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:^[1]

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

where $\sigma$ is $\tfrac12$ 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:^[2]

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

References

1. Steinbach, Olaf (2008). Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer. p. 137. ISBN 978-0-387-31312-2.
2. Steinbach, Olaf (2008). Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer. p. 182. ISBN 978-0-387-31312-2.