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The boundary element method (BEM) is a widely used method to solve biomedical electromagnetic volume conduction problems. The commonly used formulation of this method uses constant interpolation functions for the potential and flat triangular surface elements. Linear interpolation for the potential on a flat triangular mesh turned out to yield a better accuracy. In this paper, the authors introduce quadratic interpolation functions for the potential and quadratically curved surface elements, resulting from second-order spatial interpolation. Theoretically, this results in an accuracy that is inversely proportional to the third power of element size. The method is tested on a four concentric sphere geometry, representative for electroencephalogram modeling, and compared to previous solutions of this problem in literature. In addition, a cylindrical test configuration is used. It is concluded that the use of quadratic interpolation functions for the potential and of quadratically curved surface elements in BEM results in a significant increase in accuracy and in some cases even a reduction of the computation time with the same number of nodes involved in the calculations.