SG_Dee: Chebyshev sparse grid method for elliptic equations (Dirichlet)

Purpose

• "SG_Dee" provides an example of using the spectral sparge grid to solve elliptic equations with the homogeneous Dirichlet boundary condition in high dimensional cubes. The equation is discretized with a Chebyshev-Galerkin method, i.e., the sparse grid is in the frequency space (hyperbolic cross). The resultant linear system is solved with the bicgstab method.

Specifications

• Name: SG_Dee.
• Author: Feng Chen.
• Finishing date: 08/11/2009.
• Language: Fortran 90.
• Required libraries: Template.

Simple Example

• Equation: \begin{equation} \begin{aligned} &\alpha u - \beta \Delta u = f, &&\quad \text{in } \Omega, \\ &u = 0, && \quad \text{on } \partial \Omega. \end{aligned} \end{equation}
• Parameters: \begin{equation} \Omega = (-1,1)^d,\quad d=5, \quad \alpha =2, \quad \beta=3. \end{equation}
• Exact solution and input functions: \begin{equation} u(\boldsymbol{x}) = \Pi_{i=1}^d (e^{\cos\frac{\pi x_i}{2}}-1), \end{equation} then $f(\boldsymbol{x})$ is calculated accordingly.

Quick Start

• Compiling and running:

  cd ./SG_Dee
  make library
  gfortran tst_SG_Dee.f90 -llibrary -ltemplate
  ./a.out
  

• Output:

   
   dimension =           5
   dof =       61183
   max dof in 1-D =         255
   the max error is:  7.64327293492073068E-004
  ...
  

• CPU: Intel(R) Xeon(R) CPU X5550 @2.67GHz.
• OS: CentOS release 6.4 (Final).
• Compiler: gfortran 4.5.1.

References

• Jie Shen and Haijun Yu. Efficient spectral sparse grid methods and applications to high-dimensional elliptic problems, SIAM Journal on Scientific Computing, (2010).

Code Highlights

! scheme_phi_cheb2 transforms bases like phi_{k} = T_{k+2} - T_{k}
! to bases in Cheb2 Structure, in the 1 dimension case.
! -----------------------------------------------------------------
  subroutine scheme_phi_cheb2(IS, PanelLv, nop, cin, cout)
    type(InterpScheme), intent(in) :: IS
    integer, intent(in) :: PanelLv, nop
    real(dp), intent(in), dimension(:) :: cin
    real(dp), intent(out), dimension(:) :: cout
    real(dp), dimension(:), allocatable :: cwork
    integer :: i, ks, ke, PanelLen
    PanelLen = IS%m(PanelLv)
    allocate(cwork(PanelLen+2))

    do i = 1, nop
       ks = (i-1)*PanelLen + 1
       ke = i*PanelLen
       cwork(1:PanelLen) = cin(ks:ke)
       cwork(PanelLen+1:PanelLen+2) = 0d0
       cout(ks:ke) = cwork(3:PanelLen+2) - cin(ks:ke)
    end do

    deallocate(cwork)
  end subroutine scheme_phi_cheb2