# A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains

@article{Johansen1998ACG, title={A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains}, author={Hans Johansen and Phillip Colella}, journal={Journal of Computational Physics}, year={1998}, volume={147}, pages={60-85} }

We present a numerical method for solving Poisson's equation, with variable coefficients and Dirichlet boundary conditions, on two-dimensional regions. The approach uses a finite-volume discretization, which embeds the domain in a regular Cartesian grid. We treat the solution as a cell-centered quantity, even when those centers are outside the domain. Cells that contain a portion of the domain boundary use conservative differencing of second-order accurate fluxes on each cell volume. The… Expand

#### Figures and Tables from this paper

#### 440 Citations

A node-centered local refinement algorithm for Poisson's equation in complex geometries

- Mathematics
- 2004

This paper presents a method for solving Poisson's equation with Dirichlet boundary conditions on an irregular bounded three-dimensional region. The method uses a nodal-point discretization and… Expand

A Second Order Virtual Node Method for Poisson Interface Problems on Irregular Domains

- Mathematics
- 2009

We present a second order accurate, geometrically flexible and easy to implement method for solving the variable coe cient Poisson equation with interfacial discontinuities on an irregular domain. We… Expand

An Adaptive Cartesian Grid Embedded Boundary Method for the Incompressible Navier Stokes Equations in Complex Geometry

- Physics
- 2012

We present a second-order accurate projection method to solve the incompressible Navier-Stokes equations on irregular domains in two and three dimensions. We use a finite-volume discretization… Expand

A second order virtual node method for elliptic problems with interfaces and irregular domains

- Mathematics, Computer Science
- J. Comput. Phys.
- 2010

We present a second order accurate, geometrically flexible and easy to implement method for solving the variable coefficient Poisson equation with interfacial discontinuities or on irregular domains,… Expand

A finite-difference method for the variable coefficient Poisson equation on hierarchical Cartesian meshes

- Mathematics, Computer Science
- J. Comput. Phys.
- 2018

The main idea is to optimize the truncation error of the discretization scheme as a function of the local grid configuration to achieve second-order accuracy. Expand

An iterative boundary potential method for the infinite domain Poisson problem with interior Dirichlet boundaries

- Mathematics, Computer Science
- J. Comput. Phys.
- 2008

An iterative method is developed for the solution of Poisson's problem on an infinite domain in the presence of interior boundaries held at fixed potential, in three dimensions. The method combines… Expand

A Cartesian grid embedded boundary method for solving the Poisson and heat equations with discontinuous coefficients in three dimensions

- Mathematics, Computer Science
- J. Comput. Phys.
- 2011

Overall second order accuracy is achieved, as indicated by an array of tests using non-trivial interface geometries, and the ability of the method to solve problems with realistic physical parameters is demonstrated. Expand

A Cartesian grid embedded boundary method for the heat equation on irregular domains

- Mathematics
- 2001

We present an algorithm for solving the heat equation on irregular time-dependent domains. It is based on the Cartesian grid embedded boundary algorithm of Johansen and Colella (J. Comput. Phys.… Expand

A cartesian grid finite volume method for the solution of the Poisson equation with variable coefficients and embedded interfaces

- Mathematics
- 2006

We present a finite volume method for the solution of the two-dimensional elliptic equation @?.(@b(x)@?u(x))=f(x) with variable, discontinuous coefficients and solution discontinuities on irregular… Expand

A cut-cell finite element method for Poisson’s equation on arbitrary planar domains

- Mathematics
- 2021

Abstract This article introduces a cut-cell finite element method for Poisson’s equation on arbitrarily shaped two-dimensional domains. The equation is solved on a Cartesian axis-aligned grid of… Expand

#### References

SHOWING 1-10 OF 57 REFERENCES

Cartesian grid method for unsteady compressible flow in irregular regions

- Computer Science, Physics
- 1995

An adaptive Cartesian grid method for modeling time-dependent inviscid compressible flow in irregular regions using an unsplit second-order Godunov algorithm followed by a corrector applied to cells at the boundary. Expand

Adaptive mesh refinement for hyperbolic partial differential equations

- Mathematics
- 1982

We present an adaptive method based on the idea of multiple, component grids for the solution of hyperbolic partial differential equations using finite difference techniques. Based upon… Expand

A locally refined rectangular grid finite element method: application to computational fluid dynamics and computational physics

- Mathematics
- 1990

Abstract A new finite element method for solving important linear and nonlinear boundary value problems arising in computational physics is described in this paper. The method is designed to handle… Expand

A Cartesian Grid Projection Method for the Incompressible Euler Equations in Complex Geometries

- Mathematics, Computer Science
- SIAM J. Sci. Comput.
- 1997

A method for calculating time-dependent incompressible inviscid flow which combines a projection method with a "Cartesian grid" approach for representing geometry, in which the body is represented as an interface embedded in a regular Cartesian mesh. Expand

An adaptive projection method for the incompressible Euler equations

- Mathematics
- 1993

In this paper we present a method for solving the time-dependent incompressible Euler equations on an adaptive grid. The method is based on a projection formulation in which we first solve convection… Expand

A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier-Stokes Equations

- Mathematics
- 1998

In this paper we present a method for solving the equations governing time-dependent, variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids. The method is… Expand

An Adaptive Mesh Projection Method for Viscous Incompressible Flow

- Mathematics, Computer Science
- SIAM J. Sci. Comput.
- 1997

A fractional step version of Chorin's projection method for incompressible flow, with adaptive mesh refinement, which is second-order accurate in both space and time is presented. Expand

A Fast Poisson Solver for Complex Geometries

- Mathematics
- 1995

Robust fast solvers for the Poisson equation have generally been limited to regular geometries, where direct methods, based on Fourier analysis or cyclic reduction, and multigrid methods can be used.… Expand

A Multigrid Algorithm for Immersed Interface Problems

- Mathematics
- 1996

Many physical problems involve interior interfaces across which the coefficients in the problem, the solution, its derivatives, the flux, or the source term may have jumps. These interior interfaces… Expand

A Projection Method for Locally Refined Grids

- Mathematics
- 1996

A numerical method for the solution of the two-dimensional Euler equatons for incompressible flow on locally refined grids is presented. The method is a second-order Godunov-projection method adapted… Expand