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]

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

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

The exterior Calderón projector is defined to be:^[2]

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

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.