论文标题
确定派PDE的精确解决方案及其有限元近似值的概率方法
A probabilistic approach for exact solutions of determinist PDE's as well as their finite element approximations
论文作者
论文摘要
为确切的解决方案$ u $开发了一种概率方法，用于确定性偏微分方程以及其关联的近似$ u^{（k）} _ {h} $由$ p_k $ lagrange有限元元素执行。有两个限制激发了我们的方法：一方面，无法确定针对给定的部分微分方程的精确解决方案$ u $（最初促使一个方程式），另一方面，另一方面是与数值近似值$ u^{（k）} _ {h} $相关的不确定性的存在。因此，我们通过将精确解决方案$ u $及其相应的近似值$ u^{（k）} _ {h} $作为随机变量来填补此知识差距。结果，$ u $和$ u_ {h}^{（k）} $的任何功能也涉及。在本文中，我们将分析集中在$ w^{m，p} $ sobolev空间上定义的变异公式，以及相应的先验估计$ u $及其近似$ u^{（k）} _ {h} $，以考虑其相应的$ w^{m，p} - $ quort，以及$ norm and pop，以及$ p_k $有限元素。这将使我们能够得出一个新的概率分布，以评估两个lagrange有限元素之间的相对精度$ p_ {k_1} $和$ p_ {k_2}，（k_1 <k_2）$。
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 $P_{k_2}, (k_1 < k_2)$.
