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 in 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 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 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.
Revised: September 6, 2019 |
Published: December 15, 2019
Citation
Zhou Y., and V. Gupta. 2019.Interface Solutions of Partial Differential Equations with Point Singularity.Journal of Computational and Applied Mathematics 362.PNNL-SA-122662.doi:10.1016/j.cam.2018.10.006