Convergence of Finite Element Method

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

The data is chosen such that the exact solution and on . Two type of mesh refinements: regular refinement and bisection, and the following two boundary conditions are tested:

Conclusion

Contents

Non-empty Dirichlet boundary condition. Regular refinement.

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.

[N,errL2,errH1,erruIuh] = squarePoisson1;

c = {'Dof','||u-u_h||','||Du-Du_h||','||DuI-Du_h||'};
makeHtmlTable([N errL2 errH1 erruIuh],[],[],c,[],[8 8 8 8]);

Dof	||u-u_h||	||Du-Du_h||	||DuI-Du_h||
8	0.19792794	1.5035186	0.43962712
32	0.065912866	0.8373829	0.13223755
128	0.017908413	0.43160728	0.034847816
512	0.0045763652	0.21751113	0.0088349807
2048	0.0011504907	0.10897223	0.0022166851
8192	0.0002880262	0.054513305	0.00055467254
32768	7.2031833e-05	0.027260054	0.00013869956
131072	1.8009539e-05	0.013630452	3.4676855e-05

-------------------------------------------------------------------------

Pure Neumann boundary condition. Regular refinement.

The H1-norm (1st order) is still optimal. But the L2-norm is less optimal (around 1.6). |DuI-Duh| is almost 2nd order. All norms of error are bigger than the first case with non-empty Dirichlet boundary condition.

[N,errL2,errH1,erruIuh] = squarePoisson2;

c = {'Dof','||u-u_h||','||Du-Du_h||','||DuI-Du_h||'};
makeHtmlTable([N errL2 errH1 erruIuh],[],[],c,[],[8 8 8 8]);

Dof	||u-u_h||	||Du-Du_h||	||DuI-Du_h||
8	0.27270895	1.4214078	0.72068057
32	0.071292801	0.81369453	0.24349312
128	0.0235204	0.42693805	0.072718733
512	0.008005421	0.21673828	0.020371672
2048	0.0026037026	0.1090426	0.0084785408
8192	0.00081103554	0.054519495	0.0021885057
32768	0.00024410173	0.027260501	0.00056333267
131072	7.1569437e-05	0.013630467	0.00014473916

-------------------------------------------------------------------------

Pure Neumann boundary condition. Bisection.

The H1-norm (1st order) is still optimal and the L2-norm is also optimal. Due to the pattern of the mesh, |DuI-Duh| is only 1st order. No superconvergence. H1- and L2-norms of the error are comparable with the first case with non-empty Dirichlet boundary condition.

[N,errL2,errH1,erruIuh] = squarePoisson3;

c = {'Dof','||u-u_h||','||Du-Du_h||','||DuI-Du_h||'};
makeHtmlTable([N errL2 errH1 erruIuh],[],[],c,[],[8 8 8 8]);

Dof	||u-u_h||	||Du-Du_h||	||DuI-Du_h||
8	0.12163231	0.88857801	0.014388933
32	0.071071973	0.79896581	0.29112579
128	0.018295688	0.40825579	0.14540376
512	0.004607423	0.20524329	0.072696839
2048	0.0011530955	0.10296168	0.036909815
8192	0.00028856652	0.051423423	0.018244916
32768	7.2160042e-05	0.025704501	0.0090959304
131072	1.8041163e-05	0.012851348	0.0045446404

-------------------------------------------------------------------------

Non-empty Dirichlet boundary condition. Bisection.

The H1-norm (1st order) is still optimal and the L2-norm is also optimal. Due to the pattern of the mesh, |DuI-Duh| is only 1st order. No superconvergence. The same conclusion as that for pure Neumann boundary condition. But in the first two meshes, the error in H1 and L2 norm decreases slowly and |DuI-Du_h| is even increase.

[N,errL2,errH1,erruIuh] = squarePoisson4;

c = {'Dof','||u-u_h||','||Du-Du_h||','||DuI-Du_h||'};
makeHtmlTable([N errL2 errH1 erruIuh],[],[],c,[],[8 8 8 8]);

Dof	||u-u_h||	||Du-Du_h||	||DuI-Du_h||
8	0.11886104	0.8886467	1.5700925e-16
32	0.05338692	0.82766497	0.18669992
128	0.014003885	0.41519008	0.12385507
512	0.0035349094	0.20696281	0.067625649
2048	0.00088305991	0.10318994	0.035112728
8192	0.0002202738	0.051505166	0.017869086
32768	5.4975595e-05	0.025728031	0.0090112814
131072	1.3729927e-05	0.012857617	0.0045246461

Published with MATLAB® 7.9