The weak form

Finite Elements

The finite element method does not solve a PDE in the form you usually meet it. It rewrites the PDE first, in a way that looks weaker — the equation is asked to hold only "on average" against arbitrary test functions, instead of pointwise — and that weaker form is what gets discretized. The first surprise is that the weaker form is exactly equivalent to the original PDE for smooth solutions; the second is that it has nicer structure for computation. This chapter is about that rewriting.

The strong form

Take 1D Poisson's equation as our running example:

- u^{''} (x) = f (x), x \in (0, 1), u (0) = u (1) = 0.

This is the strong form. It demands that $u$ be twice differentiable everywhere on $(0, 1)$ and that the equation hold at every single point. That's a stiff demand — and one we'll relax in a moment.

Multiply by a test function and integrate

Pick any function $v (x)$ that vanishes at the boundary, $v (0) = v (1) = 0$ — call it a test function. Multiply the strong form by $v$ and integrate over the domain:

- \int_{0}^{1} u^{''} (x) v (x) d x = \int_{0}^{1} f (x) v (x) d x .

This is just an integral identity, valid for any classical solution $u$ . Now integrate the left-hand side by parts once. The boundary term $[u^{'} v]_{0}^{1}$ vanishes because $v$ is zero at both endpoints, leaving:

\int_{0}^{1} u^{'} (x) v^{'} (x) d x = \int_{0}^{1} f (x) v (x) d x for all admissible v .

This is the weak form. Read what changed. On the left, the second derivative of $u$ is gone — we now have only first derivatives, paired symmetrically between $u$ and $v$ . On the right, the source $f$ is averaged against a test function instead of being matched pointwise. The original PDE was a constraint at every point; the weak form is a constraint for every choice of $v$ .

Why the trade is good

Three things changed for the better:

Lower regularity required. The weak form only needs $u^{'}$ to be square-integrable, not $u^{''}$ . That's a much larger function space — and it includes piecewise-linear functions that are non-differentiable at the seams. Those are the basis functions FEM is going to use.
Symmetric in $u$ and $v$ . The bilinear form $a (u, v) = \int u^{'} v^{'} d x$ treats both arguments the same way. Discretizing with the same basis for trial and test functions (Galerkin's method) gives a symmetric system matrix — Poisson is self-adjoint, and the weak form preserves that.
Boundary conditions split cleanly. Dirichlet conditions (specified $u$ ) get baked into the function space — we just refuse to consider any $u$ that doesn't satisfy them. Neumann conditions (specified $u^{'}$ at the boundary) appear naturally from the boundary term we threw away in integration by parts; with a non-zero $v$ at the boundary they would contribute. We say Dirichlet is essential and Neumann is natural — and the labels reflect how each one enters the formulation.

From the weak form to a linear system

The weak form is still infinite-dimensional — $v$ ranges over an entire function space. To discretize, we restrict both $u$ and $v$ to a finite-dimensional subspace $V_{h} \subset V$ with basis ${ϕ_{1}, ϕ_{2}, \dots, ϕ_{n}}$ . Write the discrete solution as a linear combination:

u_{h} (x) = j = 1 \sum n c_{j} ϕ_{j} (x) .

Plug into the weak form, take $v = ϕ_{i}$ for each $i$ , and the integral collapses into something finite:

j = 1 \sum n c_{j} \int_{0}^{1} ϕ_{i}^{'} (x) ϕ_{j}^{'} (x) d x = \int_{0}^{1} f (x) ϕ_{i} (x) d x for i = 1, \dots, n .

That's a linear system $K c = b$ with

K_{ij} = \int_{0}^{1} ϕ_{i}^{'} (x) ϕ_{j}^{'} (x) d x, b_{i} = \int_{0}^{1} f (x) ϕ_{i} (x) d x .

Solve for the coefficient vector $c$ and you have your discrete solution $u_{h}$ . The matrix $K$ is called the stiffness matrix and the vector $b$ is the load vector — both names are vestiges of FEM's origin in structural mechanics.

What this leaves us to design: the basis functions ${ϕ_{i}}$ . The whole personality of an FEM implementation lives in that choice. Hat functions on a one-dimensional mesh — what the next chapter builds — are the simplest non-trivial choice, and they make $K$ sparse, symmetric, positive-definite, and tridiagonal. Hard to ask for a friendlier matrix.

Where this is going

Chapter 2 builds a 1D mesh and the corresponding hat-function basis, with a visualizer that lets you refine the mesh and see the basis adapt. Chapter 3 picks up where this one left off — it computes the integrals $\int ϕ_{i}^{'} ϕ_{j}^{'} d x$ and $\int f ϕ_{i} d x$ in closed form per element, assembles them into the global $K$ and $b$ , solves the system, and overlays $u_{h}$ on the analytic $u$ so you can watch the FEM solution converge as the mesh refines.