6.2. Semi-discrete form for diffusion#
In this section we go back to the Poisson equation one last time. We have seen this PDE before in 1D and 2D, but only for steady-state (time-independent) problems. Here we make one more extension by allowing time-dependent behavior to arise. We will see one common way to treat time in FEM problems, by arriving at a semi-discrete system of equations we can solve numerically.
This page also allows you to switch between tensor notation and index notation. Try it out and see how different parts of the formulation become easier to write in one or the other notation.
Strong form equation#
As always, we start the formulation with the PDE in its strong form:
where
The problem is completed by Dirichlet, Neumann and initial boundary conditions:
where
As always, we start the formulation with the PDE in its strong form:
where
The problem is completed by Dirichlet, Neumann and initial boundary conditions:
where
Weak form#
To get to the weak form we first multiply the PDE by a weight function
where
and we see that the derivative has now been transferred to
As a final step, we substitute the Neumann boundary condition of Eq. (6.5) and the constitutive relation
which is valid for any
To get to the weak form we first multiply the PDE by a weight function
where
and we see that the derivative has now been transferred to
As a final step, we substitute the Neumann boundary condition of Eq. (6.7) and the constitutive relation
which is valid for any
Semi-discrete form#
To get to a tractable FEM problem we need to restrict the space
where the shape function matrices and DOF vectors take the shape:
with
with:
We now substitute the approximations above into the weak form. The goal is to reach a so-called semi-discrete form, indicating that only spatial derivatives are discretized while time derivatives remain untouched and are treated as additional DOFs.
Substitution yields:
Making use of Eq. (1.17) we can remove
and cancel out
where the time derivative makes the new term
To get to a tractable FEM problem we need to restrict the space
where the shape function matrices and DOF vectors take the shape:
with
with:
We now substitute the approximations above into the weak form. The goal is to reach a so-called semi-discrete form, indicating that only spatial derivatives are discretized while time derivatives remain untouched and are treated as additional DOFs.
Substitution yields:
Making use of Eq. (1.17) we can remove
and cancel out
where the time derivative makes the new term
Solving the semi-discrete system#
We have reached a semi-discrete system with