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

Abstract

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 adaptive mesh refinement (AMR) on Cartesian grids, and the AMR multigrid solver of Almgren. The discrete Laplacian operator at internal boundaries comes from either linear or quadratic (Shortley-Weller) extrapolation, and the two methods are compared. It is shown that either way, solution error is second order in the mesh spacing. Error in the gradient of the solution is first order with linear extrapolation, but second order with Shortley-Weller. Examples are given with comparison with the exact solution. The method is also applied to a heavy-ion fusion accelerator problem, showing the advantage of adaptivity.

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

“…We refer the interested reader to Ng et al [95], which concluded that a linear interpolation produces second-order accurate solutions and first-order accurate gradients, while a quadratic extrapolation produces second-order accurate solutions and secondorder accurate gradients. This was first observed in [85]. We note that the location of the interface must also be found using a quadratic interpolation of the level-set function in the vicinity of the interface if secondorder accurate gradients are to be calculated.…”

“…Numerical methods for a large class of partial differential equations have been introduced using this framework; see e.g. [14,138,85] and the references therein. More recently, quadtree and octree data structures have been preferred [3], since they allow the grid to be continuously refined without being bound by blocks of uniform grids.…”

A local corrections algorithm for solving Poisson’s equation in three dimensions

“…We believe that the performance of our implementation of the global FFT solver can be improved from that seen here. We compare these results with timings for an adaptive node-centered multigrid algorithm for solving Poisson's equation with Dirichlet boundary conditions [18] on the same platform. This algorithm is run on three levels of boxes, with the fine and middle levels being the same as were used with MLC, but the coarse level being fully refined into cubes of length 32, instead of parallel slabs.…”

A supra-convergent finite difference scheme for the variable coefficient Poisson equation on non-graded grids

“…[15] and the references therein), as well as in the case of irregular domains (see e.g. [10,11,14,17,18,23,24,25] and the references therein). However, many physical problems have variations in scale and when solving these problems numerically, uniform Cartesian grids are limited in their ability to resolve small scales.…”