Identity Operator

The identity operator is implemented in two flavors BEAST.Identity() and BEAST.NCross().

Definition

The identity operator

\[\bm{\mathcal{I}} \bm b = \bm b\]

returns the function it is provided unchanged.

As bilinear form

When handed to the assemble function, the operator is interpreted as the bilinear form

\[a(\bm t, \bm b) = \int_\Gamma \bm t(\bm x) ⋅ \bm{\mathcal{I}} \bm b(\bm x) \,\mathrm{d}\bm x = \int_\Gamma \bm t(\bm x) ⋅ \bm b(\bm x) \,\mathrm{d}\bm x \,.\]

Hence, the resulting matrix $\bm A$ contains the entries

\[[\bm A]_{mn} = \int_\Gamma \bm t_m(\bm x) ⋅ \bm b_n(\bm x) \,\mathrm{d}\bm x \,.\]

API

Identity <: LocalOperator

Variant with $\bm n \times$

As a variation, the identity operator is provided with a cross product with the normal vector $\bm n$ of the surface $\Gamma$.

The bilinear form

The assemble function, interprets the operator as the bilinear form

\[a(\bm t, \bm b) = \int_\Gamma \bm t(\bm x) ⋅ \bm{\mathcal{I}} (\bm n(\bm x) \times \bm b(\bm x)) \,\mathrm{d}\bm x = \int_\Gamma \bm t(\bm x) ⋅ (\bm n(\bm x) \times \bm b(\bm x)) \,\mathrm{d}\bm x \,.\]

Hence, the resulting matrix $\bm A$ contains the entries

\[[\bm A]_{mn} = \int_\Gamma \bm t_m(\bm x) ⋅ (\bm n(\bm x) \times \bm b_n(\bm x)) \,\mathrm{d}\bm x \,.\]

API

NCross <: LocalOperator