Abstract: In this talk, I will spend first 10-15 minutes to introduce some basic ideas of Finite Element Method (FEM) for PDEs and the Immersed FEM for PDEs with discontinuous coefficient. Next I will talk about our new results on developing and analyzing a trilinear IFEM for solving three-dimensional interface problems. The proposed method can be utilized on interface-unfitted meshes such as Cartesian grids consisting of cuboids even if the 3D interface surface possesses an arbitrary shape. Fundamental estimates such as the trace inequalities, inversed inequalities, and the approximation capabilities will be established. Optimal a priori error estimates are proved in both energy and L2 norms. Numerical examples will be provided not only to verify our theoretical results but also to demonstrate the applicability of this method in tackling some real-world 3D interface models.
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