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Numerical Integration on Hyperrectangles in Isoparametric Unfitted Finite Elements

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Numerical Mathematics and Advanced Applications ENUMATH 2017 (ENUMATH 2017)

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We consider the recently introduced idea of isoparametric unfitted finite element methods and extend it from simplicial meshes to quadrilateral and hexahedral meshes. The concept of the isoparametric unfitted finite element method is the construction of a mapping from a reference configuration to a higher order accurate configuration where the reference configuration is much more accessible for higher order quadrature. The mapping is based on a level set description of the geometry and the reference configuration is a lowest order level set approximation. On simplices this results in a piecewise planar and continuous approximation of the interface. With a simple geometry decomposition quadrature rules can easily be applied based on a tesselation. On hyperrectangles the reference configuration corresponds to the zero level of a multilinear level set function which is not piecewise planar. In this work we explain how to achieve higher order accurate quadrature with only positive quadrature weights also in this case.

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The authors gratefully acknowledge funding by the German Science Foundation (DFG) within the project “LE 3726/1-1” and suggestions on a former version of this paper by Hans-Georg Raumer and an anonymous reviewer.

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Correspondence to Christoph Lehrenfeld .

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Heimann, F., Lehrenfeld, C. (2019). Numerical Integration on Hyperrectangles in Isoparametric Unfitted Finite Elements. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham.

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