Localized singular boundary method for the simulation of large-scale problems of elliptic operators in complex geometries

2022 ◽  
Vol 105 ◽  
pp. 94-106
Author(s):  
Ji Lin ◽  
Lin Qiu ◽  
Fajie Wang
Keyword(s):  
Large Scale ◽  
Elliptic Operators ◽  
Complex Geometries ◽  
Singular Boundary ◽  
Large Scale Problems ◽  
Boundary Method ◽  
Singular Boundary Method

The Local Singular Boundary Method for Solution of Two-Dimensional Advection–Diffusion Equation

2021 ◽  
pp. 2150041
Author(s):  
Karel Kovářík ◽  
Juraj Mužík
Keyword(s):  
Diffusion Equation ◽  
Large Scale ◽  
Sparse Matrix ◽  
Singular Boundary ◽  
Tracer Transport ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Boundary Method ◽  
Local Variant ◽  
Singular Boundary Method

This work focuses on the derivation of the local variant of the singular boundary method (SBM) for solving the advection-diffusion equation of tracer transport. Localization is based on the combination of SBM and finite collocation. Unlike the global variant, local SBM leads to a sparse matrix of the resulting system of equations, making it much more efficient to solve large-scale tasks. It also allows solving velocity vector variable tasks, which is a problem with global SBM. This paper compares the results on several examples for the steady and unsteady variant of the advection-diffusion equation and also examines the dependence of the accuracy of the solution on the density of the nodal grid and the size of the subdomain.

Precorrected-FFT Accelerated Singular Boundary Method for Large-Scale Three-Dimensional Potential Problems

2017 ◽  
Vol 22 (2) ◽  
pp. 460-472 ◽  
Author(s):  
Weiwei Li ◽  
Wen Chen ◽  
Zhuojia Fu
Keyword(s):  
Computational Complexity ◽  
Large Scale ◽  
Three Dimensional ◽  
Iteration Step ◽  
Singular Boundary ◽  
Matrix Vector Multiplication ◽  
Boundary Method ◽  
Potential Problems ◽  
Speed Up ◽  
Singular Boundary Method

AbstractThis study makes the first attempt to accelerate the singular boundary method (SBM) by the precorrected-FFT (PFFT) for large-scale three-dimensional potential problems. The SBM with the GMRES solver requires computational complexity, where N is the number of the unknowns. To speed up the SBM, the PFFT is employed to accelerate the SBM matrix-vector multiplication at each iteration step of the GMRES. Consequently, the computational complexity can be reduced to . Several numerical examples are presented to validate the developed PFFT accelerated SBM (PFFT-SBM) scheme, and the results are compared with those of the SBM without the PFFT and the analytical solutions. It is clearly found that the present PFFT-SBM is very efficient and suitable for 3D large-scale potential problems.

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