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Currently, many studies on the local discontinuous Galerkin method focus on the Cartesian grid with low computational efficiency and poor adaptability to complex shapes. A new immersed boundary method is presented, and this method employs the adaptive Cartesian grid to improve the adaptability to complex shapes and the immersed boundary to increase computational efficiency. The new immersed boundary method employs different boundary cells (the physical cell and ghost cell) to impose the boundary condition and the reconstruction algorithm of the ghost cell is the key for this method. The classical model elliptic equation is used to test the method. This method is tested and analyzed from the viewpoints of boundary cell type, error distribution and accuracy. The numerical result shows that the presented method has low error and a good rate of the convergence and works well in complex geometries. The method has good prospect for practical application research of the numerical calculation research.

New Immersed Boundary Method on the Adaptive Cartesian Grid Applied to the Local Discontinuous Galerkin Method