RATE OF CONVERGENCE OF MIXED FINITE ELEMENT METHOD (RT0-P0) FOR POISSON EQUATION

This example is to show the rate of convergence of mixed finite element (RT0-P0) approximation of the Poisson equation on the unit square:

$- \Delta u = f \; \hbox{in } (0,1)^2$

for the following boundary conditions:

Pure Dirichlet boundary condition. $\Gamma _D = \partial \Omega$ .
Pure Neumann boundary condition. $\Gamma _N = \partial \Omega$ .
Mix Dirichlet and Neumann boundary condition. $u=g_D \hbox{ on }\Gamma_D, \quad \nabla u\cdot n=g_N \hbox{ on }\Gamma_N.$

Written by Ming Wang and improved by Long Chen.

Setting
Pure Neumann boundary condition.
Pure Dirichlet boundary condition.
Mix Dirichlet and Neumann boundary condition.
Conclusion

Setting

[node,elem] = squaremesh([0,1,0,1],0.25);
pde = sincosNeumanndata;
option.L0 = 1;
option.maxIt = 4;
option.printlevel = 1;
option.elemType = 'RT0';
option.refType = 'red';

Pure Neumann boundary condition.

option.solver = 'mg';

option.solver = 'tripremixpoisson';
bdFlag = setboundary(node,elem,'Neumann');
mfemPoisson(node,elem,pde,bdFlag,option);

Triangular Preconditioner Preconditioned GMRES 
#dof:      336,  #nnz:     1103, V-cycle:  1, iter: 12,   err = 9.64e-09,   time = 0.01 s
Triangular Preconditioner Preconditioned GMRES 
#dof:     1312,  #nnz:     4639, V-cycle:  1, iter: 13,   err = 5.74e-09,   time = 0.06 s
Triangular Preconditioner Preconditioned GMRES 
#dof:     5184,  #nnz:    19007, V-cycle:  1, iter: 14,   err = 2.97e-09,   time =  0.1 s
Triangular Preconditioner Preconditioned GMRES 
#dof:    20608,  #nnz:    76927, V-cycle:  1, iter: 14,   err = 7.88e-09,   time = 0.16 s
Table: Error
 #Dof       h       ||u-u_h||    ||u_I-u_h||  ||sigma-sigma_h||||sigma-sigma_h||_{div}
  336   1.25e-01   1.35519e-01   5.10491e-02   1.00885e+00   1.01710e+01
 1312   6.25e-02   6.55869e-02   1.07858e-02   5.03896e-01   5.14701e+00
 5184   3.12e-02   3.27265e-02   2.54075e-03   2.51860e-01   2.58126e+00
20608   1.56e-02   1.63622e-02   6.24929e-04   1.25918e-01   1.29160e+00

Table: CPU time
 #Dof   Assemble     Solve      Error      Mesh    
  336   0.00e+00   1.00e-02   0.00e+00   0.00e+00
 1312   1.00e-02   6.00e-02   1.00e-02   0.00e+00
 5184   2.00e-02   1.00e-01   2.00e-02   0.00e+00
20608   4.00e-02   1.60e-01   7.00e-02   1.00e-02

Pure Dirichlet boundary condition.

option.solver = 'mg';

option.solver = 'tripremixpoisson';
bdFlag = setboundary(node,elem,'Dirichlet');
mfemPoisson(node,elem,pde,bdFlag,option);

Triangular Preconditioner Preconditioned GMRES 
#dof:      336,  #nnz:     1232, V-cycle:  1, iter: 13,   err = 5.14e-09,   time = 0.01 s
Triangular Preconditioner Preconditioned GMRES 
#dof:     1312,  #nnz:     4896, V-cycle:  1, iter: 14,   err = 4.17e-09,   time = 0.04 s
Triangular Preconditioner Preconditioned GMRES 
#dof:     5184,  #nnz:    19520, V-cycle:  1, iter: 14,   err = 4.55e-09,   time = 0.11 s
Triangular Preconditioner Preconditioned GMRES 
#dof:    20608,  #nnz:    77952, V-cycle:  1, iter: 14,   err = 5.28e-09,   time = 0.18 s
Table: Error
 #Dof       h       ||u-u_h||    ||u_I-u_h||  ||sigma-sigma_h||||sigma-sigma_h||_{div}
  336   1.25e-01   1.29702e-01   3.08718e-02   1.00257e+00   1.01710e+01
 1312   6.25e-02   6.53059e-02   7.92226e-03   5.03081e-01   5.14701e+00
 5184   3.12e-02   3.27071e-02   1.99320e-03   2.51757e-01   2.58126e+00
20608   1.56e-02   1.63602e-02   4.99086e-04   1.25905e-01   1.29160e+00

Table: CPU time
 #Dof   Assemble     Solve      Error      Mesh    
  336   0.00e+00   1.00e-02   1.00e-02   1.00e-02
 1312   1.00e-02   4.00e-02   1.00e-02   0.00e+00
 5184   2.00e-02   1.10e-01   3.00e-02   1.00e-02
20608   5.00e-02   1.80e-01   7.00e-02   1.00e-02

Mix Dirichlet and Neumann boundary condition.

option.solver = 'uzawapcg';
bdFlag = setboundary(node,elem,'Dirichlet','~(x==0)','Neumann','x==0');
mfemPoisson(node,elem,pde,bdFlag,option);

Uzawa-type MultiGrid Preconditioned PCG 
#dof:      336,  #nnz:     1200, V-cycle:  1, iter: 16,   err = 6.09e-09,   time = 0.02 s
Uzawa-type MultiGrid Preconditioned PCG 
#dof:     1312,  #nnz:     4832, V-cycle:  1, iter: 16,   err = 7.58e-09,   time = 0.06 s
Uzawa-type MultiGrid Preconditioned PCG 
#dof:     5184,  #nnz:    19392, V-cycle:  1, iter: 16,   err = 9.33e-09,   time = 0.18 s
Uzawa-type MultiGrid Preconditioned PCG 
#dof:    20608,  #nnz:    77696, V-cycle:  1, iter: 16,   err = 9.13e-09,   time = 0.45 s
Table: Error
 #Dof       h       ||u-u_h||    ||u_I-u_h||  ||sigma-sigma_h||||sigma-sigma_h||_{div}
  336   1.25e-01   1.30305e-01   3.49802e-02   1.00524e+00   1.01710e+01
 1312   6.25e-02   6.53848e-02   8.95903e-03   5.03425e-01   5.14701e+00
 5184   3.12e-02   3.27171e-02   2.25324e-03   2.51800e-01   2.58126e+00
20608   1.56e-02   1.63615e-02   5.64155e-04   1.25911e-01   1.29160e+00

Table: CPU time
 #Dof   Assemble     Solve      Error      Mesh    
  336   1.00e-02   2.00e-02   0.00e+00   0.00e+00
 1312   2.00e-02   6.00e-02   1.00e-02   1.00e-02
 5184   2.00e-02   1.80e-01   2.00e-02   0.00e+00
20608   7.00e-02   4.50e-01   9.00e-02   1.00e-02

Conclusion

The optimal rates of convergence for u and sigma are observed, namely, 1st order for L2 norm of u, L2 norm of sigma and H(div) norm of sigma. The 2nd order convergent rates between two discrete functions |uI-uh| and |sigmaI-sigmah| are known as superconvergence.