- We are concerned with... - Mathematical models in physics as the minimisation of a potential (energy). - You know they usually involve partial differential equations. - You can probably think of a couple of examples. - You go very far with analytical methods, at least, in biophysics. - Comment on FEM vs FDM vs FDM. Not in the scope. - FEM deals better with complicated geometry, general BCs and variable or non-linear material properties.

- Maybe you are familiar with MP and DE, but not so much with VF - However, VF is the best one for mathematicians to work with - After illustrating them in the most simple example - Revisit the concepts in an abstract way and state general w.p.th.

- 💡 Using physical principles (energy minimisation), we can mathematically state physical problems as minimisation problems in an *infinite dimensional* space of functions.

We can now consider the variation of the functional $J$ at $u$ with respect to a variation of $v \in \mathcal{C}_0^1([0,1])$ times $\alpha \in \mathbb{R}$.

Any perturbation of the equilibrium configuration requires to supply energy to the system

Note that the solution of (BVP) is point-wise, whereas (VF) and (MP) is in the weak sense

The solution of a clamped elastic structure in an L-shaped domain does not have a solution in classical sense, since the stresses in the inner corner are infinite

Cauchy sequence wikipedia

- A symmetric positive definite bilinear form is an inner product. - The proof is done on V, then apply equivalence.

Comment on the hypothesis on f.

- Recasting the problem into a discrete vector subspace - If (V) is well-posed, then (G) is well-posed (previous theorem for finite dim, also)

Piecewise polynomials

Very general geometry

You do not want them to retain this.

You do not want them to retain this.

You do not want them to retain this.

Note that the basis functions are "globally" continuous.

- The model can be smooth? - Split window to associate with mathematical formulation - 3 ways to verify model - Visual inspection - Consistency check - Exact convergence test - Approx. convergence test - Overkill solution - Solution does not change

3. [FEniCS documentation](https://fenicsproject.org/documentation/) 🠒 Check out the books 4. [Wolfgang Bangerth's video lectures](https://www.math.colostate.edu/~bangerth/videos.html) from [deal.ii](www.dealii.org) 5. M. J. Gander and F. Kwok. *Numerical Analysis of PDEs Using Maple and MATLAB*. SIAM, 2018.