When using an Advanced attribute node, you have an option to choose the matrix symmetry. But how do you know which problems are symmetric? When the discretization of a PDE problem results in a symmetric Jacobian (stiffness) matrix (and a symmetric mass matrix for time-dependent or eigenvalue problems), you can often apply faster and less memory-consuming algorithms to solve the resulting linear systems. PDEs with symmetric discretization typically occur in models involving acoustics, diffusion, electromagnetics, heat transfer by conduction, and structural mechanics. In contrast, problems in fluid mechanics, convection-diffusion, and convection-conduction typically involve nonsymmetric Jacobian matrices.
where T denotes the transpose and the bar denotes the complex conjugate.
The Lagrange multiplier vector Λ is typically undetermined, and COMSOL Multiphysics does not solve for it. Similarly, the constraint
NU = M often contains the same equation several times. To handle this problem, the COMSOL software turns to a constraint-handling method that uses elimination. The solver computes a solution
Ud to the constraint
NU = M as well as a matrix
Null, whose columns form a basis for the null space of
N. For unidirectional constraints (
NF ≠ NT) a matrix
Nullf is also computed, whose columns form a basis for the null space of
NFT. Then it obtains the solution as
U = Null Un + Ud. Here
Un is the solution of
Kc Un = Lc, where
Here Kc is the eliminated stiffness matrix.
