# Numerical approximation of poroelasticity with random coefficients using Polynomial Chaos and Hybrid High-Order methods

## TL;DR

This paper develops a numerical method combining Polynomial Chaos and Hybrid High-Order discretizations to efficiently approximate the behavior of poroelastic materials with uncertain properties, validated through convergence and uncertainty propagation studies.

## Contribution

It introduces a non-intrusive Polynomial Chaos approach integrated with Hybrid High-Order methods for stochastic poroelasticity problems, supporting complex meshes and arbitrary approximation orders.

## Key findings

- Convergence of Polynomial Chaos approximation with sparse spectral projection.
- Effective propagation of input uncertainty to the solution.
- Validation through numerical experiments on an injection-extraction problem.

## Abstract

In this work, we consider the Biot problem with uncertain poroelastic coefficients. The uncertainty is modelled using a finite set of parameters with prescribed probability distribution. We present the variational formulation of the stochastic partial differential system and establish its well-posedness. We then discuss the approximation of the parameter-dependent problem by non-intrusive techniques based on Polynomial Chaos decompositions. We specifically focus on sparse spectral projection methods, which essentially amount to performing an ensemble of deterministic model simulations to estimate the expansion coefficients. The deterministic solver is based on a Hybrid High-Order discretization supporting general polyhedral meshes and arbitrary approximation orders. We numerically investigate the convergence of the probability error of the Polynomial Chaos approximation with respect to the level of the sparse grid. Finally, we assess the propagation of the input uncertainty onto the solution considering an injection-extraction problem.