Stable generalized finite element method (SGFEM) for parabolic interface problems. (English) Zbl 1464.65135
Summary: We study a stable generalized finite element method (SGFEM) for parabolic interface problems with smooth interfaces. The standard finite element (FE) space based on a simple mesh is enriched by a one-side distance function to simulate gradient jumps of exact solutions on the interfaces. The mesh is (quasi)uniform and independent of the interfaces so that mesh matching or refinement along the interfaces is not needed. The main idea to derive the SGFEM is to modify the one-side distance function by subtracting its FE interpolant. The SGFEM approximates the parabolic interface problem with an optimal convergence rate and is stable in that its scaled condition number is of the same order as the FEM \((O(h^{- 2})\), \(h\) is a mesh-size parameter) and does not depend on the relative positions of the mesh and interfaces. The proposed SGFEM is a conforming method and is free from any penalty parameters or stability schemes. Moreover, the components of the approximation functions of the SGFEM are independent when extended to vector-valued interface problems. The optimal convergence rates of SGFEM for curved interfaces are proven theoretically for the backward Euler and Crank-Nicolson formulations. The convergence and conditioning are tested by numerical experiments.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65F35 | Numerical computation of matrix norms, conditioning, scaling |
Software:
IIMPACKReferences:
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