  Numerical models based on partial differential equations (PDE), or
integro-differential equations, are ubiquitous in engineering and science,
making it possible to understand or design systems for which physical
experiments would be expensive-sometimes impossible-to carry out. Such models
usually construct an approximate solution of the underlying continuous
equations, using discretization methods such as finite differences or the
finite elements method. The resulting discretization error introduces a form of
uncertainty on the exact but unknown value of any quantity of interest (QoI),
which affects the predictions of the numerical model alongside other sources of
uncertainty such as parametric uncertainty or model inadequacy. The present
article deals with the quantification of this discretization uncertainty.A
first approach to this problem, now standard in the V\&V (Verification and
Validation) literature, uses the grid convergence index (GCI) originally
proposed by P. Roache in the field of computational fluid dynamics (CFD), which
is based on the Richardson extrapolation technique. Another approach, based on
Bayesian inference with Gaussian process models, was more recently introduced
in the statistical literature. In this work we present and compare these two
paradigms for the quantification of discretization uncertainty, which have been
developped in different scientific communities, and assess the potential of
the-younger-Bayesian approach to provide a replacement for the well-established
GCI-based approach, with better probabilistic foundations. The methods are
illustrated and evaluated on two standard test cases from the literature
(lid-driven cavity and Timoshenko beam).
