A weakly elliptic singularity is a isolated normal surface singularity whose arithmetic genus is equal to one. This thesis focuses on the weakly elliptic singularities, which are complete intersection. By the method of elliptic sequence which is introduced by Stephen S.-T Yau, I study the weighted dual graph of such singularities and give the complete lists of weighted dual graphs of complete intersection weakly elliptic singularises.;Our classification strategy is very simple. By the tools of elliptic sequence, we first introduce several lemmas in chapter 4, which give the subtle structure and numerical restriction for the weakly dual graphs. Then we list all the possible structure of weighted dual graphs in terms of its associated elliptic sequence. In the chapter 5, we start with minimally elliptic singularities with Z2 = -1, -2, -3, -4 and apply the theorems 4.8 to theorem 4.11, then all the weighted dual graphs were listed.
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