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