Solving Stochastic PDEs Using FEniCS and UQtk

TL;DR

This paper demonstrates how to implement the intrusive spectral stochastic finite element method for stochastic PDEs efficiently using FEniCS and UQTk, reducing coding efforts and facilitating broader adoption.

Contribution

It shows how to alleviate implementation challenges of intrusive SSFEM using FEniCS and UQTk, with detailed algorithmic guidance and code snippets.

Findings

Implementation of intrusive SSFEM is simplified with FEniCS and UQTk.

Provides detailed algorithmic steps and code for practical use.

Facilitates wider adoption of intrusive stochastic PDE methods.

Abstract

The intrusive (sample-free) spectral stochastic finite element method (SSFEM) is a powerful numerical tool for solving stochastic partial differential equations (PDEs). However, it is not widely adopted in academic and industrial applications because it demands intrusive adjustments in the PDE solver, which require substantial coding efforts compared to the non-intrusive (sampling) SSFEM. Using an example of stochastic PDE, in this article, we demonstrate that the implementational challenges of the intrusive approach can be alleviated using FEniCS -- a general purpose finite element package and UQTk -- a collection of libraries and tools for the quantification of uncertainty. Furthermore, the algorithmic details and code snippets are provided to assist computational scientists in implementing these methods for their applications. This article is extracted from the author's thesis [1].