Weak Constraints
The standard method to enforce constraints in the COMSOL Multiphysics software applies the constraints pointwise at node points in the mesh. At each node point, only local values of the dependent variables are affected by the constraint, making the constraints independent of each other. The solvers can therefore eliminate both the constrained degrees of freedom and the constraint force terms, effectively reducing the system of equations and decreasing the number of degrees of freedom being solved for.
Weak constraints enforce the constraint in a local average sense, using shape functions as weights. Reaction terms are explicitly included in the system of equations, which is extended with Lagrange multiplier variables. These Lagrange multipliers in general have a physical meaning and an interpretation as a constraint force or flux. Whereas a standard constraint decreases the number of degrees of freedom by the number of unique constraints, weak constraints increase the degrees of freedom by the same number.
A weak constraint is respected only on average over each Lagrange multiplier shape function, rather than pointwise at mesh nodes. When it is possible to satisfy the constraint everywhere on each mesh element, standard and weak constraints in general lead to the same solution. Conversely, when constraints are contradictory or impossible to satisfy everywhere, standard and weak constraints can distribute the error differently, and therefore lead to slightly different solutions.
Weak constraints can be of use in the following situations:
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Standard constraints must never contain time derivatives of the dependent variables. Weak constraints do not have this limitation because they allow the same variables as any other term in the combined system of equations. Note that reaction terms cannot be applied symmetrically to time derivatives but must be selectively applied to individual variables.
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When the reaction force or flux is needed during a solution, because it enters into a coefficient somewhere, the Lagrange multiplier from a weak constraint can provide an accurate value (see Computing Accurate Fluxes). The corresponding variables computed from derivatives of the dependent variables are not as accurate and can, if used, introduce considerable errors in the solution.
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When constraints are strongly nonlinear, weak constraints can allow faster and more robust convergence. For nonlinear constraints, the true linearized subproblem solved in each solution step depends on the value of the Lagrange multiplier variables from the previous step. When using standard constraints, this information is discarded between solution steps. Using weak constraints, the Lagrange multiplier values are instead retained between steps because they are part of the solution vector.
Compared to standard, eliminated, constraints, weak constraints can also have the following drawbacks:
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Discontinuous constraints result in (theoretically) infinite Lagrange multipliers. In practice, large oscillations result.
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Pointwise and weak constraints on the same set of variables on adjacent boundaries (that is, boundaries that share common node points in the mesh) do not work. This means that if all boundaries must be constrained on a solid and you want to use a weak constraint on one boundary segment (one face), the weak constraint must be used on the entire boundary of the solid (if the boundary is connected).
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Lagrange multipliers are in some cases difficult to interpret. For example, Lagrange multipliers from Dirichlet conditions in axial symmetry are not equal to the reaction flux per area but rather per length and full revolution. For separate Weak Constraints nodes in axial symmetry, the default quadrature settings include a multiplication by 2π r, making the Lagrange multiplier represent flux per area.
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Because extra unknowns are introduced for the Lagrange multipliers, the size of the problem increases compared to the standard constraint elimination method.
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The Lagrange multiplier variables added by the weak constraints have a different unit than the main system variables and can therefore be of a different order of magnitude. This can lead to scaling problems. Usually the automatic variable scaling in the solvers is sufficient, but there are cases when manual scaling is needed.
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Weak constraints introduce zeros on the main diagonal of the Jacobian matrix of the discretized system, which therefore cannot be positive definite. This makes certain linear solvers and preconditioners unavailable for solving problems with weak constraints. In particular, the conjugate gradients iterative solver does not work, and neither does the SOR class of preconditioners and smoothers. Instead, try another iterative solver and use the Vanka algorithm with the Lagrange multipliers as the Vanka variables, or use the incomplete LU factorization algorithm as preconditioner.
Constraint Reaction Terms