An Immersed Finite Element Method with Integral Equation Correction

We propose a robust immersed finite element method in which an integral equation formulation is used to enforce essential boundary conditions. The solution of a boundary value problem is expressed as the superposition of a finite element solution and an integral equation solution. For computing the finite element solution, the physical domain is embedded into a slightly larger Cartesian (box-shaped) domain and is discretized using a block-structured mesh. The defect in the essential boundary conditions, which occurs along the physical domain boundaries, is subsequently corrected with an integral equation method. In order to facilitate the mapping between the finite element and integral equation solutions, the physical domain boundary is represented with a signed distance function on the block-structured mesh. As a result, only a boundary mesh of the physical domain is necessary and no domain mesh needs to be generated, except for the non-boundary-conforming block-structured mesh. The overall approach is first presented for the Poisson equation and then generalised to incompressible viscous flow equations. As an example for fluid-structure coupling, the settling of a heavy rigid particle in a closed tank is considered