Computable error estimates for finite element approximations of elliptic partial differential equations with rough stochastic data

TL;DR

This paper develops practical error estimates for finite element solutions of elliptic PDEs with rough stochastic data, enabling more accurate and efficient numerical simulations in complex geophysical applications.

Contribution

It introduces goal-oriented, low-cost error estimates for PDEs with irregular stochastic coefficients, addressing limitations of standard methods.

Findings

Error estimates effectively capture high-frequency solution content.

Numerical experiments confirm the estimates' accuracy in 1D and 2D.

Applications demonstrated in geophysical subsurface flow modeling.

Abstract

We derive computable error estimates for finite element approximations of linear elliptic partial differential equations (PDE) with rough stochastic coefficients. In this setting, the exact solutions contain high frequency content that standard a posteriori error estimates fail to capture. We propose goal-oriented estimates, based on local error indicators, for the pathwise Galerkin and expected quadrature errors committed in standard, continuous, piecewise linear finite element approximations. Derived using easily validated assumptions, these novel estimates can be computed at a relatively low cost and have applications to subsurface flow problems in geophysics where the conductivities are assumed to have lognormal distributions with low regularity. Our theory is supported by numerical experiments on test problems in one and two dimensions.