New Cartesian grid methods for interface problems 
                    using the finite element formulation

                           Zhilin Li
      Center For Research in Scientific Computation & Mathematics
                  North Carolina State University
                         Raleigh, NC 27695, USA


New finite element methods based on Cartesian triangulations are 
presented for one and two dimensional elliptic interface problems involving 
discontinuities in the coefficients in the solution.  The triangulations 
in these methods do not need to fit the interfaces. The basis functions 
in these methods are constructed to satisfy the interface jump conditions
either exactly or approximately.  Both non-conforming and conforming 
finite element spaces are considered.  Corresponding interpolation 
functions are proved to be second order accurate in the maximum norm.

For non-homogeneous jump conditions, we have developed a new strategy to
transform the original interface problem to a new one with homogeneous jump
conditions using the level set function.