The numerical solution of second-order boundary value problems on nonuniform meshes
Mathematics of Computation
Supra-convergent schemes on irregular grids
Mathematics of Computation
Applications of spatial data structures: Computer graphics, image processing, and GIS
Applications of spatial data structures: Computer graphics, image processing, and GIS
The design and analysis of spatial data structures
The design and analysis of spatial data structures
Journal of Computational Physics
A fast adaptive vortex method in three dimensions
Journal of Computational Physics
SIAM Journal on Numerical Analysis
A fast Poisson solver for complex geometries
Journal of Computational Physics
Computation of three dimensional dendrites with finite elements
Journal of Computational Physics
A simple level set method for solving Stefan problems
Journal of Computational Physics
Journal of Computational Physics
A Cartesian grid embedded boundary method for Poisson's equation on irregular domains
Journal of Computational Physics
Tree methods for moving interfaces
Journal of Computational Physics
A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)
Journal of Computational Physics
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
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
A Level Set Approach for the Numerical Simulation of Dendritic Growth
Journal of Scientific Computing
Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries
Journal of Computational Physics
Simulating water and smoke with an octree data structure
ACM SIGGRAPH 2004 Papers
Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes
Journal of Computational Physics
A node-centered local refinement algorithm for Poisson's equation in complex geometries
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Journal of Scientific Computing
A Multigrid Method on Non-Graded Adaptive Octree and Quadtree Cartesian Grids
Journal of Scientific Computing
An adaptive discretization of incompressible flow using a multitude of moving Cartesian grids
Journal of Computational Physics
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We present finite difference schemes for solving the variable coefficient Poisson and heat equations on irregular domains with Dirichlet boundary conditions. The computational domain is discretized with non-graded Cartesian grids, i.e., grids for which the difference in size between two adjacent cells is not constrained. Refinement criteria is based on proximity to the irregular interface such that cells with the finest resolution is placed on the interface. We sample the solution at the cell vertices (nodes) and use quadtree (in 2D) or octree (in 3D) data structures as efficient means to represent the grids. The boundary of the irregular domain is represented by the zero level set of a signed distance function. For cells cut by the interface, the location of the intersection point is found by a quadratic fitting of the signed distance function, and the Dirichlet boundary value is obtained by quadratic interpolation. Instead of using ghost nodes outside the interface, we use directly this intersection point in the discretization of the variable coefficient Laplacian. These methods can be applied in a dimension-by-dimension fashion, producing schemes that are straightforward to implement. Our method combines the ability of adaptivity on quadtrees/octrees with a quadratic treatment of the Dirichlet boundary condition on the interface. Numerical results in two and three spatial dimensions demonstrate second-order accuracy for both the solution and its gradients in the L 1 and L 驴 norms.