Augmented immersed interface methods have been developed recently for interface problems and problems on irregular domains including CFD applications with free boundaries and moving interfaces. In an augmented method, one or several augmented variables are introduced along the interface or boundary so that one can get efficient discretizations. The augmented variables should be chosen such that the interface or boundary conditions are satisfied. The key to the success of the augmented methods often relies on the interpolation scheme to couple the augmented variables with the governing differential equations through the interface or boundary conditions. This has been done using a least squares interpolation (under-determined) for which the singular value decomposition (SVD) is used to solve for the interpolation coefficients. In this paper, based on properties of the finite element method, a new augmented immersed finite element method (IFEM) that does not need the interpolations is proposed for elliptic interface problems that have a piecewise constant coefficient. Thus the new augmented method is more efficient and simple than the old one that uses interpolations. The method then is extended to Poisson equations on irregular domains with a Dirichlet boundary condition. Numerical experiments with arbitrary interfaces/irregular domains and large jump ratios are provided to demonstrate the accuracy and the efficiency of the new augmented methods. Numerical results also show that the number of GMRES iterations is independent of the mesh size and nearly independent of the jump in the coefficient.
Interface problem Piecewise constant coefficient Immersed finite element Augmented immersed finite element method Poisson equation on irregular domain Fast poisson solver Least squares interpolation using SVD
Mathematics Subject Classification (2010)
65N15 65N30 35J60
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Hou, T., Wu, X., Cai, Z.: Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comp. 68, 913–943 (1999)CrossRefMathSciNetMATHGoogle Scholar
Hou, T., Wu, X., Zhang, Y.: Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation. Commun. Math. Sci. 2, 185–205 (2004)CrossRefMathSciNetMATHGoogle Scholar
Mayo, A., Greenbaum, A.: Fast parallel iterative solution of Poisson’s and the biharmonic equations on irregular regions. SIAM J. Sci. Statist. Comput. 13, 101–118 (1992)CrossRefMathSciNetMATHGoogle Scholar
Nielsen, B.F.: Finite element discretizations of elliptic problems in the presence of arbitrarily small ellipticity: An error analysis. SIAM J. Numer. Anal. 36, 368–392 (1999)CrossRefMathSciNetGoogle Scholar
Oevermann, M., Klein, R.: A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces. J. Comput. Phys. 219, 749–769 (2006)CrossRefMathSciNetMATHGoogle Scholar
Saad, Y., Schultz, M.: Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7, 856–869 (1986)CrossRefMathSciNetMATHGoogle Scholar