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.
Original language | English |
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Pages (from-to) | 1993-2011 |
Number of pages | 19 |
Journal | Numerical Methods for Partial Differential Equations |
Volume | 38 |
Issue number | 6 |
DOIs | |
State | Published - Nov 2022 |
Funding
Office of Science, ERKJ352; ERKJ359; Ralph E. Powe Junior Faculty Enhancement Awards, Laboratory Directed Research and Development, DE‐AC05‐00OR22725; U.S. National Science Foundation, Computational Mathematics, 1620027 Funding information This material is based upon work supported in part by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under contracts ERKJ352, ERKJ359; the U.S. National Science Foundation, Computational Mathematics program under award 1620027; and by the Laboratory Directed Research and Development program at the Oak Ridge National Laboratory, which is operated by UT‐Battelle, LLC., for the U.S. Department of Energy under Contract DE‐AC05‐00OR22725. Dr Lin Mu's research is partially supported by Ralph E. Powe Junior Faculty Enhancement Awards.
Keywords
- domain decomposition
- high dimensionality
- non-intrusive model reduction
- parametric PDEs
- random fields
- sharp transitions
- uncertainty quantification