The Numerical Methods for Oscillating Singularities in Elliptic Boundary Value Problems

The singularities near the crack tips of homogeneous materials are monotone of type rα and rα logδr (depending on the boundary conditions along nonsmooth domains). However, the singularities around the interfacial cracks of the heterogeneous bimaterials are oscillatory of type rα sin(ε log r). The method of auxiliary mapping (MAM), introduced by Babus̆ka and Oh, was proven to be successful in dealing with rα type singularities. However, the effectiveness of MAM is reduced in handling oscillating singularities. This paper deals with oscillating singularities as well as the monotone singularities by extending MAM through introducing the power auxiliary mapping and the exponential auxiliary mapping.