Calderón projector explained

In applied mathematics, the Calderón projector is a pseudo-differential operator used widely in boundary element methods. It is named after Alberto Calderón.

Definition

The interior Calderón projector is defined to be:^[1]

l{C}=\left(\begin{array}{cc}(1-\sigma) Id - K & V \\ W &\sigma Id + K '\end{array}\right),

where

\sigma is \tfrac 12 almost everywhere, Id is the identity boundary operator, K is the double layer boundary operator, V is the single layer boundary operator, K ' is the adjoint double layer boundary operator and W is the hypersingular boundary operator.

The exterior Calderón projector is defined to be:^[1]

l{C}=\left(\begin{array}{cc}\sigma Id + K &- V \\- W &(1-\sigma) Id - K '\end{array}\right).

References

1. Book: Steinbach, Olaf . 2008 . Numerical Approximation Methods for Elliptic Boundary Value Problems . limited . Springer . 978-0-387-31312-2.