Introduction
Introduction#
This library is conceived to be a collection of solvers to be used in OpenFOAM-v6.
Four algorithms have been implemented (with their extensions):
Proper Orthogonal Decomposition (POD): with projection for the online phase
Empirical Interpolation Method (EIM)
Generalised Empirical Interpolation Method (GEIM)
Parameterised-Background Data-Weak (PBDW) formulation
Moreover, some useful routines have been implemented in the src/MOR/MOR folder to compute the following:
Scalar product in \(L^2\) for scalar and vector fields:
\[\begin{equation*}
\langle \phi, \psi\rangle = \int_\Omega \phi \cdot \psi\,d\Omega \qquad \qquad
\langle \mathbf{u}, \mathbf{v}\rangle = \int_\Omega \mathbf{u} \cdot \mathbf{v}\,d\Omega
\end{equation*}\]
Norm in \(L^2\) for scalar and vector fields:
\[\begin{equation*}
\|\phi\|_{L^2}^2=\int_\Omega \phi^2\,d\Omega \qquad \qquad
\|\mathbf{u}\|_{L^2}^2=\int_\Omega \mathbf{u}\cdot \mathbf{u}\,d\Omega
\end{equation*}\]
Norm in \(H^1\) for scalar and vector fields:
\[\begin{equation*}
\|\phi\|_{H^1}^2=\int_\Omega \phi^2\,d\Omega + \int_\Omega \nabla\phi\cdot \nabla \phi\,d\Omega \qquad \qquad
\|\mathbf{u}\|_{H^1}^2=\int_\Omega \mathbf{u}\cdot \mathbf{u}\,d\Omega + \int_\Omega \nabla\mathbf{u}: \nabla \mathbf{u}\,d\Omega
\end{equation*}\]
Norm in \(L^\infty\) for scalar and vector fields:
\[\begin{equation*}
\|\phi\|_{L^\infty} =\max\limits_\Omega |\phi|\qquad \qquad
\|\mathbf{u}\|_{L^\infty} =\max\limits_\Omega \|\mathbf{u}\|_2
\end{equation*}\]