# 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.