A non‐intrusive domain‐decomposition model reduction method for linear steady‐state partial differential equations with random coefficients

Abstract

Domain decomposition methods have been proved to be an effective strategy to reduce the dimension of parametric partial differential equations (PDEs). However, existing domain decomposition methods for parametric PDEs are usually intrusive, which means domain decomposition based solvers need to be implemented from scratch for each target parametric PDE. To address this issue, we develop a new non-intrusive domain-decomposition model reduction method for linear steady-state PDEs with random-field coefficients. As a variant of our previous work by Mu and Zhang, the new method only needs access to the final linear system, that is, the global stiffness matrix and the right hand side, of a deterministic PDE solver, in order to build a domain-decomposition-based reduced model without intrusive implementation from scratch. The key idea is to remove the interface condition between sub-domains and rely on the correlation between columns of the linear system to couple the sub-domains. The non-intrusive feature enables the applicability of the proposed method to a broader class of uncertainty quantification problems, where many legacy codes/solvers can be fully reused by our method. Two numerical examples including diffusion equations with random diffusivity and convection-dominated transport with random velocity, are provided to demonstrate the effectiveness and efficiency of our method.