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

We introduce a method for solving the variable coefficient Poisson equation on non-graded Cartesian grids that yields second order accuracy for the solutions and their gradients. We employ quadtree (in 2D) and octree (in 3D) data structures as an efficient means to represent the Cartesian grid, allowing for constraint-free grid generation. The schemes take advantage of sampling the solution at the nodes (vertices) of each cell. In particular, the discretization at one cell's node only uses nodes of two (2D) or three (3D) adjacent cells, producing schemes that are straightforward to implement. Numerical results in two and three spatial dimensions demonstrate supra-convergence in the L∞ norm.

[1]  P. Colella,et al.  A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier-Stokes Equations , 1998 .

[2]  P. Colella,et al.  An Adaptive Level Set Approach for Incompressible Two-Phase Flows , 1997 .

[3]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[4]  Ronald Fedkiw,et al.  Simulating water and smoke with an octree data structure , 2004, ACM Trans. Graph..

[5]  Zhilin Li.  A Fast Iterative Algorithm for Elliptic Interface Problems , 1998 .

[6]  P. Colella,et al.  A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains , 1998 .

[7]  Ann S. Almgren,et al.  An adaptive level set approach for incompressible two-phase flows , 1997 .

[8]  Michael J. Aftosmis,et al.  Adaptive Cartesian mesh generation , 1998 .

[9]  Héctor D. Ceniceros,et al.  Study of the long-time dynamics of a viscous vortex sheet with a fully adaptive nonstiff method , 2004 .

[10]  R. LeVeque,et al.  A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources , 2006 .

[11]  M. Berger,et al.  Adaptive mesh refinement for hyperbolic partial differential equations , 1982 .

[12]  Hanan Samet,et al.  Applications of spatial data structures - computer graphics, image processing, and GIS , 1990 .

[13]  P. Colella,et al.  A Fast Adaptive Vortex Method in Three Dimensions , 1994 .

[14]  Mikhail Shashkov,et al.  Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes , 2004 .

[15]  A. Mayo.  The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions , 1984 .

[16]  S. Popinet.  Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries , 2003 .

[17]  L. Greengard,et al.  A Fast Poisson Solver for Complex Geometries , 1995 .

[18]  Gene H. Golub,et al.  Matrix computations , 1983 .

[19]  A. Chorin.  A Numerical Method for Solving Incompressible Viscous Flow Problems , 1997 .

[20]  P. Colella,et al.  A node-centered local refinement algorithm for Poisson's equation in complex geometries , 2004 .

[21]  W. Cheney,et al.  Numerical Analysis: Mathematics of Scientific Computing , 1991 .

[22]  R. Fedkiw,et al.  A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem , 2005 .

[23]  A. B. Strong,et al.  A Fully Conservative Second-Order Finite Difference Scheme for Incompressible Flow on Nonuniform Grids , 2002 .

[24]  P. G. Ciarlet,et al.  Introduction to Numerical Linear Algebra and Optimisation , 1989 .

[25]  Ann S. Almgren,et al.  A fast adaptive vortex method using local corrections , 1991 .

[26]  Andrew B. White,et al.  Supra-convergent schemes on irregular grids , 1986 .

[27]  Thomas A. Manteuffel,et al.  The numerical solution of second-order boundary value problems on nonuniform meshes , 1986 .

[28]  D. P. Young,et al.  A locally refined rectangular grid finite element method: application to computational fluid dynamics and computational physics , 1990 .

[29]  Li-Tien Cheng,et al.  A second-order-accurate symmetric discretization of the Poisson equation on irregular domains , 2002 .

[30]  Hanan Samet,et al.  The Design and Analysis of Spatial Data Structures , 1989 .

[31]  P. Colella,et al.  Local adaptive mesh refinement for shock hydrodynamics , 1989 .

[32]  S. Osher,et al.  Spatially adaptive techniques for level set methods and incompressible flow , 2006 .