The Numerical Methods for Oscillating Singularities in Elliptic Boundary Value Problems

Numerical methods and error analysis for one dimensional elliptic problems containing singularities

“…However, it fails to remove the pollution effect caused by "log x." Unlike the two-dimensional counterpart ( [25]), an exponential auxiliary mapping defined by () ϭ exp() is of no use in one-dimensional variational equation.…”

“…In this case, MAM has no advantages in the framework of the h-FEM. However, in many practical examples of 2D singularities (corner singularity, jump boundary singularity, interface singularity, and so on), the coefficients of high degree terms in asymptotic expansions ) 10␤ is reduced, MAM in the framework of the p-FEM is able to yield improved results (see [25]). …”

Extension of the method of auxiliary mapping for three-dimensional elliptic boundary value problems

“…Thus, by the choice of mapping size to be ¿1; no components in Q are singular for all three cases. However, in Reference [30], it was observed that these properties do not hold for one-dimensional case.…”

“…The rate of convergence is sketched in the following (a) The following inequality was proved in Reference [23] v 1; S 6K v 1;ˆ S (30) where K = max{4 ; 3 }, and ¿1 is the largest mapping size of three auxiliary mappings. Thus, we have…”

The reproducing singularity particle shape functions for problems containing singularities

“…Thus, in this paper, to deal with singularity problems in the framework of meshfree particle methods, using RPP shape functions of order 2N and special mappings similar to [4,9,18,19], we construct special particle shape functions that reproduce three kinds of singularities:…”