AI 中文总结
针对完美导电性和线性弹性问题中夹杂物近接触时电场和应力高度集中的难题，提出基于解高阶导数渐近估计的高阶有限元方法，证明其收敛性，通过数值例子展示收敛速度及夹杂物靠近时解梯度的爆炸行为。
AI 中文摘要
在完美导电性和线性弹性问题中，电场和应力总是在相邻完美（刚性）夹杂物之间的狭窄区域内高度集中，并在夹杂物之间的距离趋近于零时趋于无穷大。针对此类集中问题设计具有严格误差分析的高阶数值方法仍然是未解决的问题。在本文中，我们提出了第一种用于解决这些问题的高阶有限元方法。我们的方法基于解的高阶导数的渐近估计，采用分级网格和根据这些导数估计专门设计的辅助基函数。我们证明所提出的方法以$O(h^p)$的$H^1$误差界收敛，且误差界与夹杂物之间的距离（可能趋近于零）无关，其中$h$是网格尺寸，$p$是有限元的次数。给出了二维和三维的数值例子来证明数值解的收敛速度。特别是，当夹杂物相互靠近时，展示了解的梯度的爆炸行为。
英文摘要
In perfect conductivity and linear elasticity problems, the electric field and stress always become highly concentrated within narrow regions between adjacent perfect (rigid) inclusions, and blow up as the distance between inclusions approaches zero. The design of high-order numerical methods with rigorous error analysis for such concentration problems remains open. In this paper, we present the first high-order finite element method for solving these problems. Our approach is based on asymptotic estimates of high-order derivatives of solutions, employing a graded mesh and auxiliary basis functions specifically designed from these derivative estimates. We prove that the proposed method converges with an $H^1$-error bound of $O(h^p)$ and the error bound is independent of the distance (possibly approaching zero) between inclusions, where $h$ is the mesh size and $p$ is the degree of finite elements. Numerical examples in both two and three dimensions are presented to demonstrate the convergence rates of the numerical solutions. In particular, the blow-up behaviors of the gradients of solutions are demonstrated when the inclusions approach each other.
