Interface solutions of partial differential equations with point singularity

Abstract

Abstract Partial differential equations (PDEs) such as those that describe linear elastic fracture mechanics admit analytic solutions with low regularity at crack tips. Existing numerical methods for such PDEs suffer from the constraint of this low regularity and fail to deliver an optimal high-order rate of convergence. We approach this problem by (i) choosing an artificial interface to enclose the center of low regularity and (ii) representing the solution in the interior of artificial interface as unknown linear combination of known low-regularity modes of the solution. This results in an interface formulation of the original PDE, and the linear combination is represented in the interface conditions. By enforcing the smooth component of the numerical solution in the interior domain to be approximately zero, a least squares problem is obtained for the unknown coefficients. The solution of this least squares problem will provide the approximate interface conditions for the numerical solution of the PDE in the exterior domain. The potential of our interface formulation is favorably demonstrated by numerical experiments on one- and two-dimensional Poisson equations with low-regularity solutions. High-order numerical solutions of unknown coefficients and PDEs are obtained. This proves the potential of the proposed interface formulation as the theoretical basis for solving linear elastic fracture mechanics problems. We address the relations between our interface formulation and domain decomposition methods, as well as a regularization strategy for the Poisson–Boltzmann equation with singular charge density.