GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Fast parallel iterative solution of Poisson's and the biharmonic equations on irregular regions
SIAM Journal on Scientific and Statistical Computing - Special issue on iterative methods in numerical linear algebra
The rapid evaluation of volume integrals of potential theory on general regions
Journal of Computational Physics
On the numerical solution of the biharmonic equation in the plane
Proceedings of the eleventh annual international conference of the Center for Nonlinear Studies on Experimental mathematics : computational issues in nonlinear science: computational issues in nonlinear science
Laplace's equation and the Dirichlet-Neumann map in multiply connected domains
Journal of Computational Physics
SIAM Journal on Numerical Analysis
A fast Poisson solver for complex geometries
Journal of Computational Physics
Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension
SIAM Journal on Scientific Computing
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
An immersed boundary method with formal second-order accuracy and reduced numerical viscosity
Journal of Computational Physics
The blob projection method for immersed boundary problems
Journal of Computational Physics
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. VII: partial differential equations
The immersed interface method for the Navier-Stokes equations with singular forces
Journal of Computational Physics
A Cartesian grid embedded boundary method for the heat equation on irregular domains
Journal of Computational Physics
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
Journal of Computational Physics
Maximum Principle Preserving Schemes for Interface Problems with Discontinuous Coefficients
SIAM Journal on Scientific Computing
The Immersed Interface/Multigrid Methods for Interface Problems
SIAM Journal on Scientific Computing
Immersed Interface Methods for Neumann and Related Problems in Two and Three Dimensions
SIAM Journal on Scientific Computing
A Method for Computing Nearly Singular Integrals
SIAM Journal on Numerical Analysis
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Regularization Techniques for Numerical Approximation of PDEs with Singularities
Journal of Scientific Computing
New Formulations for Interface Problems in Polar Coordinates
SIAM Journal on Scientific Computing
Convergence of the ghost fluid method for elliptic equations with interfaces
Mathematics of Computation
SIAM Journal on Scientific Computing
A ghost-cell immersed boundary method for flow in complex geometry
Journal of Computational Physics
New Geometric Immersed Interface Multigrid Solvers
SIAM Journal on Scientific Computing
A Grid-Based Boundary Integral Method for Elliptic Problems in Three Dimensions
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Numerical approximations of singular source terms in differential equations
Journal of Computational Physics
A numerical method for solving variable coefficient elliptic equation with interfaces
Journal of Computational Physics
Journal of Computational Physics
Discretization of Dirac delta functions in level set methods
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
Meshfree Particle Methods
Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces
Journal of Computational Physics
A weak formulation for solving elliptic interface problems without body fitted grid
Journal of Computational Physics
A kernel-free boundary integral method for implicitly defined surfaces
Journal of Computational Physics
Hi-index | 31.46 |
This paper presents a class of kernel-free boundary integral (KFBI) methods for general elliptic boundary value problems (BVPs). The boundary integral equations reformulated from the BVPs are solved iteratively with the GMRES method. During the iteration, the boundary and volume integrals involving Green's functions are approximated by structured grid-based numerical solutions, which avoids the need to know the analytical expressions of Green's functions. The KFBI method assumes that the larger regular domain, which embeds the original complex domain, can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast Fourier transform (FFT) based Poisson/Helmholtz solvers or those based on geometric multigrid iterations are applicable. The structured grid-based solutions are obtained with standard finite difference method (FDM) or finite element method (FEM), where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme. Numerical results demonstrating the efficiency and accuracy of the KFBI methods are presented. It is observed that the number of GMRES iterations used by the method for solving isotropic and moderately anisotropic BVPs is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals. With the standard second-order FEMs and FDMs, the KFBI method shows a second-order convergence rate in accuracy for all of the tested Dirichlet/Neumann BVPs when the anisotropy of the diffusion tensor is not too strong.