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The object of this paper is to present the numerical algorithm to obtain a finite difference solution for an elliptic transmission problem by use of the direct method of lines ([Nak65],[KK98],[KK99]). Let Π be a rectangular domain in R2, Ω1 be an open subset of Π and Ω2= Π\Ω1, Γ=∂ Ω1 (see Figure 1). Then the elliptic transmission problem is formulated as follows. And it is well known that (3) and (4) are called the conditions of transmission (cf.[DL90],[Lio71]).