Probabilistic Approach to Characterize Quantitative Uncertainty in Numerical Approximations

This paper proposes a statistical and probabilistic approach to compare and analyze the errors of two different approximation methods. We introduce the principle of numerical uncertainty in such a process, and we illustrate it by considering the discretization difference between two different approximation orders, e.g., first and second order Lagrangian finite element. Then, we derive a probabilistic approach to define and to qualify equivalent results. We illustrate our approach on a model problem on which we built the two above mentioned finite element approximations. We consider some variables as physical “predictors”, and we characterize how they influence the odds of the approximation methods to be locally “same order accurate”.