The aim of this paper is to provide a new perspective on finite element accuracy. Starting from a geometrical reading of the Bramble- Hilbert lemma, we recall the two probabilistic laws we got in previous works that estimate the relative accuracy, considered as a random variable, between two finite elements P-k and P-m (k < m). Then we analyze the asymptotic relation between these two probabilistic laws when the difference m - k goes to infinity. New insights which qualify the relative accuracy in the case of high order finite elements are also obtained.

We derive an explicit k-dependence in $W^{m,p}$ error estimates for $P_k$ Lagrange finite elements. Two laws of probability are established to measure the relative accuracy between $P_{k_1}$ and $P_{k_2}$ finite elements, ( $k_1<k_2$), in terms of $W^{m,p}$-norms. We further prove a weak asymptotic relation in $D^{\prime }\left(\mathbb{R}\right)$ between these probabilistic laws when difference $k_2-k_1$ goes to infinity. Moreover, as expected, one finds that $P_{k_2}$ finite element is surely more accurate than $P_{k_1}$, for sufficiently small values of the mesh size h. Nevertheless, our results also highlight cases where $P_{k_1}$ is more likely accurate than $P_{k_2}$, for a range of values of h. Hence, this approach brings a new perspective on how to compare two finite elements, which is not limited to the rate of convergence.

A probabilistic approach is developed for the exact solution u to a deterministic partial differential equation as well as for its associated approximation $u_h^{(k)}$ performed by $P_k$ 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 $u_h^{(k)}$. We, thus, fill this knowledge gap by considering the exact solution u together with its corresponding approximation $u_h^{(k)}$ as random variables. By a method of consequence, any function where u and $u_h^{(k)}$ are involved are modeled as random variables as well. In this paper, we focus our analysis on a variational formulation defined on $W^{m,p}$ Sobolev spaces and the corresponding a priori estimates of the exact solution u and its approximation $u_h^{(k)}$ in order 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 $P_{k_2},(k_1<k_2)$.

This paper is devoted to a new first order Taylor-like formula, where the corresponding remainder is strongly reduced in comparison with the usual one, which appears in the classical Taylor’s formula. To derive this new formula, we introduce a linear combination of the first derivative of the concerned function, which is computed at $n+1$ equally spaced points between the two points, where the function has to be evaluated. We show that an optimal choice of the weights in the linear combination leads to minimizing the corresponding remainder. Then, we analyze the Lagrange $P_1$- interpolation error estimate and the trapezoidal quadrature error, in order to assess the gain of the accuracy we obtain using this new Taylor-like formula.

The aim of this paper is to provide new perspectives on relative finite element accuracy which is usually based on the asymptotic speed of convergence comparison when the mesh size h goes to zero. Starting from a geometrical reading of the error estimate due to the Bramble-Hilbert lemma, we derive two probability distributions that estimate the relative accuracy, considered as a random variable, between two Lagrange finite elements P-k and P-m (k < m). We establish mathematical properties of these probabilistic distributions and we get new insights which, among others, show that P-k or P-m is more likely accurate than the other, depending on the value of the mesh size h.