EquilibratedFlux.jl
This package is based on Gridap.jl to provide post-processing tools to calculate reconstructed fluxes associated to the given approximate solution of a PDE.
For simplicity, we consider here the Poisson equation
\[\begin{align} - \Delta u &= f &&\text{in }\Omega\\ u &= g &&\text{on }\partial\Omega. \end{align}\]
We suppose we have already computed a conforming approximation $u_h \in V_h\subset H^1_0(\Omega)$ to the solution $u$ in Gridap.jl by solving
\[(\nabla u_h, \nabla v_h) = (f, v_h)\quad\forall v_h\in V_h,\]
The EquilibratedFlux.jl
library then provides the tools to compute a reconstructed flux associated to $u_h$. This flux, obtained by postprocessing, is an approximation to the numerical flux, i.e.
\[\sigma_h \approx -\nabla u_h.\]
This flux has the important property of being "conservative over faces" in the sense that
\[\sigma_h \in \mathbf{H}(\mathrm{div},\Omega).\]
We provide two functions to obtain such an object: build_equilibrated_flux
and build_averaged_flux
both provide reconstructed fluxes, which we denote by $\sigma_{\mathrm{eq},h}$ and $\sigma_{\mathrm{ave},h}$ respectively.
In addition to the properties listed above, the equilibrated flux $\sigma_{\mathrm{eq},h}$ satisfies the so-called equilibrium condition, i.e., for piecewise polynomial $f$, we have
\[\nabla\cdot\sigma_{\mathrm{eq},h} = f.\]
The reconstructed flux is the main ingredient in computing a posteriori error estimators. See the first tutorial for a complete demonstration of how to do this.
EquilibratedFlux.build_averaged_flux
— Methodbuild_averaged_flux(𝐀ₕ, model)
TODO: relevant docstring
EquilibratedFlux.build_equilibrated_flux
— Methodbuild_equilibrated_flux(𝐀ₕ, f, model, RT_order; measure = nothing, weight= 1.0)
TODO: relevant docstring