  In this paper, we discuss the application of the Generalized Finite Element
Method (GFEM) to approximate the solutions of quasilinear elliptic equations
with multiple interfaces in one dimensional space. The problem is characterized
by spatial discontinuity of the elliptic coefficient that depends on the
unknown solution. It is known that unless the partition of the domain matches
the discontinuity configuration, accuracy of standard finite element techniques
significantly deteriorates and standard refinement of the partition may not
suffice. The GFEM is a viable alternative to overcome this predicament. It is
based on the construction of certain enrichment functions supplied to the
standard space that capture effects of the discontinuity. This approach is
called stable (SGFEM) if it maintains an optimal rate of convergence and the
conditioning of GFEM is not worse than that of the standard FEM. A convergence
analysis is derived and performance of the method is illustrated by several
numerical examples. Furthermore, it is known that typical global formulations
such as FEMs do not enjoy the numerical local conservation property that is
crucial in many conservation law-based applications. To remedy this issue, a
Lagrange multiplier technique is adopted to enforce the local conservation. A
numerical example is given to demonstrate the performance of proposed
technique.
