A Probabilistic Approach for Solutions of Deterministic PDE’s as Well as Their Finite Element Approximations - Joël Chaskalovic (Axioms, 2021)
A probabilistic approach is developed for the exact solution u to a deterministic partial differential equation as well as for its associated approximation <semantics><msubsup>uh(k)</msubsup></semantics> performed by <semantics>Pk</semantics> Lagrange finite element. Two limitations motivated our approach: On the one hand, the inability to determine the exact solution u relative 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 <semantics><msubsup>uh(k)</msubsup></semantics>. We, thus, fill this knowledge gap by considering the exact solution u together with its corresponding approximation <semantics><msubsup>uh(k)</msubsup></semantics> as random variables. By a method of consequence, any function where u and <semantics><msubsup>uh(k)</msubsup></semantics> are involved are modeled as random variables as well. In this paper, we focus our analysis on a variational formulation defined on <semantics>Wm,p</semantics> Sobolev spaces and the corresponding a priori estimates of the exact solution u and its approximation <semantics><msubsup>uh(k)</msubsup></semantics> in order to consider their respective <semantics>Wm,p</semantics>-norm as a random variable, as well as the <semantics>Wm,p</semantics> approximation error with regards to <semantics>Pk</semantics> finite elements. This will enable us to derive a new probability distribution to evaluate the relative accuracy between two Lagrange finite elements <semantics>Pk1</semantics> and <semantics>Pk2,(k1<k2)</semantics>.