Table 1.

Comparison of average number of Newton–Raphson iterations and Jacobian computations per load step as well as their respective average times between Newton–Raphson and Shamanskii–Newton–Raphson schemes. The efficiency of the SNR scheme with (PEN-MOD) and without (PEN) the introduced modifications on the penalty method is presented as well. The total solve time per load step and the total time per load step are illustrated as well. This comparison was performed on the passive inflation test (Section 3.1.2) on mesh₂ ($k={10}^7$ for the penalty, weakly penalized and Perturbed Lagrangian (PL) methods), the simulations were run on a single processor and a direct solver was used.

	$\mathit{J}$ compute time [s]a	$\mathit{J}$ computationsa	$\mathit{R}$ compute time [s]a	$\mathit{R}$ Computations a	Solve time [s]a	Total time [s]a
Newton–Raphson
PEN	181.55	4	1.93	4	47.51	231.13
WP	242.05	3.81	2.42	3.81	41.46	286.06
PL	246.77	3.88	2.49	3.88	45.36	294.75
Shamanskii–Newton–Raphson
PEN	25.80	0.46	11.67	10.82	7.2	44.81
PEN-MOD	10.14	0.113	14.07	9.51	1.76	26.08
WP-MOD	3.95	0.047	13.19	9.79	1.18	18.47
PL	4.66	0.053	13.43	9.88	1.37	19.61

^aTimes/iterations given as the average per load step.