In the perfect conductivity problem (i.e., the conductivity problem with perfectly conducting inclusions), the gradient of the electric field is often very large in a narrow region between two inclusions and blows up as the distance between the inclusions tends to zero. The rigorous error analysis for the computation of such perfect conductivity problems with close-to-touching inclusions of general geometry still remains open in three dimensions. We address this problem by establishing new asymptotic estimates for the second-order partial derivatives of the solution with explicit dependence on the distance $\varepsilon $ between the inclusions, and use the asymptotic estimates to design a class of graded meshes and finite element spaces to solve the perfect conductivity problem with possibly close-to-touching inclusions. In particular, we propose a special finite element basis function that resolves the asymptotic singularity of the solution by making the interpolation error bounded in $W^{1,\infty }$ in a neighborhood of the close-to-touching point, even though the solution itself is blowing up in $W^{1,\infty }$. This is crucial in the error analysis for the numerical approximations. We prove that the proposed method yields optimal-order convergence in the $H^1$ norm, uniformly with respect to the distance $\varepsilon $ between the inclusions, in both two and three dimensions for general convex smooth inclusions, which are possibly close-to-touching. Numerical experiments are presented to support the theoretical analysis and to illustrate the convergence of the proposed method for different shapes of inclusions in both two- and three-dimensional domains.

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收敛有限元方法解决具有紧密接触夹杂物的完美电导率问题

在完美电导率问题（即具有完美导电夹杂物的电导率问题）中，电场梯度在两个夹杂物之间的狭窄区域中通常非常大，并且随着夹杂物之间的距离趋于零而爆炸。对于计算这种具有接近接触的一般几何形状的夹杂物的完美电导率问题的严格误差分析在三个维度上仍然是开放的。我们通过为解的二阶偏导数建立新的渐近估计来解决这个问题，该估计明确依赖于包含物之间的距离 $\varepsilon $，并使用渐近估计来设计一类分级网格和有限元空间解决可能接近接触的内含物的完美导电性问题。特别是，我们提出了一种特殊的有限元基函数，通过使插值误差在接近接触点的邻域中限制在 $W^{1,\infty }$ 内来解决解的渐近奇异性，即使解决方案本身在 $W^{1,\infty }$ 中爆炸。这对于数值近似的误差分析至关重要。我们证明，所提出的方法在 $H^1$ 范数下产生最优阶收敛，对于一般凸光滑包含物，在二维和三维上，相对于包含物之间的距离 $\varepsilon $ 一致，这些距离可能很接近- 触摸。数值实验旨在支持理论分析，并说明所提出的方法对于二维和三维域中不同形状的夹杂物的收敛性。