A Variable-Separation Method for Nonlinear Partial Differential Equations With Random Inputs

Abstract

In this paper, we consider a variable-separation (VS) method to solve the nonlinear partial differential equations (PDEs) with random inputs. The aim of the VS method is to get a sep- arated representation of the Galerkin solution for nonlinear PDEs with random inputs. An essential ingredient of the proposed method is the construction of the optimal stochastic basis functions. The nonlinearity can affect the computation efficiency and may bring challenges for the construction of the optimal stochastic basis functions. In order to overcome the difficulty, we develop the VS method such that the optimal stochastic basis functions are generated in an incremental constructive man- ner. At each enrichment step, a stochastic basis function is determined by the linearized equation deduced from the nonlinear problems at hand. The computation of the VS method decomposes into an offline phase and an online phase. The linearization of the construction for stochastic basis functions can significantly improve the computation efficiency in both offline and online stages. We first describe the VS method for nonlinear stochastic problems in a general framework. Then two nonlinear mod- els with random inputs are considered to formulate the details and methodologies of the proposed method, namely, the nonlinear elliptic equations and the steady Navier--Stokes equations.