Convergent finite element methods for the perfect conductivity problem with close-to-touching inclusions

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.