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This paper presents a new approach to the finite element open boundary problems using high order boundary sensitivity. The finite element method (FEM) gives the solution with boundary truncation error in the open boundary problems like air-core magnet. And it doesn't inform us of the precision of solution. In order to overcome the weakness of FEM, the high-order energy sensitivity method is proposed. It is based on the sensitivity analysis technique. It can be used for the check on the precision of solution and suggest the proper boundary. It was applied to the finite element analysis of air-core magnet to validate the proposed method.