  We present a novel probabilistic finite element method (FEM) for the solution
and uncertainty quantification of elliptic partial differential equations based
on random meshes, which we call random mesh FEM (RM-FEM). Our methodology
allows to introduce a probability measure on standard piecewise linear FEM. We
present a posteriori error estimators based uniquely on probabilistic
information. A series of numerical experiments illustrates the potential of the
RM-FEM for error estimation and validates our analysis. We furthermore
demonstrate how employing the RM-FEM enhances the quality of the solution of
Bayesian inverse problems, thus allowing a better quantification of numerical
errors in pipelines of computations.
