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In this paper, we will present composite boundary elements (CBE) for classical Fredholm boundary integral equations. These new boundary elements allow the low-dimensional discretisation of boundary integral equations where the minimal number of degrees of freedom is independent of the, possibly, huge number of charts which are necessary to describe a complicated surface.The applications are threefold: (a) The coarse-grid discretisation by composite boundary elements allow the use of multigrid algorithms for solving the fine-grid discretisation independently of the number of patches which are necessary to describe the surface. (b) If the accuracy requirements are moderate, the composite boundary elements allow the low-dimensional discretisation of the integral equation. (c) A posteriori error indicators can be applied already to a low-dimensional discretisation, which do not resolve the domain, to obtain a problem-adapted discretisation.