LEGENDRE: Spectral methods using Legendre polynomials

Purpose

• The library "Legendre" provides spectral methods using Legendre polynomials. It includes core subroutines and test functions for several variants:
  • standard collocation method;
  • optimal collocation method;
  • standard Galerkin method;
  • optimal Galerkin method;
  • quasi-optimal Galerkin method;
  • solvers for 2-D and 3-D rectangular domains;
  • solvers for nonhomogeneous boundary conditions;
  • solvers for periodic-nonperiodic boundary conditions;
  • solvers for systems of coupled elliptic equations.

Specifications

• Name: Legendre.
• Author: Feng Chen.
• Version: 19.
• Language: Fortran 90.
• Required libraries: BLAS, LAPACK.

Simple Example

• Equation: \begin{equation} \begin{aligned} &\alpha u -(\beta_1 u_{xx} + \beta_2 u_{yy}) = f, &&\quad \text{in } \Omega, \\ &\partial_{\boldsymbol{n}} u = 0, && \quad \text{on } \partial \Omega. \end{aligned} \end{equation}
• Parameters: \begin{equation} \Omega = (-1,1)^2 , \quad \alpha =2, \quad \beta_1=3, \quad \beta_2=4. \end{equation}
• Exact solution and input functions: \begin{equation} u(x,y) = \cos(\pi x) \cos(\pi y). \end{equation} $f(x,y)$ is given according to $u(x,y)$.

Quick Start

• Compiling and running:

  cd ./Legendre
  make library
  gfortran tst_Legendre.f90 -llibrary -llapack -lblas
  ./a.out
  

• Output:

   
  Nx = Ny = 256, optimal Legendre-Galerkin method
  The computational time is  0.12000000     seconds.
  Solver error is  4.32986979603811051E-015
  Transform error is  6.45816733424453560E-013
  Diff-1 error is  4.91152708606358846E-009
  Diff-2 error is  4.87690250101431529E-009
  ...
  

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

References

• Jie Shen and Tao Tang. Spectral and high-order methods with applications, Science Press of China, (2006).
• Feng Chen and Jie Shen. Efficient spectral-Galerkin methods for systems of coupled second-order equations and their applications, Journal of Computational Physics, Volume 231, Issue 15, 5016-5028, (2012).

Core Subroutines

D1_E1_Legen_Col_Solve.f90
D1_E1_Legen_Gal_Opti_Solve.f90
D1_E1_Legen_Gal_Orth_RHSData.f90
D1_E1_Legen_Gal_Orth_Solve.f90
D1_E1_Legen_Gal_Quasi_Solve.f90
D1_EL_Join_Legen_Col_Solve.f90
D1_EL_Join_Legen_Gal_Orth_Solve.f90
D1_EL_Join_Legen_Gal_Orth_System.f90
D1_EL_Legen_Gal_Opti_Solve.f90
D1_EL_Legen_Gal_Opti_Solve_ND.f90
D1_EL_Legen_Gal_Orth_RHSData.f90
D1_EL_Legen_Gal_Quasi_Solve.f90
D1_EL_ND.f90
D1_EL_ND_Init.f90
D1_Legen_Gal_Opti_BasisCoefficients.f90
D1_Legen_Gal_Opti_BasisTransform.f90
D1_Legen_Gal_Opti_EigenData.f90
D1_Legen_Gal_Opti_Mass.f90
D1_Legen_Gal_Opti_RHS.f90
D1_Legen_Gal_Orth_EigenData.f90
D1_Legen_Gal_Orth_Matrices.f90
D1_Legen_Gal_Quasi_Init.f90
D2_E1_Fourier_Gal_Legen_Col_Solve.f90
D2_E1_Legen_Gal_Opti_InvData.f90
D2_E1_Legen_Gal_Opti_Solve.f90
D2_E1_Legen_Gal_Quasi_Solve.f90
D2_EL_Join_Fourier_Gal_Legen_Col_Solve.f90
D2_EL_Legen_Gal_Opti_Solve.f90
D2_EL_Legen_Gal_Opti_Solve_ND.f90
D2_EL_Legen_Gal_Quasi_Solve.f90
D2_EL_ND.f90
D2_EL_ND_Init.f90
D2_Legen_Gal_Opti_BasisTransform.f90
D2_Legen_Gal_Opti_RHS.f90
D3_E1_Legen_Gal_Opti_Solve.f90
D3_E1_Legen_Gal_Quasi_Solve.f90
D3_EL_Legen_Gal_Opti_Solve.f90
D3_EL_Legen_Gal_Opti_Solve_ND.f90
D3_EL_Legen_Gal_Quasi_Solve.f90
D3_EL_ND.f90
D3_EL_ND_Init.f90
D3_Legen_Gal_Opti_BasisTransform.f90
D3_Legen_Gal_Opti_RHS.f90