Calderón projector

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:

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

where $\sigma$ is ${\tfrac {1}{2}}$ almost everywhere, ${\mathsf {Id}}$ is the identity boundary operator, ${\mathsf {K}}$ is the double layer boundary operator, ${\mathsf {V}}$ is the single layer boundary operator, ${\mathsf {K}}'$ is the adjoint double layer boundary operator and ${\mathsf {W}}$ is the hypersingular boundary operator.

The exterior Calderón projector is defined to be:

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