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
Improved volume conservation in the computation of flows with immersed elastic boundaries
Journal of Computational Physics
A fast Poisson solver for complex geometries
Journal of Computational Physics
The embedded curved boundary method for orthogonal simulation meshes
Journal of Computational Physics
A Cartesian grid embedded boundary method for Poisson's equation on irregular domains
Journal of Computational Physics
Computation of solid-liquid phase fronts in the sharp interface limit on fixed grids
Journal of Computational Physics
Superconvergence of the Shortley-Weller approximation for Dirichlet problems
Journal of Computational and Applied Mathematics
A boundary condition capturing method for Poisson's equation on irregular domains
Journal of Computational Physics
A second-order-accurate symmetric discretization of the Poisson equation on irregular domains
Journal of Computational Physics
The CASL algorithm for quasi-geostrophic flow in a cylinder
Journal of Computational Physics
Journal of Computational Physics
Journal of Scientific Computing
Journal of Computational Physics
A fast parallel Poisson solver on irregular domains applied to beam dynamics simulations
Journal of Computational Physics
A second order virtual node method for elliptic problems with interfaces and irregular domains
Journal of Computational Physics
The Immersed Interface Technique for Parabolic Problems with Mixed Boundary Conditions
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Hi-index | 31.47 |
The Poisson equation subject to Dirichlet boundary conditions on an irregular domain can be treated by embedding the region in a rectangular domain and solving using finite differences over the rectangle. The crucial issue is the discretization of the boundaries of the irregular domain. In the past, both linear and quadratic boundary treatments have been used and error bounds have been derived in both cases, showing that the linear case gives uniform second-order accuracy, whereas the quadratic case gives third-order accuracy at the boundaries and second-order accuracy internally. Thus, it has been recommended that the linear boundary treatment be used, as it is simpler, gives rise to a symmetric matrix formulation and has uniform accuracy. The present work shows that this argument is inadequate, because the coefficients of the error terms also play an important role. We demonstrate this in the 1-D case by determining explicit expressions for the error for both the linear and quadratic boundary treatments. It is shown that for the linear case the coefficient of error is in general large enough to dominate the calculation and that therefore it is necessary to use a quadratic boundary treatment in order to obtain errors comparable with those obtained for a regular domain. We go on to show that the 1-D expressions for error can be used to approximate the boundary error for 2-D problems, and that for the linear treatment, the boundary error again dominates.