  Linear elastic fracture mechanics admit analytic solutions that have low
regularity at crack tips. Current numerical methods for partial differential
equations (PDEs) of this type suffer from the constraint of such low
regularity, and fail to deliver optimal high order rate of convergence. We
approach the problem by (i) choosing an artificial interface to enclose the
center of the low regularity; and (ii) representing the solution in the
interior of artificial interface as unknown linear combination of known modes
of low regular solutions. This gives rise to an interface formulation of the
original PDE, and the linear combination are represented the interface
conditions. By enforcing the smooth component of numerical solution in the
interior domain to be approximately zero, a least square problem is obtained
for the unknown coefficients. The solution of this least square problem will
provide 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 1-D and 2-D Poisson equations with low
regular 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 indicate 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.
