  The estimation of the probability of rare events is an important task in
reliability and risk assessment. We consider failure events that are expressed
in terms of a limit-state function, which depends on the solution of a partial
differential equation (PDE). In many applications, the PDE cannot be solved
analytically. We can only evaluate an approximation of the exact PDE solution.
Therefore, the probability of rare events is estimated with respect to an
approximation of the limit-state function. This leads to an approximation error
in the estimate of the probability of rare events. Indeed, we prove an error
bound for the approximation error of the probability of failure, which behaves
like the discretization accuracy of the PDE multiplied by an approximation of
the probability of failure, the first order reliability method (FORM) estimate.
This bound requires convexity of the failure domain. For non-convex failure
domains, we prove an error bound for the relative error of the FORM estimate.
Hence, we derive a relationship between the required accuracy of the
probability of rare events estimate and the PDE discretization level. This
relationship can be used to guide practicable reliability analyses and, for
instance, multilevel methods.
