Journal of Computational Physics
SIAM Journal on Numerical Analysis
A hybrid method for moving interface problems with application to the Hele-Shaw flow
Journal of Computational Physics
A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)
Journal of Computational Physics
The immersed finite volume element methods for the elliptic interface problems
Mathematics and Computers in Simulation - Special issue from IMACS sponsored conference: “Modelling '98”
High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces
Journal of Computational Physics
Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities
Journal of Computational Physics
BDDC methods for discontinuous Galerkin discretization of elliptic problems
Journal of Complexity
Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems
Journal of Computational Physics
On computational issues of immersed finite element methods
Journal of Computational Physics
Journal of Computational Physics
Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces
Journal of Computational Physics
SIAM Journal on Numerical Analysis
A weak Galerkin finite element method for second-order elliptic problems
Journal of Computational and Applied Mathematics
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Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried out to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both L"2 and L"~ norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order O(h) to O(h^1^.^5) for the solution itself in L"~ norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order O(h^1^.^7^5) to O(h^2) in the L"~ norm for C^1 or Lipschitz continuous interfaces associated with a C^1 or H^2 continuous solution.