In this work, the uncertainty associated with the finite element discretization error is modeled following the Bayesian paradigm. First, a continuous formulation is derived, where a Gaussian process prior over the solution space is updated based on observations from a finite element discretization. To avoid the computation of intractable integrals, a second, finer, discretization is introduced that is assumed sufficiently dense to represent the true solution field. A prior distribution is assumed over the fine discretization, which is then updated based on observations from the coarse discretization. This yields a posterior distribution with a mean that serves as an estimate of the solution, and a covariance that models the uncertainty associated with this estimate. Two particular choices of prior are investigated: a prior defined implicitly by assigning a white noise distribution to the right-hand side term, and a prior whose covariance function is equal to the Green's function of the partial differential equation. The former yields a posterior distribution with a mean close to the reference solution, but a covariance that contains little information regarding the finite element discretization error. The latter, on the other hand, yields posterior distribution with a mean equal to the coarse finite element solution, and a covariance with a close connection to the discretization error. For both choices of prior a contradiction arises, since the discretization error depends on the right-hand side term, but the posterior covariance does not. We demonstrate how, by rescaling the eigenvalues of the posterior covariance, this independence can be avoided.

翻译：本研究遵循贝叶斯范式对有限元离散误差相关的不确定性进行建模。首先，推导了连续形式的数学表达，其中基于有限元离散的观测数据更新解空间上的高斯过程先验。为避免计算不可解积分，引入第二个更细化的离散化网格，该网格被假定足够密集以表征真实解场。在细化网格上设定先验分布，随后基于粗网格观测数据对其进行更新，由此得到后验分布：其均值作为解的估计值，协方差则建模该估计值的不确定性。本文考察两种特定的先验选择：一种通过为右端项赋予白噪声分布而隐式定义的先验，另一种协方差函数等于偏微分方程格林函数的先验。前者产生的后验均值接近参考解，但协方差包含关于有限元离散误差的信息较少；后者则产生均值等于粗网格有限元解的后验分布，且其协方差与离散误差密切相关。两种先验选择均存在矛盾——离散误差本应依赖于右端项，但后验协方差却未体现此依赖关系。我们展示了如何通过重新缩放后验协方差的特征值来规避这种独立性。