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New Cartesian grid methods for interface problems using the finite element formulation. (English) Zbl 1055.65130

Authors’ summary: New finite element methods based on Cartesian triangulations are presented for two dimensional elliplic interface problems involving discontinuities in the coefficients. The triangulations in these methods do not need to fit the interfaces. The basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximalely. Both non-conforming and conforming finite element spaces are considered.
Corresponding interpolation functions are proved to be second older accurate in the maximum norm. The conforming finite element method has been shown to be convergent. With Cartesian triangulations, these new methods can be used as finite difference methods. Numerical examples are provided to support the methods and the theoretical analysis.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35R05 PDEs with low regular coefficients and/or low regular data
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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