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A sharp interface method using enriched finite elements for elliptic interface problems. (English) Zbl 1464.65176

Many applications in engineering and biology involve immersed interfaces moving in time within the computational domain. Interface problems are discretized using fitted or unfitted finite elements. The mesh generation process in the case of fitted elements for moving interfaces becomes expensive. In order to avoid expensive re-meshing, the immersed boundary method (IBM) of Peskin is used. In this paper, enriched finite element spaces are used which yield an interface fitted scheme for elliptic interface problems. The features of the interface are approximated properly by introducing local modifications, enrichments, or extensions of the finite element spaces. The derived immersed interface method avoids costly re-meshing. It was shown that the proposed method has the same order of approximation as comparable enriched methods for a general elliptic interface problem. Moreover, the discrete operator inherits the symmetry and positive definiteness of the continuous operator since it is a fitted method. The stability and convergence of the solution is proven and the numerical tests demonstrate optimal order of convergence. The method is compared with the enrichment of a CutFEM and of Peskins original IBM by looking at the local finite element spaces.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65F35 Numerical computation of matrix norms, conditioning, scaling
35J15 Second-order elliptic equations
35A15 Variational methods applied to PDEs
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