Data mining and probabilistic models for error estimate analysis of finite element method

Abstract

In this paper, we propose a new approach based on data mining techniques and probabilistic models to compare and analyze finite element results of partial differential equations. We focus on the numerical errors produced by linear and quadratic finite element approximations. We first show how error estimates contain a kind of numerical uncertainty in their evaluation, which may influence and even damage the precision of finite element numerical results. A model problem, derived from an elliptic approximate Vlasov–Maxwell system, is then introduced. We define some variables as physical predictors, and we characterize how they influence the odds of the linear and quadratic finite elements to be locally “same order” accurate. Beyond this example, this approach proposes a method to compare, between several approximation methods, the accuracy of numerical results.