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2D Poisson Multigrid Solver

2D Poisson Equation Problem


with Dirichlet Boundary Condition


The exact solution is


The residual is


Exact Solver on Coarsest Level

The coarsest grid should be solved to exact solution
The program provides two solvers

  1. Gauss-Seidel with residual L2-norm less than tolerance
  2. LU decomposition
  3. CG Solver
tol = 1.0d-6
exact_solver = 1

Set Level

Set arbitrary level. The coarsest grid should greater than 3

if (Level_Nx(Level_num).le.3) then
    write(*,*) Level_num," level is high for ",Nx," grid "
    stop
end if

Set Multigrid Solver

Gauss-Seidel scheme is selected to be smoother

V Cycle Scheme

subroutine MG_Vcycle(Nx,Ny,dx,dy,RHS,U,Level_num,tol,exact_solver,I_cycle)
    implicit none
    integer,intent(in) :: Nx,Ny,Level_num,exact_solver
    real*8 ,intent(in) :: dx,dy,tol
    real*8,dimension(0:Nx,0:Ny), intent(in)    :: RHS
    real*8,dimension(0:Nx,0:Ny), intent(inout) :: U
    integer,dimension(Level_num) :: Level_Nx,Level_Ny
    real*8 ,dimension(Level_num) :: Level_dx,Level_dy
    real*8  :: rms0,rms,rmsc
    integer :: Max_Iter,Relax_Iter
    integer :: i,j,k,Level,iter
    integer, intent(inout) :: I_cycle

Full Cycle Scheme

subroutine MG_Fcycle(Nx,Ny,dx,dy,RHS,U,Level_num,tol,exact_solver,I_cycle)
    implicit none
    integer,intent(in) :: Nx,Ny,Level_num,exact_solver
    real*8 ,intent(in) :: dx,dy,tol
    real*8,dimension(0:Nx,0:Ny), intent(in)    :: RHS
    real*8,dimension(0:Nx,0:Ny), intent(inout) :: U
    integer,dimension(Level_num) :: Level_Nx,Level_Ny
    real*8 ,dimension(Level_num) :: Level_dx,Level_dy
    real*8  :: rms0,rms,rmsc
    integer :: Max_Iter,Relax_Iter
    integer :: i,j,k,Level,iter,inner_cycle
    integer, intent(out) :: I_cycle

Multigrid V cycle Result

The following tables are the result of Multigrid V cycle with 5 and 6 levels
The P- is parallel version implemented by OpenMP (Use 8 threads for test)

Solver Np Cycle CPU Time
MG5 1024 6 0.8668
P-MG5 1024 6 0.2620
MG6 1024 6 0.3376
P-MG6 1024 6 0.1236
Solver Np Cycle CPU Time
MG5 2048 6 9.7714
P-MG5 2048 6 1.9416
MG6 2048 6 1.7439
P-MG6 2048 6 0.6016
MG7 2048 6 1.2188
P-MG7 2048 6 0.4571
Solver Np Cycle CPU Time
MG5 4096 5 123.5220
P-MG5 4096 6 21.8231
MG6 4096 6 13.2918
P-MG6 4096 6 3.2305
MG7 4096 6 5.2741
P-MG7 4096 6 1.8986
Solver Np Cycle CPU Time
MG6 8192 5 135.0223
P-MG6 8192 6 26.9059
MG7 8192 6 27.4078
P-MG7 8192 6 8.2924

Compile and Run

Use PGI Compiler

make

Run

./Run

Reference

See wikipedia