# A Little Bit of Theory

We will introduce fundamental Nutils concepts based on the 1D homogeneous Laplace problem,

\[ u''(x) = 0 \]

with boundary conditions \( u(0) = 0 \) and \( u'(1) = 1 \). Even though the solution is trivially found to be \( u(x) = x \), the example serves to introduce many key concepts in the Nutils paradigm, concepts that can then be applied to solve a wide class of physics problems.

## Weak Form

A key step to solving a problem using the Finite Element Method is to cast it into weak form.

Let \( Ω \) be the unit line \( [0,1] \) with boundaries \( Γ_\text{left} \) and \( Γ_\text{right} \), and let \( H_0(Ω) \) be a suitable function space such that any \( u ∈ H_0(Ω) \) satisfies \( u = 0 \) in \( Γ_\text{left} \). The Laplace problem is solved uniquely by the element \( u ∈ H_0(Ω) \) for which \( R(v, u) = 0 \) for all test functions \( v ∈ H_0(Ω) \), with \( R \) the bilinear functional

\[ R(v, u) := ∫_Ω \frac{∂v}{∂x_i} \frac{∂u}{∂x_i} \ dV - ∫_{Γ_\text{right}} v \ dS. \]

## Discrete Solution

The final step before turning to code is to make the problem discrete.

To restrict ourselves to a finite dimensional subspace we adopt a set of Finite Element basis functions \( φ_n ∈ H_0(Ω) \). In this space, the Finite Element solution is established by solving the linear system of equations \( R_n(\hat{u}) = 0 \), with residual vector \( R_n(\hat{u}) := R(φ_n, \hat{u}) \), and discrete solution

\[ \hat{u}(x) = φ_n(x) \hat{u}_n. \]

Note that discretization inevitably implies approximation, i.e. \( u ≠ \hat{u} \) in general. In this case, however, we choose \( {φ_n} \) to be the space of piecewise linears, which contains the exact solution. We therefore expect our Finite Element solution to be exact.