A Little Bit of Theory

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

$$u^{\prime \prime }(x)=0$$

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

$$R(v,u):={\int }_{\Omega }\frac{\partial v}{\partial x_i}\frac{\partial u}{\partial x_i}dV-{\int }_{{\Gamma }_{\text{right}}}vdS.$$

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 ${\phi }_n\in H_0(\Omega )$. In this space, the Finite Element solution is established by solving the linear system of equations $R_n(\hat{u})=0$, with residual vector $R_n(\hat{u}):=R({\phi }_n,\hat{u})$, and discrete solution

$$\hat{u}(x)={\phi }_n(x){\hat{u}}_n.$$

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