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]
[math]\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), }[/math]
where [math]\displaystyle{ \sigma }[/math] is [math]\displaystyle{ \tfrac12 }[/math] almost everywhere, [math]\displaystyle{ \mathsf{Id} }[/math] is the identity boundary operator, [math]\displaystyle{ \mathsf{K} }[/math] is the double layer boundary operator, [math]\displaystyle{ \mathsf{V} }[/math] is the single layer boundary operator, [math]\displaystyle{ \mathsf{K}' }[/math] is the adjoint double layer boundary operator and [math]\displaystyle{ \mathsf{W} }[/math] is the hypersingular boundary operator.
The exterior Calderón projector is defined to be:[2]
- [math]\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). }[/math]
References
- ↑ Steinbach, Olaf (2008). Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer. p. 137. ISBN 978-0-387-31312-2. https://archive.org/details/numericalapproxi00stei_284.
- ↑ Steinbach, Olaf (2008). Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer. p. 182. ISBN 978-0-387-31312-2. https://archive.org/details/numericalapproxi00stei.
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