LMKernel

LMKernel enables stabilization of Lagrange multiplier (LM) methods in which the Lagrange multiplier residual equation does not have any on-diagonal dependence, e.g. dependence on itself. This is a saddle point problem. Use of LMKernel is akin to a pressure-stabilized Petrov-Galerkin formulation for incompressible Navier-Stokes in which the strong form of the momentum equation is added to the mass equation. Because the momentum equation contains the Lagrange multiplier (pressure in this case), addition of it to the mass equation introduces on-diagonal dependence for the LM. For species transport, a user may wish to introduce a Lagrange multiplier that ensures the species concentration is always non-negative. This LM will appear in the (primal) species mass transport equation in the following way. Let's assume that the mass transport equation (without the LM) looks like:

where is the primal variable and is some arbitrary source or sink. When trying to enforce positivity of we introduce the LM into the primal equation like so:

(1)

where is the value of the Lagrange multiplier. The corresponding LM equation will be a nonlinear complementarity problem (NCP) function of the form

This NCP function ensures the following conditions:

(2)

Examining Eq. (2) it's clear that positive values of (required by Eq. (2)) will introduce a source in the primal equation. will "automatically" adjust itself such that never drops below zero. When the LM is no longer present in its own residual equation and we have a saddle point problem. However, by adding the strong form of the species mass transport equation to the LM residual equation, on-diagonal dependence is restored because of the presence of the LM in the primal equation. This removes the saddle point and allows use of a much wider range of preconditioners.