Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
SIAM Journal on Numerical Analysis
On the implementation of mixed methods as nonconforming methods for second-order elliptic problems
Mathematics of Computation
Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension
SIAM Journal on Scientific Computing
A hybrid method for moving interface problems with application to the Hele-Shaw flow
Journal of Computational Physics
A Fast Iterative Algorithm for Elliptic Interface Problems
SIAM Journal on Numerical Analysis
A Covolume Method Based on Rotated Bilinears for the Generalized Stokes Problem
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems
Mathematics of Computation
A rectangular immersed finite element space for interface problems
Scientific computing and applications
Convergence of the ghost fluid method for elliptic equations with interfaces
Mathematics of Computation
P1 Nonconforming Finite Element Multigrid Method for Radiation Transport
SIAM Journal on Scientific Computing
A numerical method for solving variable coefficient elliptic equation with interfaces
Journal of Computational Physics
The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains (Frontiers in Applied Mathematics)
Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces
Journal of Computational Physics
A sharp interface finite volume method for elliptic equations on Cartesian grids
Journal of Computational Physics
Fast solvers for 3D Poisson equations involving interfaces in a finite or the infinite domain
Journal of Computational and Applied Mathematics
Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces
Journal of Computational Physics
Journal of Computational Physics
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We study some numerical methods for solving a second order elliptic problem with interface. We introduce an immersed finite element method based on the “broken” $P_1$-nonconforming piecewise linear polynomials on interface triangular elements having edge averages as degrees of freedom. These linear polynomials are broken to match the homogeneous jump condition along the interface which is allowed to cut through the element. We prove optimal orders of convergence in the $H^1$- and $L^2$-norm. Next we propose a mixed finite volume method in the context introduced in [S. H. Chou, D. Y. Kwak, and K. Y. Kim, Math. Comp., 72 (2003), pp. 525-539] using the Raviart-Thomas mixed finite element and this “broken” $P_1$-nonconforming element. The advantage of this mixed finite volume method is that once we solve the symmetric positive definite pressure equation (without Lagrangian multiplier), the velocity can be computed locally by a simple formula. This procedure avoids solving the saddle point problem. Furthermore, we show optimal error estimates of velocity and pressure in our mixed finite volume method. Numerical results show optimal orders of error in the $L^2$-norm and broken $H^1$-norm for the pressure and in the $H(\mathrm{div})$-norm for the velocity.