The flow of making block matrix for dual mortar

About

Concept

\begin{bmatrix}
K_1 & 0 & B_1^T\\
0 & K_2 & B_2^T\\
B_1 & B_2 & 0
\end{bmatrix}

\begin{bmatrix}
u_1 \\ u_2 \\ \lambda
\end{bmatrix}
=
\begin{bmatrix}
f_1 \\ f_2 \\ 0
\end{bmatrix}

In primal mortar, \mu takes FE basis

D_{ij}=\int_{\Gamma_c}\mu_i\phi^{(s)}_jd\Gamma,\qquad
M_{ij}=\int_{\Gamma_c}\mu_i\phi^{(m)}_jd\Gamma

where \phi is trace basis of each surface s=slave and m=master.

So the constraints are

Du_s - Mu_m = 0

In dual mortar, \mu is made with basis \phi that is on slave side.

\int_{\Gamma_c} \mu_i  \phi_j  d\Gamma_c = \delta_{ij}

when we regard 1 as slave and 2 as master, typically

B_1 = I,
\qquad
B_2 = -D^{-1}M

when we use dual basis with scaling,

B_1 = D_{\text{diag}},
\qquad
B_2 = -M

If we use above, the connection will be reformulated as

B_m = D^{-1} M\\
B_s = I

And the constraints will be

u_s - Pu_m = 0,
\qquad
P:=D^{-1}M

This P is projection of dual mortar.

The implementation flow

1. Create supermesh / intersection quadrature

2. At each crossing cell, evaluate the slave-side trace basis \phi^{(s)} and master-side \phi^{(m)}

3. Assemble local blocks

   D_e = \int \phi^{(s)}\phi^{(s)T} d\Gamma,\qquad
   M_e = \int \phi^{(s)}\phi^{(m)T} d\Gamma

4. Project Block

P_e = D_e^{-1}M_e

5. Assemble to global