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We present a method of CutFEM type for the Poisson problem with eitherDirichlet or Neumann boundary conditions. The computational mesh is obtainedfrom a background (typically uniform Cartesian) mesh by retaining only theelements intersecting the domain where the problem is posed. The resulting meshdoes not thus fit the boundary of the problem domain. Several finite elementmethods (XFEM, CutFEM) adapted to such meshes have been recently proposed. Theoriginality of the present article consists in avoiding integration over theelements cut by the boundary of the problem domain, while preserving theoptimal convergence rates, as confirmed by both the theoretical estimates andthe numerical results.

CutFEM without cutting the mesh cells: a new way to impose Dirichlet and Neumann boundary conditions on unfitted meshes