Improved difference schemes for the Dirichlet problem of Poisson's equation in the neighbourhood of corners

1978 ◽  
Vol 30 (3) ◽  
pp. 315-332 ◽  
Author(s):  
C. Zenger ◽  
H. Gietl
Keyword(s):  
Dirichlet Problem ◽  
Difference Schemes ◽  
Poisson’S Equation ◽  
Poisson's Equation

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.

On the computational complexity of the Dirichlet Problem for Poisson's Equation

2016 ◽  
Vol 27 (8) ◽  
pp. 1437-1465 ◽  
Author(s):  
AKITOSHI KAWAMURA ◽  
FLORIAN STEINBERG ◽  
MARTIN ZIEGLER
Keyword(s):  
Dirichlet Problem ◽  
Computational Complexity ◽  
Complexity Theory ◽  
Complexity Class ◽  
Poisson’S Equation ◽  
Poisson's Equation ◽  
Precise Sense ◽  
Real Functions ◽  
Riemann Integration

The last years have seen an increasing interest in classifying (existence claims in) classical mathematical theorems according to their strength. We pursue this goal from the refined perspective of computational complexity. Specifically, we establish that rigorously solving the Dirichlet Problem for Poisson's Equation is in a precise sense ‘complete’ for the complexity class ${\#\mathcal{P}}$ and thus as hard or easy as parametric Riemann integration (Friedman 1984; Ko 1991. Complexity Theory of Real Functions).

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