A two-level stochastic collocation method for semilinear elliptic equations with random coefficients

TL;DR

This paper introduces a two-level stochastic collocation method combining coarse and fine discretizations in physical and random spaces, achieving similar accuracy to direct methods but with improved computational efficiency for semilinear elliptic equations with randomness.

Contribution

The paper develops a novel two-level stochastic collocation approach that reduces computational cost while maintaining accuracy for solving semilinear elliptic equations with random coefficients.

Findings

Achieves the same accuracy as direct stochastic collocation methods.

Reduces computational cost for nonlinear problems with randomness.

Numerical experiments confirm theoretical efficiency and accuracy.

Abstract

In this work, we propose a novel two-level discretization for solving semilinear elliptic equations with random coefficients. Motivated by the two-grid method for deterministic partial differential equations (PDEs) introduced by Xu \cite{xu1994novel}, our two-level stochastic collocation method utilizes a two-grid finite element discretization in the physical space and a two-level collocation method in the random domain. In particular, we solve semilinear equations on a coarse mesh $\mathcal{T}_H$ with a low level stochastic collocation (corresponding to the polynomial space $\mathcal{P}_{\boldsymbol{P}}$) and solve linearized equations on a fine mesh $\mathcal{T}_h$ using high level stochastic collocation (corresponding to the polynomial space $\mathcal{P}_{\boldsymbol{p}}$). We prove that the approximated solution obtained from this method achieves the same order of accuracy as that from solving the original semilinear problem directly by stochastic collocation method with $\mathcal{T}_h$ and $\mathcal{P}_{\boldsymbol{p}}$. The two-level method is computationally more efficient than the standard stochastic collocation method for solving nonlinear problems with random coefficients. Numerical experiments are provided to verify the theoretical results.