Interface Formulation and High Order Numerical Solutions of PDEs with Low Regularity

TL;DR

This paper introduces an interface formulation method to improve high order numerical solutions of PDEs with low regularity, such as those in fracture mechanics, by enclosing low regularity regions and solving for interface conditions.

Contribution

It proposes a novel interface formulation approach that enables high order convergence for PDEs with low regularity solutions, demonstrated on Poisson equations.

Findings

Successful high order solutions for 1-D and 2-D Poisson equations with low regularity.

The method effectively enforces interface conditions via least squares.

Potential applications to linear elastic fracture mechanics problems.

Abstract

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.