$\vec\Phi = K_i \nabla u_i$ is the flux across the interface. For example, if $u$ is the thermal energy density and $K$ the thermal conductivity, then $\vec\Phi$ is the thermal energy flux. Energy conservation then dictates that whatever flows into the interface on one side ($\vec n_1 \cdot K_1 \nabla \vec\Phi_1$) better be equal to what flows out on the ...
In addition to Wolfgang Bangerth's explanation of temperature and concentration, let me give an other application where such interface conditions arise: (linear) elasticity, which has a similar structure to the elliptic equation in your example. Consider the situation with no interface first. Then the equilibrium, constitutive and kinematic equations of ...
Why not try drawing the basis?
You use Raviart-Thomas basis to approximate a vector field so a quiver plot makes sense.
This basis function (one out of three) is associated with the left edge. It is a kind of flux towards the left edge which is orthogonal to the normals of the other edges and magnitude is zero on the opposite vertex.
In the global basis you ...
The following paper is a good starting point.
Arnold, D. N. (1990). Mixed finite element methods for elliptic problems. Computer methods in applied mechanics and engineering, 82(1-3), 281-300. Author copy: https://www-users.cse.umn.edu/~arnold/papers/mixed.pdf
It presents the general problem and why one would like to use one formulation or the other. Also, ...
I think, one should distinguish two goals here:
Understand, at a superficial level, mixed FEMs.
Only concerned about implementing them.
I will start with the second use case, as I feel from your question that this is what you are after.
just tell me what is exact form of the test functions? how to deduce the final algebraic equations? how to address ...
Take a look at hyper.deal, which was written for this kind of thing. In general, you may want to look at the literature on the "radiative transfer problem" (say, the papers by Marv Adams and colleagues) on how to solve these kinds of problems.
I'd argue that RT0 basis are interpolants, but for functions that are observed/measured in terms of flux across facets (as opposed to functions that are measured by point sampling). Each RT0 function has unit flux across one particular facet, zero across the others, and smooth behavior over the interior. Similarly, the Nedelec basis interpolates functions ...
If I understand your question right then yes, you're correct.
The most common approach to enforcing Dirichlet boundary conditions with the finite element method is to modify the linear system of equations, which could be called a post-processing step after matrix assembly to paraphrase you.
Nitsche's method circumvents the need for this post-processing step ...
The Raviart-Thomas element has only 3 degrees of freedom on each triangle, namely the normal components of the vectors at the midpoints of the three edges. So the root of your misunderstanding is that $\mathbf u$ is a vector of size 3, matching the element matrix size.
You are correct, all degrees-of-freedom are constrained weakly so there is no need to post process the matrix. Here is an example with $f=10$ and $u_0(x) = \sin(2 \pi x)$:
The example source code runs after pip install scikit-fem==4.0.1:
import numpy as np
from skfem import *
from skfem.helpers import grad, dot
from skfem.models import laplace, unit_load
There's nothing wrong with PIC (to my eye), though you could experiment with other methods. For example, your problem has features similar to those of the lattice Boltzmann method. Another possibility is tensor trains.