We present a finite volume scheme for solving elliptic boundary value problems with solutions that have one or a few small regions with high activity. The scheme results from combining the local defect correction method (LDC), introduced in , with standard finite volume discretizations on a global coarse and on local fine uniform grids. The iterative discretization method that is obtained in this way yields a discrete approximation of the continuous solution on a composite grid. For the LDC method in its standard form, the discrete conservation property, which is one of the main attractive features of a finite volume method, is lost for the composite grid approximation. For the modified LDC method we present here, discrete conservation holds for the composite grid solution, too.