Primal versus dual-mixed formulation in the elastic case

All our computations for the elasto-plasticity problem are based on its primal variational formulation as a nonlinear boundary value problem. We think that this approach is superior over the dual-mixed formulation with respect to total costs. The better accuracy of the latter is by far compensated by the higher solution efficiency of the primal method. For supporting this view, we recall from [31] the result of a test calculation for the above model problem ,,disc with crack'' in the elastic purely case. The dual-mixed finite element scheme uses continuous (isoparametric) bilinear trial functions for both unknowns, deformation and stresses, with a suitable least-squares stabilization. The grid control is also by a residual-based a posteriori error estimator obtained through solving a dual problem.

The results are shown in Table 7 for uniformly refined and in Table 8 for adaptively refined meshes. Clearly, with the same number of unknowns, the dual-mixed scheme provides a significantly higher accuracy than the primal scheme. However, one has to consider that computing the solution of the former one may be much more expensive due to the indefinit character of the corresponding linear systems.

Tabelle 7: Comparison between Rel_primal and Rel_dual on uniformly refined meshes

L	N	Relprimal	Reldual
1	256	0.0283	0.0157
2	1024	0.0181	0.0086
3	4096	0.0113	0.0046
4	16384	0.0070	0.0025
5	65536	0.0043	0.0013

Tabelle 8: Comparison between Rel_primal and Rel_dual on adaptively refined meshes

L	N	Relprimal	N	Reldual
1	256	0.0283	256	0.0157
2	484	0.0180	475	0.0086
3	1060	0.0113	856	0.0078
4	2113	0.0070	1618	0.0041
5	4435	0.0044	2755	0.0021
6	8830	0.0027	4447	0.0019
7	15886	0.0017	7339	0.0010
8	29947	0.0010	11323	0.0005
9	52288	0.0006	17065	0.0003