RATE OF CONVERGENCE OF QUADRATIC ELEMENT FOR POISSON EQUATION

This example is to show the rate of convergence of quadratic 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] = squaremesh([0,1,0,1],0.25);
mesh = struct('node',node,'elem',elem);
option.L0 = 2;
option.maxIt = 4;
option.printlevel = 1;
option.plotflag = 1;
option.elemType = 'P2';

Non-empty Dirichlet boundary condition.

pde = sincosdata;
mesh.bdFlag = setboundary(node,elem,'Dirichlet','~(x==0)','Neumann','x==0');
femPoisson(mesh,pde,option);

Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:     4225,  #nnz:    39184, smoothing: (1,1), iter: 11,   err = 7.55e-10,   time = 0.023 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    16641,  #nnz:   160272, smoothing: (1,1), iter: 11,   err = 7.19e-10,   time = 0.077 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    66049,  #nnz:   648208, smoothing: (1,1), iter: 11,   err = 6.86e-10,   time = 0.27 s

 #Dof       h       ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}

 1089   6.25e-02   5.70490e-05   8.40318e-03   6.97560e-04   8.88533e-05
 4225   3.12e-02   7.13419e-06   2.10745e-03   1.10743e-04   1.14531e-05
16641   1.56e-02   8.92199e-07   5.27421e-04   1.82456e-05   1.45475e-06
66049   7.81e-03   1.11557e-07   1.31907e-04   3.09580e-06   1.83386e-07

 #Dof   Assemble     Solve      Error      Mesh    

 1089   1.00e-02   2.09e-03   1.00e-02   0.00e+00
 4225   5.00e-02   2.28e-02   4.00e-02   0.00e+00
16641   1.70e-01   7.72e-02   6.00e-02   3.00e-02
66049   4.50e-01   2.71e-01   2.10e-01   1.40e-01

Pure Neumann boundary condition.

option.plotflag = 0;
pde = sincosNeumanndata;
% pde = sincosdata;
mesh.bdFlag = setboundary(node,elem,'Neumann');
femPoisson(mesh,pde,option);

Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:     4225,  #nnz:    41458, smoothing: (1,1), iter: 13,   err = 5.23e-10,   time = 0.028 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    16641,  #nnz:   164850, smoothing: (1,1), iter: 14,   err = 2.47e-10,   time = 0.076 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    66049,  #nnz:   657394, smoothing: (1,1), iter: 14,   err = 5.84e-10,   time = 0.28 s

 #Dof       h       ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}

 1089   6.25e-02   4.45898e-04   6.57576e-02   1.35282e-02   1.75150e-03
 4225   3.12e-02   5.64820e-05   1.67107e-02   2.25947e-03   2.36781e-04
16641   1.56e-02   7.10333e-06   4.20307e-03   3.84722e-04   3.06054e-05
66049   7.81e-03   8.90408e-07   1.05337e-03   6.65670e-05   3.88530e-06

 #Dof   Assemble     Solve      Error      Mesh    

 1089   1.00e-02   3.36e-03   1.00e-02   0.00e+00
 4225   3.00e-02   2.83e-02   1.00e-02   1.00e-02
16641   1.00e-01   7.56e-02   7.00e-02   2.00e-02
66049   5.00e-01   2.79e-01   1.80e-01   5.00e-02

Pure Robin boundary condition.

option.plotflag = 0;
pde = sincosRobindata;
mesh.bdFlag = setboundary(node,elem,'Robin');
femPoisson(mesh,pde,option);

Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:     4225,  #nnz:    41473, smoothing: (1,1), iter: 11,   err = 9.54e-10,   time = 0.019 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    16641,  #nnz:   164865, smoothing: (1,1), iter: 12,   err = 1.74e-10,   time = 0.063 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    66049,  #nnz:   657409, smoothing: (1,1), iter: 12,   err = 1.97e-10,   time = 0.28 s

 #Dof       h       ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}

 1089   6.25e-02   4.46184e-04   6.57577e-02   1.36110e-02   1.73084e-03
 4225   3.12e-02   5.64917e-05   1.67107e-02   2.26778e-03   2.34617e-04
16641   1.56e-02   7.10364e-06   4.20307e-03   3.85477e-04   3.04111e-05
66049   7.81e-03   8.90418e-07   1.05337e-03   6.66341e-05   3.86933e-06

 #Dof   Assemble     Solve      Error      Mesh    

 1089   1.00e-02   2.36e-03   1.00e-02   0.00e+00
 4225   1.00e-02   1.88e-02   2.00e-02   0.00e+00
16641   9.00e-02   6.31e-02   5.00e-02   1.00e-02
66049   3.10e-01   2.78e-01   2.30e-01   5.00e-02

Conclusion

The optimal rate of convergence of the H1-norm (2nd order) and L2-norm (3rd order) is observed. The order of |DuI-Duh| is 2.5 order and thus superconvergence exists.