We develop a finite element method for the Laplace--Beltrami operator on asurface described by a set of patchwise parametrizations. The patches provide apartition of the surface and each patch is the image by a diffeomorphism of asubdomain of the unit square which is bounded by a number of smooth trimcurves. A patchwise tensor product mesh is constructed by using a structuredmesh in the reference domain. Since the patches are trimmed we obtain cutelements in the vicinity of the interfaces. We discretize the Laplace--Beltramioperator using a cut finite element method that utilizes Nitsche's method toenforce continuity at the interfaces and a consistent stabilization term tohandle the cut elements. Several quantities in the method are convenientlycomputed in the reference domain where the mappings impose a Riemannian metric.We derive a priori estimates in the energy and $L^2$ norm and also presentseveral numerical examples confirming our theoretical results.
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