A two-cell linear finite element best approximation for Poisson

Given: Countable Choice and the Axiom of Choice; $I=(0,1)$; a scalar field $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$; $f=1$, the piecewise-linear function $\varphi$, and the standard Poisson form $a(v,w)={\int }_0^1{v}^{\mathrm{\prime }}\overline{w^{\mathrm{\prime }}}dx$ (with conjugation the identity over $\mathbb{R}$).

For a Hermitian coercive form on a Hilbert space, the conforming Galerkin solution is the best approximant in the induced energy norm (Hermitian Galerkin solutions are best in the energy norm).

The lemma gives explicit real $C_c^{\mathrm{\infty }}$ approximants for $\varphi$ and $p$; these are test functions over either scalar field, so both functions belong to the corresponding $H_0^1(0,1;\mathbb{K})$ (The two-cell hat and quadratic functions belong to H¹₀).

The weak Dirichlet Poisson theorem applies on an open nonempty domain bounded in one direction when a Poincar'e constant for $W_0^{1,2}$ is specified; under Axiom of Choice and Countable Choice it gives a unique weak solution for each bounded conjugate-linear datum (Existence and uniqueness for the weak Dirichlet Poisson problem).

The real or complex $L^2$ integral pairing is well defined, symmetric or conjugate symmetric, and satisfies Cauchy--Schwarz (The complex $L^2$ pairing is well-defined and satisfies Cauchy–Schwarz).

The classical second derivative of a smooth function is its weak second derivative, with the bilinear test identity (Classical derivatives agree with weak derivatives).

The constant function $1$ has $L^2(0,1)$ norm $1$; its integral pairing defines a bounded conjugate-linear functional on $H_0^1(0,1)$ by Cauchy--Schwarz and $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2}\le \mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}$ (A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included, The complex $L^2$ pairing is well-defined and satisfies Cauchy–Schwarz, Integer-order Sobolev spaces and their norms, The negative Sobolev space $H^{-1}(\mathrm{\Omega })$).

Countable Choice and the Axiom of Choice are retained from the cited weak-Poisson and coercive-form suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

$H_0^1(0,1)$ is the closure of $C_c^{\mathrm{\infty }}(0,1)$ in the $W^{1,2}$ norm, and test functions have their classical derivatives as weak derivatives (Zero-boundary Sobolev space as a norm closure, Integer-order Sobolev spaces and their norms).

The real second fundamental theorem evaluates the integral of a continuous derivative, continuous real functions on compact intervals are Riemann integrable, and a bounded Riemann-integrable function has the same Lebesgue integral (The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$, Every continuous function on a closed nondegenerate rectangle in ${\mathbb{R}}^m$ is Riemann integrable, A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral). For integrable real or complex functions, restricting the integral to a measurable set corresponds to multiplying by its indicator; this follows componentwise from the positive/negative-part definition (Integral over a measurable subset, Integrable real and complex functions, and their integrals).

${\lambda }_1([0,x])=x$ for $0\le x\le 1$ and ${\lambda }_1((0,1))=1$; the integral of an indicator is the measure of its set (A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included, The nonnegative Lebesgue integral).

The nonnegative integral is monotone (Monotonicity and nonnegative homogeneity of the nonnegative integral).