How to use analytical solutions?

General
1

code (Error control):

def u_ex(mod):
    return lambda x: mod.cos(2*mod.pi*x[0])*mod.cos(2*mod.pi*x[1])

https://jsdokken.com/dolfinxtutorial/chapter4/convergence.html)

So from what I gathered it the function is meant to make use of an analytical solution for u. There is also a description in Chapter 20 of dolfinx book where a different forumulation of analytical solution is shewn for Navier stokes where there is also an analytical solution for f, u and rho where the u_ex code shewn it here is said to be for poisson’s.

So then the first question I have is since in both cases a 2D representation is being used to display analytical solutions is there a different analytical formulation for a 3D problem?

Also I looked around for more information on how these analytical solutions get formulated. So far I found a bit of talk about them here:

Detailed Explanation of the Finite Element Method (FEM).

Is there a place to look to as a reference or manual how to develop analytic solutions like this for use with dolfinx for all the different types of problems that may arise?

2

Any introductory course on partial differential equations should equip you with the skills to form a manufactured solution.

As a classic example, consider the Poisson problem. Let \Omega = (0, 1)^2 such that we seek u: \Omega \rightarrow \mathbb{R} which satisfies

\begin{align} - \nabla^2 u &= f &&\text{in } \Omega, \\ u &= 0 &&\text{on } \partial \Omega, \end{align}

where f is to be determined.

First we must choose a function u which satisfies the boundary conditions and is at least twice differentiable u \in C^2(\Omega). Arbitrarily we choose the function

u = \sin(\pi x) \sin(\pi y).

Now we may determine f by inserting into the original equation

f = -\nabla^2 (\sin(\pi x) \sin(\pi y)) = 2 \pi^2 \sin(\pi x) \sin(\pi y).

And we’re done. We have our problem defined for the manufactured solution.

You can follow through the same process for the cube \Omega = (0, 1)^3 where you should see, for example,

\begin{align} u &= \sin(\pi x) \sin(\pi y) \sin(\pi z), \\ f &= 3 \pi^2 \sin(\pi x) \sin(\pi y) \sin(\pi z). \end{align}
2 Likes