document: As mentioned previously, the weakly penalized system can be thought of as a generalized formulation which can result in the PL, penalty or statically condensed PL formulations depending on the choice of the projection operator. The equivalence of these methods under the weakly penalized regime, allows us to combine and take advantage of the good characteristics of each method. For instance, the weakly penalized formulation combines the simplified structure of the penalty method with the convergence characteristics of the PL formulation. However, due to the stiffness of the linear system at high values of the bulk modulus, the penalized formulations (classic penalty/weakly penalized) exhibit deteriorated nonlinear convergence. This stands in stark contrast to the PL method which (for inf–sup stable schemes) exhibits fast convergence even for high bulk modulus. However, we observe that, when the choice of πh provides equivalence with the discrete PL method, poor nonlinear convergence is observed though, in principle, the convergence should be similar. Examining the update formulae for both weakly penalized and PL approaches (see Appendix C), we observe that deteriorated convergence stems from: (1) initial residual amplification, and (2) the amplification of the residual.
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extractive_keyphrases: amplification of the residual convergence generalized formulation inf–sup stable schemes initial residual amplification penalized formulations penalty method pl weakly penalized
abstractive_keyphrases: classic penalty/weakly penalized deteriorated convergence nonlinear convergence pl formulation pl, penalty or statically condensed pl formulations poor nonlinear convergence weakly penalized and pl approaches weakly penalized formulation