A probabilistic approach for exact solutions of determinist PDE's as well as their finite element approximations

Abstract

A probabilistic approach is developed for the exact solution $u$ to a determinist partial differential equation as well as for its associated approximation $u^{(k)}_{h}$ performed by $P_k$ Lagrange finite element. Two limitations motivated our approach: on the one hand, the inability to determine the exact solution $u$ to a given partial differential equation (which initially motivates one to approximating it) and, on the other hand, the existence of uncertainties associated with the numerical approximation $u^{(k)}_{h}$. We thus fill this knowledge gap by considering the exact solution $u$ together with its corresponding approximation $u^{(k)}_{h}$ as random variables. By way of consequence, any function where $u$ and $u_{h}^{(k)}$ are involved as well. In this paper, we focus our analysis to a variational formulation defined on $W^{m,p}$ Sobolev spaces and the corresponding a priori estimates of the exact solution $u$ and its approximation $u^{(k)}_{h}$ to consider their respective $W^{m,p}-$norm as a random variable, as well as the $W^{m,p}$ approximation error with regards to $P_k$ finite elements. This will enable us to derive a new probability distribution to evaluate the relative accuracy between two Lagrange finite elements $P_{k_1}$ and $P_{k_2}, (k_1 < k_2)$.