Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
The rapid evaluation of volume integrals of potential theory on general regions
Journal of Computational Physics
SIAM Journal on Numerical Analysis
A fast Poisson solver for complex geometries
Journal of Computational Physics
A Fast Iterative Algorithm for Elliptic Interface Problems
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
A Cartesian grid embedded boundary method for the heat equation on irregular domains
Journal of Computational Physics
Journal of Computational Physics
A Method for Computing Nearly Singular Integrals
SIAM Journal on Numerical Analysis
Three-dimensional elliptic solvers for interface problems and applications
Journal of Computational Physics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Convergence of the ghost fluid method for elliptic equations with interfaces
Mathematics of Computation
A Grid-Based Boundary Integral Method for Elliptic Problems in Three Dimensions
SIAM Journal on Numerical Analysis
A kernel-independent adaptive fast multipole algorithm in two and three dimensions
Journal of Computational Physics
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)
Fourth order accurate evaluation of integrals in potential theory on exterior 3D regions
Journal of Computational Physics
A kernel-free boundary integral method for elliptic boundary value problems
Journal of Computational Physics
A grid based particle method for moving interface problems
Journal of Computational Physics
Hi-index | 31.45 |
The kernel-free boundary integral (KFBI) method is a structured grid method for general elliptic partial differential equations. Unlike the standard boundary integral method, it avoids direct evaluation of volume and boundary integrals, which needs to know analytical expressions for the integral kernels. To evaluate a boundary or volume integral, the KFBI method first solves a corrected interface problem on a structured grid and then the numerical solution on the structured grid is interpolated to get approximate values of the integral at points on the boundary. Selection of control points of the boundary plays a key role in the KFBI method since both the correction for the interface equations and the interpolation with the structured grid based solution involve calculation of tangential derivatives of boundary data while stability and efficiency of the numerical differentiation critically depend on the distribution of control points. This work proposes a new point selection method, based on an overlapping surface decomposition of the boundary, which is implicitly defined by a level set function. The points selected are intersection points of the boundary with the grid lines of an underlying Cartesian grid. By the method, the interpolation stencils can be easily chosen to be locally uniform along a coordinate axis in two space dimensions and locally uniform on a coordinate plane in three space dimensions, which allows efficient numerical differentiation and boundary reconstruction/representation. An additional equilibrating process of boundary data further guarantees stable numerical differentiation. Numerical results demonstrating the method with examples in both two and three space dimensions are presented.