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A higher degree immersed finite element method based on a Cauchy extension for elliptic interface problems. (English) Zbl 1420.65122

Summary: This article develops and analyzes a \(p\)th degree immersed finite element (IFE) method for solving the elliptic interface problems with meshes independent of the coefficient discontinuity in the involved partial differential equations. The proposed \(p\)th degree IFE functions are macro polynomials constructed by weakly solving a Cauchy problem locally on each interface element according to the interface jump conditions. To alleviate the discontinuous effects of IFE functions, penalties on both the edges of interface elements and the interface itself are employed in the proposed IFE scheme. New techniques are introduced to analyze the proposed IFE functions in a format of macro polynomials, including their existence, the optimal approximation capabilities of the resulting IFE spaces, and trace inequalities. These results are then further applied to prove that the proposed IFE method converges optimally in both the \(L^2\) and \(H^1\) norms.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35R05 PDEs with low regular coefficients and/or low regular data
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

Software:

CutFEM; PROST; IIMPACK; OASES
PDFBibTeX XMLCite
Full Text: DOI

References:

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