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

Axioms

Joël Chaskalovic¹ 

[1] Jean Le Rond d’Alembert Institute, Sorbonne University, Place Jussieu, CEDEX 05, 75252 Paris, France;

关键词: error estimates;    finite elements;    Bramble–Hilbert lemma;    Sobolev spaces;   

DOI  :  10.3390/axioms10040349

来源: DOAJ

【 摘 要 】

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 $P_{k_2},(k_1<k_2)$.

【 授权许可】

Unknown