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Finite element methods and their convergence for elliptic and parabolic interface problems. (English) Zbl 0909.65085
The finite element method for solving second-order both elliptic and parabolic interface problems is studied. It is proved that the method converges as the usual non-interface elliptic and parabolic problems, both for the energy-norm and the \(L\)-norm. The resultant linear systems are always symmetric and positive definite when the original partial differential equations are self-adjoint and uniformly elliptic. The authors approximate the smooth interface by a polygon, and the interface function by its interpolant. Here the approximation problem seems similar to the classical finite element method.

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35K20 Initial-boundary value problems for second-order parabolic equations
65F10 Iterative numerical methods for linear systems
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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