The P1 AFEM approximation class

Assume Countable Choice. The space $H_0^1(\mathrm{\Omega })$ is the closure of $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ in the $H^1$ norm, as in Zero-boundary Sobolev space as a norm closure and The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

Fix a compatible initial triangulation ${\mathcal{T}}_0$, its 2D NVB family $\mathbb{T}({\mathcal{T}}_0)$ from Compatible 2D newest-vertex-bisection meshes, and a weak solution $u\in H_0^1(\mathrm{\Omega })$ of the symmetric $A=I$, $b=0$ model in Residual indicators and data oscillation for P1 AFEM. For a conforming refinement $\mathcal{T}$, let $V(\mathcal{T})$ be its continuous P1 zero-boundary space and define $E(\mathcal{T};u)={\inf }_{v\in V(\mathcal{T})}\mathrm{\parallel }\mathrm{\nabla }(u-v){\mathrm{\parallel }}_{L^2(\mathrm{\Omega })},E_N(u)={\inf }_{\begin{array}{c}\mathcal{T}\in \mathbb{T}({\mathcal{T}}_0)\\ \mathrm{\#}\mathcal{T}-\mathrm{\#}{\mathcal{T}}_0\le N\end{array}}E(\mathcal{T};u)\mathrm{.}$ For $s>0$, the approximation seminorm and class are $\mathrm{\mid }u{\mathrm{\mid }}_{{\mathbb{A}}_s}={\sup }_{N\ge 0}(N+1{)}^s{E}_N(u),{\mathbb{A}}_s=\{u\in H_0^1(\mathrm{\Omega }):\mathrm{\mid }u{\mathrm{\mid }}_{{\mathbb{A}}_s}<\mathrm{\infty }\}\mathrm{.}$ The infimum is over a nonempty finite-budget set because ${\mathcal{T}}_0$ is allowed. This definition concerns best energy approximation, not the number of cells selected by an algorithm. For the symmetric Laplace Galerkin solution, Céa quasi-optimality for coercive conforming Galerkin problems supplies the unique exact solution, while Galerkin orthogonality for conforming approximations gives orthogonality in the gradient inner product. Expanding the square shows that the exact Galerkin error equals $E(\mathcal{T};u)$.

The model, P1 spaces, and energy norm are those of Broken and conforming polynomial spaces on a matching mesh and Residual indicators and data oscillation for P1 AFEM. This class is used only in the restricted P1 Laplace optimal-complexity result on this page.