A second-order-accurate symmetric discretization of the Poisson equation on irregular domains
Journal of Computational Physics
Journal of Computational Physics
Reactive autophobic spreading of drops
Journal of Computational Physics
Accurate numerical methods for micromagnetics simulations with general geometries
Journal of Computational Physics
Three-dimensional elliptic solvers for interface problems and applications
Journal of Computational Physics
Solving a Nonlinear Problem in Magneto-Rheological Fluids Using the Immersed Interface Method
Journal of Scientific Computing
Convergence of the ghost fluid method for elliptic equations with interfaces
Mathematics of Computation
A fast solver for the Stokes equations with distributed forces in complex geometries
Journal of Computational Physics
Journal of Computational Physics
A numerical method for solving variable coefficient elliptic equation with interfaces
Journal of Computational Physics
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
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
Journal of Computational Physics
A coupling interface method for elliptic interface problems
Journal of Computational Physics
Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities
Journal of Computational Physics
A kernel-free boundary integral method for elliptic boundary value problems
Journal of Computational Physics
Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems
Journal of Computational Physics
Journal of Computational Physics
A well-conditioned augmented system for solving Navier-Stokes equations in irregular domains
Journal of Computational Physics
Journal of Computational and Applied Mathematics
A sharp interface finite volume method for elliptic equations on Cartesian grids
Journal of Computational Physics
A low numerical dissipation immersed interface method for the compressible Navier-Stokes equations
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
An interpolation matched interface and boundary method for elliptic interface problems
Journal of Computational and Applied Mathematics
A second order virtual node method for elliptic problems with interfaces and irregular domains
Journal of Computational Physics
Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Stochastic Galerkin methods for elliptic interface problems with random input
Journal of Computational and Applied Mathematics
An Analysis of a Broken $P_1$-Nonconforming Finite Element Method for Interface Problems
SIAM Journal on Numerical Analysis
Augmented strategies for interface and irregular domain problems
NAA'04 Proceedings of the Third international conference on Numerical Analysis and its Applications
Journal of Computational Physics
A kernel-free boundary integral method for implicitly defined surfaces
Journal of Computational Physics
Journal of Computational and Applied Mathematics
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A fast, second-order accurate iterative method is proposed for the elliptic equation \[ \grad\cdot(\beta(x,y) \grad u) =f(x,y) \] in a rectangular region $\Omega$ in two-space dimensions. We assume that there is an irregular interface across which the coefficient $\beta$, the solution u and its derivatives, and/or the source term f may have jumps. We are especially interested in the cases where the coefficients $\beta$ are piecewise constant and the jump in $\beta$ is large. The interface may or may not align with an underlying Cartesian grid. The idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by LeVeque and Li [ SIAM J. Numer. Anal., 4 (1994), pp. 1019--1044]. In order to take advantage of fast Poisson solvers on a rectangular region, an intermediate unknown function, the jump in the normal derivative across the interface, is introduced. Our discretization is equivalent to using a second-order difference scheme for a corresponding Poisson equation in the region, and a second-order discretization for a Neumann-like interface condition. Thus second-order accuracy is guaranteed. A GMRES iteration is employed to solve the Schur complement system derived from the discretization. A new weighted least squares method is also proposed to approximate interface quantities from a grid function. Numerical experiments are provided and analyzed. The number of iterations in solving the Schur complement system appears to be independent of both the jump in the coefficient and the mesh size.