  For boundary value problem of an elliptic equation with variable coefficients
describing the physical field distribution in inhomogeneous media, the Levi
function can represent the solution in terms of volume and surface potentials,
with the drawback that the volume potential involving in the solution
expression requires heavy computational costs as well as the solvability of the
integral equations with respect to the density pair. We introduce an modified
integral expression for the solution to an elliptic equation in divergence form
under the Levi function framework. The well-posedness of the linear integral
system with respect to the density functions to be determined is rigorously
proved. Based on the singularity decomposition for the Levi function, we
propose two schemes to deal with the volume integrals so that the density
functions can be solved efficiently. One method is an adaptive discretization
scheme (ADS) for computing the integrals with continuous integrands, leading to
the uniform accuracy of the integrals in the whole domain, and consequently the
efficient computations for the density functions. The other method is the dual
reciprocity method (DRM) which is a meshless approach converting the volume
integrals into boundary integrals equivalently by expressing the volume density
as the combination of the radial basis functions determined by the interior
grids. The proposed schemes are justified numerically to be of satisfactory
computation costs. Numerical examples in 2-dimensional and 3-dimensional cases
are presented to show the validity of the proposed schemes.
