A Probabilistic Approach for Solutions of Deterministic PDE’s as Well as Their Finite Element Approximations

A probabilistic approach is developed for the exact solution u to a deterministic partial differential equation as well its associated approximation uh(k) performed by Pk Lagrange finite element. Two limitations motivated our approach: On one hand, inability determine relative given (which initially motivates approximating it) and, on other existence of uncertainties with numerical uh(k). We, thus, fill this knowledge gap considering together corresponding random variables. By method consequence, any function where and are involved modeled variables well. In paper, we focus analysis variational formulation defined Wm,p Sobolev spaces priori estimates in order consider their respective Wm,p-norm variable, error regards elements. This will enable us derive new probability distribution evaluate accuracy between two elements Pk1 Pk2,(k1<k2).