arising in finite-element methods. In this example, we will consider the Laplace equation on a compact smooth $d$-dimensional closed submanifold $\Omega \subseteq {\mathbf{R}}^d$ with compact smooth boundary $\partial \Omega$. Let $$V := \{f \in C^{\infty}(\Omega, {\mathbf{R}}) {\ |\ }f,\, \|\nabla f\| \in L^2(\Omega, {\mathbf{R}}), \ f \vert_{\partial \Omega} = 0 \}$$ be the collection of all smooth real-valued functions on $\Omega$ that are square-integrable, have square-integrable derivative, and vanish on the boundary $\partial \Omega$. Here the inner product is given by $$\smash{\langle u, v \rangle = \int_{\Omega} u(x) \, v(x) {\, \mathrm{d}}x + \int_{\Omega} \nabla u(x) \cdot \nabla v(x) {\, \mathrm{d}}x.}$$ Let $g \in V$ be given. We consider the problem of finding an $f \in V$ such that $$\label{eq:laplace}
\Delta f(x) = -g(x), \qquad (x \in \Omega).$$ The first step in the finite-element method is to rewrite the above problem into its *weak formulation*.[^4] Let $h \in V$ and suppose $f$ is a solution to , then using integration by parts $$\int_{\Omega} g(x) \, h(x) {\, \mathrm{d}}x = -\int_{\Omega} \Delta f(x) \, h(x) {\, \mathrm{d}}x 
= -0 + \int_{\Omega} \nabla f(x) \cdot \nabla h(x) {\, \mathrm{d}}x.$$ Hence, every solution $f$ necessarily satisfies the weak formulation of this problem: $$\label{eq:laplaceweak}
