Marching cubes: A high resolution 3D surface construction algorithm
SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques
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
Journal of Computational Physics
A boundary condition capturing method for Poisson's equation on irregular domains
Journal of Computational Physics
The blob projection method for immersed boundary problems
Journal of Computational Physics
Maximum Principle Preserving Schemes for Interface Problems with Discontinuous Coefficients
SIAM Journal on Scientific Computing
hypre: A Library of High Performance Preconditioners
ICCS '02 Proceedings of the International Conference on Computational Science-Part III
Three-dimensional elliptic solvers for interface problems and applications
Journal of Computational Physics
Convergence of the ghost fluid method for elliptic equations with interfaces
Mathematics of Computation
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
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)
An extended pressure finite element space for two-phase incompressible flows with surface tension
Journal of Computational Physics
Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities
Journal of Computational Physics
A fourth-order auxiliary variable projection method for zero-Mach number gas dynamics
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
Journal of Computational Physics
An Analysis of a Broken $P_1$-Nonconforming Finite Element Method for Interface Problems
SIAM Journal on Numerical Analysis
Journal of Computational Physics
An improved immersed boundary method with direct forcing for the simulation of particle laden flows
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
A weak formulation for solving elliptic interface problems without body fitted grid
Journal of Computational Physics
An asymptotic solution approach for elliptic equations with discontinuous coefficients
Journal of Computational Physics
Hi-index | 31.50 |
We present a second order sharp interface finite volume method for the solution of the three-dimensional elliptic equation @?.(@b(x-)@?u(x-))=f(x-) with variable coefficients on Cartesian grids. In particular, we focus on interface problems with discontinuities in the coefficient, the source term, the solution, and the fluxes across the interface. The method uses standard piecewise trilinear finite elements for normal cells and a double piecewise trilinear ansatz for the solution on cells intersected by the interface resulting always in a compact 27-point stencil. Singularities associated with vanishing partial volumes of intersected grid cells are removed by a two-term asymptotic approach. In contrast to the 2D method presented by two of the authors in [M. Oevermann, R. Klein, A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces, Journal of Computational Physics 219 (2006) 749-769] we use a minimization technique to determine the unknown coefficients of the double trilinear ansatz. This simplifies the treatment of the different cut-cell types and avoids additional special operations for degenerated interface topologies. The resulting set of linear equations has been solved with a BiCGSTAB solver preconditioned with an algebraic multigrid. In various testcases - including large @b-ratios and non-smooth interfaces - the method achieves second order of accuracy in the L"~ and L"2 norm.