Improved volume conservation in the computation of flows with immersed elastic boundaries
Journal of Computational Physics
A level set approach for computing solutions to incompressible two-phase flow
Journal of Computational Physics
SIAM Journal on Numerical Analysis
A Cartesian grid embedded boundary method for Poisson's equation on irregular domains
Journal of Computational Physics
A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)
Journal of Computational Physics
SIAM Journal on Numerical Analysis
A boundary condition capturing method for Poisson's equation on irregular domains
Journal of Computational Physics
Convergence of the ghost fluid method for elliptic equations with interfaces
Mathematics of Computation
A Boundary Condition--Capturing Multigrid Approach to Irregular Boundary Problems
SIAM Journal on Scientific Computing
A numerical method for solving variable coefficient elliptic equation with interfaces
Journal of Computational Physics
Journal of Computational Physics
The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains (Frontiers in Applied Mathematics)
Journal of Computational Physics
Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces
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
SIAM Journal on Numerical Analysis
A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth
Journal of Scientific Computing
The adaptive immersed interface finite element method for elliptic and Maxwell interface problems
Journal of Computational Physics
A sharp interface finite volume method for elliptic equations on Cartesian grids
Journal of Computational Physics
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
Hi-index | 31.45 |
A typical elliptic interface problem is casted as piecewise defined elliptic partial differential equations (PDE) in different regions which are coupled together with interface conditions, such as jumps in solution and flux across the interface. In many situations, such as the interface is moving, the challenge is how to solve such a problem accurately, robustly and efficiently without generating a body fitted mesh. The key issue is how to capture complex geometry of the interface and jump conditions across the interface effectively on a fixed mesh while the interface is not aligned with the mesh and the PDE is not valid across the interface. In this work we present a systematic formulation and further study of a second order accurate numerical method proposed in Hou and Liu (2005) [16] for elliptic interface problem. The key idea is to decompose the solution into two parts, a singular part and a regular part. The singular part captures the interface conditions while the regular part belongs to an appropriate space in the whole domain, which can be solved by a standard finite element formulation. In a general setup the two parts are coupled together. We give an explicit study of the construction of the singular part and the discretized system for the regular part. One key advantage of using weak formulation is that one can avoid assuming/using more regularity than necessary of the solution and the interface. We present the numerical algorithm and numerical tests in 3D to demonstrate the accuracy and other properties of our method.