A Neumann–Neumann preconditioned iterative substructuring approach for computing solutions to Poisson’s equation with prescribed jumps on an embedded boundary

2013 ◽  
Vol 235 ◽  
pp. 683-700 ◽  
Author(s):  
Gregory H. Miller ◽  
Elbridge Gerry Puckett
Keyword(s):  
Poisson’S Equation ◽  
Poisson's Equation ◽  
Iterative Substructuring ◽  
Embedded Boundary ◽  
Preconditioned Iterative

scholarly journals Grid independent convergence using multilevel circulant preconditioning: Poisson’s equation

2021 ◽  
Author(s):  
Henrik Brandén
Keyword(s):  
Boundary Integral Equations ◽  
Linear Equations ◽  
Iterative Solution ◽  
Boundary Integral ◽  
Poisson’S Equation ◽  
Dirichlet Boundary Conditions ◽  
Poisson's Equation ◽  
Spatial Dimensions ◽  
Preconditioned Iterative ◽  
Circulant Preconditioning

AbstractWe consider the iterative solution of the discrete Poisson’s equation with Dirichlet boundary conditions. The discrete domain is embedded into an extended domain and the resulting system of linear equations is solved using a fixed point iteration combined with a multilevel circulant preconditioner. Our numerical results show that the rate of convergence is independent of the grid’s step sizes and of the number of spatial dimensions, despite the fact that the iteration operator is not bounded as the grid is refined. The embedding technique and the preconditioner is derived with inspiration from theory of boundary integral equations. The same theory is used to explain the behaviour of the preconditioned iterative method.

scholarly journals A Family of Sixth-Order Compact Finite-Difference Schemes for the Three-Dimensional Poisson Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Yaw Kyei ◽  
John Paul Roop ◽  
Guoqing Tang
Keyword(s):  
Finite Difference ◽  
High Performance ◽  
Three Dimensional ◽  
Difference Schemes ◽  
Poisson’S Equation ◽  
Sixth Order ◽  
Finite Difference Schemes ◽  
Poisson's Equation ◽  
Compact Finite Difference ◽  
Compact Finite Difference Schemes

We derive a family of sixth-order compact finite-difference schemes for the three-dimensional Poisson's equation. As opposed to other research regarding higher-order compact difference schemes, our approach includes consideration of the discretization of the source function on a compact finite-difference stencil. The schemes derived approximate the solution to Poisson's equation on a compact stencil, and thus the schemes can be easily implemented and resulting linear systems are solved in a high-performance computing environment. The resulting discretization is a one-parameter family of finite-difference schemes which may be further optimized for accuracy and stability. Computational experiments are implemented which illustrate the theoretically demonstrated truncation errors.

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