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

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^{(k)}_{h} 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_{k1} and P_{k2},
(k_{1} < k_{2}).