“…Therefore, a lot of research has been conducted to construct accurate methods employing meshes which are not aligned with Γ but possess a "simple" structure and are fixed throughout the simulation; see for instance the immersed boundary method [42], the immersed interface method [33,37], immersed finite elements [36,39,45], the fictitious domain method [5,27,28,40], the unfitted finite element method [7,19,31,34], the finite cell method [41], unfitted discontinuous Galerkin methods [9], or composite finite elements [30,38], and the references provided there. In this work we will focus on a diffuse interface method for solving (1), see for instance [1,17,23,24,32,35,43]. In this method the sharp interface condition u = g on Γ is replaced by suitable conditions on u − g on a diffuse layer centered around Γ.…”