Dual extremum principles have been used previously to obtain error bounds for a wide class of boundary value problems. Here the dual extremum principles are formulated associated with the set of operator equations of the form L//j// phi //j equals T//j*k//jT//j// phi //j equals f//j(// phi //j) in D//j, j equals 1,2, with the boundary conditions// phi //j equals alpha //j equals alpha //j on GAMMA //j equals PARTIAL D//j minus GAMMA , j equals 1,2, and the interface matching conditions sigma //1(// phi //2 minus // phi //1 minus g) equals 0 on GAMMA , sigma //1*k//1T//1// phi //1 minus sigma //2*k//2T//2// phi //2 equals h(x,y) on GAMMA .