Solving Elliptic equations with sharp-edged interfaces is a challenging problem for most existing methods, especially when the solution is highly oscillatory. Nonetheless, it has wide applications in engineering and science. In the first part of this talk, I will present a non-traditional finite element method for solving matrix coefficient elliptic equations with shape-edged interfaces with 2nd order accuracy in L-infinity norm. In the second part of this talk, I will switch to a topics on interface shape classification, where the goal is to characterize a shape using the least amount of data, to generate a shape library and to match a given shape to one of the shapes in the library after certain rotation, scaling and shifting. I will present a novel method by using the response matrix data, motivated by a direct imaging method for inverse problems.