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

Abstract : A probabilistic approach is developed for the exact solution u to a deterministic partial differential equation as well as for its associated approximation u(k)h performed by Pk 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 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 a method of consequence, any function where u and u(k)h are involved are modeled as random variables as well. In this paper, we focus our analysis on a variational formulation defined on Wm,p Sobolev spaces and the corresponding a priori estimates of the exact solution u and its approximation u(k)h in order to consider their respective Wm,p-norm as a random variable, as well as the Wm,p approximation error with regards to Pk finite elements. This will enable us to derive a new probability distribution to evaluate the relative accuracy between two Lagrange finite elements Pk1 and Pk2,(k1
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Submitted on : Friday, January 7, 2022 - 10:31:16 AM
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Joël Chaskalovic. A Probabilistic Approach for Solutions of Deterministic PDE’s as Well as Their Finite Element Approximations. Axioms, MDPI, 2021, 10 (4), pp.349. ⟨10.3390/axioms10040349⟩. ⟨hal-03516325⟩

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