  This paper develops meshless methods for probabilistically describing
discretisation error in the numerical solution of partial differential
equations. This construction enables the solution of Bayesian inverse problems
while accounting for the impact of the discretisation of the forward problem.
In particular, this drives statistical inferences to be more conservative in
the presence of significant solver error. Theoretical results are presented
describing rates of convergence for the posteriors in both the forward and
inverse problems. This method is tested on a challenging inverse problem with a
nonlinear forward model.
