011, 2019.

• G. Zhang, L. Mu, A non-intrusive domain-decomposition model reduction method for linear steady-state PDEs with random coefficients, Numerical Methods for Partial Differential Equations, (2021), 1--19. (https://doi.org/10.1002/num.22856).