scholarly journals Bounds for Derivatives in the Dirichlet Problem for Poisson’s Equation

1962 ◽  
Vol 10 (2) ◽  
pp. 370-380 ◽  
Author(s):  
J. H. Bramble ◽  
L. E. Payne
Keyword(s):  
Dirichlet Problem ◽  
Poisson’S Equation ◽  
Poisson's Equation

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).

scholarly journals Domain decomposition based on a direct method for solving the three-dimensional Poisson's equation in nonstationary astrophysical problems

2015 ◽  
pp. 94-98
Author(s):  
Н.В. Снытников
Keyword(s):  
Dirichlet Problem ◽  
Parallel Algorithm ◽  
Direct Method ◽  
Three Dimensional ◽  
Poisson’S Equation ◽  
Variable Separation ◽  
Poisson's Equation ◽  
Academy Of Sciences ◽  
Variable Separation Method ◽  
A Direct Method

Предложен новый параллельный алгоритм для решения трехмерного уравнения Пуассона в контексте нестационарных задач астрофизики. Алгоритм основан на декомпозиции трехмерной области по двум направлениям, в применении прямого метода решения задачи Дирихле в каждой подобласти и в комбинации метода сопряжения подобластей для двумерного экранированного уравнения Пуассона с методом разделения переменных. Тестовые эксперименты проводились на суперкомпьютерах Межведомственного суперкомпьютерного центра (МСКЦ) и Сибирского суперкомпьютерного центра (ССКЦ). A new parallel algorithm for solving the three-dimensional Poisson's equation in the context of nonstationary problems of astrophysics is proposed. This algorithm is based on a decomposition of the 3D domain in two directions, on the application of a direct method for solving the Dirichlet problem in each subdomain, and on a combination of subdomains coupling for the screened Poisson's equation with the variable separation method. Test experiments were conducted on supercomputers installed at the Joint Supercomputing Center of Russian Academy of Sciences (Moscow) and at the Siberian Supercomputing Center (Novosibirsk).

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