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