RATE OF CONVERGENCE OF LINEAR ELEMENT FOR POISSON EQUATION

This example is to show the rate of convergence of linear finite element approximation of the Poisson equation on the unit square:

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

for the following boundary condition:

Non-empty Dirichlet boundary condition. $u=g_D \hbox{ on }\Gamma_D, \quad \nabla u\cdot n=g_N \hbox{ on }\Gamma_N.$
Pure Neumann boundary condition. $\nabla u\cdot n=g_N \hbox{ on } \partial \Omega$ .
Robin boundary condition. $g_R u + \nabla u\cdot n=g_N \hbox{ on }\partial \Omega$

Setting
Non-empty Dirichlet boundary condition.
Pure Neumann boundary condition.
Pure Robin boundary condition.
Conclusion

Setting

[node,elem] = cubemesh([0,1,0,1,0,1],0.5);
pde = sincosdata3;
option.L0 = 1;
option.maxIt = 4;
option.elemType = 'P1';
option.printlevel = 1;
option.plotflag = 1;

Non-empty Dirichlet boundary condition.

bdFlag = setboundary3(node,elem,'Dirichlet','~(x==0)','Neumann','x==0');
% bdFlag = setboundary3(node,elem,'Dirichlet');
femPoisson3(node,elem,pde,bdFlag,option);

Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:     4913,  #nnz:    23790, smoothing: (1,1), iter: 11,   err = 3.72e-09,   time = 0.017 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    35937,  #nnz:   209374, smoothing: (1,1), iter: 11,   err = 5.13e-09,   time = 0.11 s
Table: Error
 #Dof    ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}
  125   6.46061e-02   9.23927e-01   1.17911e-01   6.04325e-02
  729   1.86268e-02   4.80477e-01   3.91991e-02   1.95770e-02
 4913   4.86895e-03   2.42898e-01   1.06348e-02   5.17072e-03
35937   1.23196e-03   1.21798e-01   2.72075e-03   1.31609e-03

Table: CPU time
 #Dof   Assemble     Solve      Error      Mesh    
  125   3.02e-03   1.36e-04   1.68e-03   1.82e-03
  729   1.44e-02   1.03e-03   6.74e-03   1.25e-02
 4913   1.52e-01   1.69e-02   3.83e-02   6.24e-02
35937   1.07e+00   1.05e-01   4.91e-01   6.07e-01

Pure Neumann boundary condition.

bdFlag = setboundary3(node,elem,'Neumann');
femPoisson3(node,elem,pde,bdFlag,option);

Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:     4913,  #nnz:    32650, smoothing: (1,1), iter: 13,   err = 2.43e-09,   time = 0.023 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    35937,  #nnz:   245018, smoothing: (1,1), iter: 14,   err = 4.76e-09,   time = 0.14 s
Table: Error
 #Dof    ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}
  125   7.70680e-02   8.69504e-01   3.38890e-01   1.33687e-01
  729   2.35461e-02   4.70688e-01   1.05354e-01   4.13393e-02
 4913   6.27721e-03   2.41377e-01   3.23192e-02   1.41357e-02
35937   1.59840e-03   1.21576e-01   9.72436e-03   5.74640e-03

Table: CPU time
 #Dof   Assemble     Solve      Error      Mesh    
  125   3.85e-03   3.02e-04   1.51e-03   2.45e-03
  729   1.59e-02   2.63e-03   8.20e-03   1.13e-02
 4913   1.19e-01   2.34e-02   3.67e-02   5.64e-02
35937   1.09e+00   1.43e-01   5.25e-01   5.96e-01

Pure Robin boundary condition.

pdeRobin = sincosRobindata3;
bdFlag = setboundary3(node,elem,'Robin');
femPoisson3(node,elem,pdeRobin,bdFlag,option);

Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:     4913,  #nnz:    35729, smoothing: (1,1), iter: 11,   err = 2.75e-09,   time = 0.024 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    35937,  #nnz:   257313, smoothing: (1,1), iter: 11,   err = 8.49e-09,   time = 0.13 s
Table: Error
 #Dof    ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}
  125   6.40633e-02   8.75746e-01   2.90399e-01   1.03297e-01
  729   1.93507e-02   4.71714e-01   9.10331e-02   4.09455e-02
 4913   5.14258e-03   2.41519e-01   2.57055e-02   1.83236e-02
35937   1.30834e-03   1.21594e-01   6.98457e-03   6.58871e-03

Table: CPU time
 #Dof   Assemble     Solve      Error      Mesh    
  125   5.38e-03   2.74e-04   1.35e-03   4.41e-03
  729   3.01e-02   3.61e-03   2.03e-02   7.46e-03
 4913   1.32e-01   2.45e-02   4.42e-02   6.84e-02
35937   1.08e+00   1.32e-01   5.11e-01   6.07e-01

Conclusion

The optimal rate of convergence of the H1-norm (1st order) and L2-norm (2nd order) is observed. The 2nd order convergent rate between two discrete functions |DuI-Duh| is known as superconvergence.