RATE OF CONVERGENCE OF WEAK GALERKIN FINITE ELEMENT METHOD

This example is to show the rate of convergence of CR nonconforming 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:

  1. Non-empty Dirichlet boundary condition. $u=g_D \hbox{ on }\Gamma_D, \quad \nabla u\cdot n=g_N \hbox{ on }\Gamma_N. \Gamma _N = \{(x,y): x=0, y\in [0,1]\}, \; \Gamma _D = \partial \Omega \backslash \Gamma _N$.
  2. Pure Neumann boundary condition. $\Gamma _N = \partial \Omega$.
  3. Robin boundary condition. $g_R u + \nabla u\cdot n=g_N \hbox{ on }\partial \Omega$

Contents

clear all; close all;
[node,elem] = squaremesh([0,1,0,1],0.25);
option.L0 = 2;
option.maxIt = 4;
option.printlevel = 1;
option.elemType = 'WG';
option.plotflag = 0;

Non-empty Dirichlet boundary condition.

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

Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    20608,  #nnz:   107058, iter: 22,   err = 7.5614e-09,   time = 0.68 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    82176,  #nnz:   462050, iter: 22,   err = 7.5865e-09,   time =  1.4 s

display('   ||u-u_h||   ||Du-Du_h||  ||DuI-Du_h||  ||uI-u_h||_{max}');
format shorte
display([err.L2 err.H1 err.uIuhH1 err.uIuhMax]);

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

ans =

   1.9879e-03   1.6246e-01   6.3330e-02   3.2079e-03
   4.9724e-04   8.1266e-02   3.1525e-02   8.0320e-04
   1.2433e-04   4.0637e-02   1.5745e-02   2.0084e-04
   3.1083e-05   2.0319e-02   7.8704e-03   5.0220e-05

Pure Neumann boundary condition.

pde = sincosNeumanndata;
% bdFlag = setboundary(node,elem,'Dirichlet','(x==0)','Neumann','~(x==0)');
bdFlag = setboundary(node,elem,'Neumann');
err = femPoisson(node,elem,pde,bdFlag,option);

Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    20608,  #nnz:   108249, iter: 22,   err = 7.6491e-09,   time = 0.57 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    82176,  #nnz:   464729, iter: 22,   err = 8.7841e-09,   time =  1.4 s

display('   ||u-u_h||   ||Du-Du_h||  ||DuI-Du_h||  ||uI-u_h||_{max}');
format shorte
display([err.L2 err.H1 err.uIuhH1 err.uIuhMax]);

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

ans =

   9.9726e-03   6.4852e-01   2.6172e-01   1.7953e-02
   2.4845e-03   3.2489e-01   1.2710e-01   4.3312e-03
   8.8151e-04   2.1364e-01   1.1662e-01   9.8067e-02
   2.2042e-04   1.0687e-01   5.8311e-02   4.9073e-02

Pure Robin boundary condition.

pdeRobin = sincosRobindata;
bdFlag = setboundary(node,elem,'Robin');
err = femPoisson(node,elem,pdeRobin,bdFlag,option);

Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    20608,  #nnz:   108256, iter: 21,   err = 6.2675e-09,   time =  0.6 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    82176,  #nnz:   464736, iter: 21,   err = 6.9218e-09,   time =  1.4 s

display('   ||u-u_h||   ||Du-Du_h||  ||DuI-Du_h||  ||uI-u_h||_{max}');
format shorte
display([err.L2 err.H1 err.uIuhH1 err.uIuhMax]);

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

ans =

   9.7330e-03   6.4838e-01   2.6094e-01   1.6932e-02
   2.4444e-03   3.2488e-01   1.2708e-01   4.3121e-03
   6.1181e-04   1.6253e-01   6.3104e-02   1.0824e-03
   1.5300e-04   8.1274e-02   3.1497e-02   2.7081e-04

Conclusion

The optimal rate of convergence of the H1-norm (1st order) and L2-norm (2nd order) is observed. No superconvergence for |DuI-Duh|.

MGCG converges uniformly in all cases.


Published with MATLAB® 7.12