Adaptive finite-element methods

Assume Countable Choice as in the cited Sobolev and Lax–Milgram suppliers. Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be a bounded polygon, $b\in {\mathbb{R}}^2$ constant, and $f\in L^2(\mathrm{\Omega };\mathbb{R})$. Here $H_0^1$ is the zero-boundary closure of Zero-boundary Sobolev space as a norm closure. There is a unique real-valued $u\in H_0^1(\mathrm{\Omega };\mathbb{R})$ satisfying ${\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\cdot \mathrm{\nabla }v+(b\cdot \mathrm{\nabla }u)vdx={\int }_{\mathrm{\Omega }}fvdx(v\in H_0^1(\mathrm{\Omega }))\mathrm{.}$ No $H^2$ regularity is asserted.

Assume Countable Choice for the $H_0^1$ convention of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

Fix a finite conforming triangular mesh $\mathcal{T}$. For each interior vertex $z$, choose the incident triangle with least fixed mesh index and denote it $K_z$. On the reference triangle $\widehat{K}$, let ${\widehat{\lambda }}_1,{\widehat{\lambda }}_2,{\widehat{\lambda }}_3$ be its barycentric coordinates. The Gram matrix $G_{ij}={\int }_{\widehat{K}}{\widehat{\lambda }}_i{\widehat{\lambda }}_j$ is positive definite because ${\int }_{\widehat{K}}{p}^2>0$ for every nonzero $p\in P_1(\widehat{K})$. Thus there are unique dual affine polynomials ${\widehat{\psi }}_i\in P_1(\widehat{K})$ with ${\int }_{\widehat{K}}{\widehat{\lambda }}_j{\widehat{\psi }}_i={\delta }_{ij}$. Pull them to $K_z$ by the affine map using the normalized measure so ${\int }_{K_z}{\phi }_y{\psi }_{z,K_z}={\delta }_{zy}$ for its three local nodal hats.

For $v\in H_0^1(\mathrm{\Omega })$ set $c_z(v)={\int }_{K_z}v{\psi }_{z,K_z}$ at interior vertices and $c_z(v)=0$ at boundary vertices. Define $I_{\mathcal{T}}v={\sum }_{z\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}}{c}_z(v){\phi }_z,$ where ${\phi }_z$ is the continuous piecewise-affine nodal hat. The sum is finite and belongs to the conforming zero-boundary P1 space. This is a moment quasi-interpolant; it is not a point-evaluation operator on arbitrary Sobolev classes.

The Sobolev spaces and zero-boundary closure use Integer-order Sobolev spaces and their norms and Zero-boundary Sobolev space as a norm closure. The mesh and finite-element conventions use Matching affine simplicial meshes and shape regularity and Broken and conforming polynomial spaces on a matching mesh. Reproduction and bounds are established by a separate theorem on this page.

A compatible labeled initial mesh is a finite matching affine simplicial triangulation ${\mathcal{T}}_0$ of a polygon in ${\mathbb{R}}^2$ as in Matching affine simplicial meshes and shape regularity, equipped with labels on its edges such that each triangle has one edge labeled $0$ and two labeled $1$, and every shared edge has the same label from both incident triangles. Matching means that distinct cell interiors are disjoint and any nonempty cell intersection is a whole common face: in two dimensions it may be a vertex or an edge. For a descendant triangle of generation $g(K)$, its three edge labels, in any order, are $(g(K)+1,g(K)+1,g(K))$. The unique edge labeled $g(K)$ is its reference edge.

To bisect $K$, join the midpoint of its reference edge to the opposite vertex. The two resulting triangles have generation $g(K)+1$. In each child, the two new half-edges and the bisector receive label $g(K)+2$; the one inherited nonreference edge retains label $g(K)+1$ and is the child's reference edge. A finite bisection forest is the collection of the initial triangles and all descendants reached by finitely many such bisections; its leaves form a triangulation when all shared-edge subdivisions agree. The admissible family $\mathbb{T}({\mathcal{T}}_0)$ consists of the matching conforming leaf triangulations generated by these compatible bisections. A refinement $\mathcal{T}\le {\mathcal{T}}_{\ast }$ means every cell of ${\mathcal{T}}_{\ast }$ lies in a cell of $\mathcal{T}$.

For deterministic operational ordering, fix a finite starting mesh in this family and a total order of its cells. Enumerate those operational roots as $K_1,\dots ,K_N$ in that order; this starting mesh may already be a geometric descendant of ${\mathcal{T}}_0$, and operational numbering does not reset its generations or edge labels. Give root $K_r$ identifier $(r,\mathrm{\varnothing })$ and order its three vertices lexicographically by their two real coordinates. If a parent has identifier $(r,w)$, with $w$ a finite binary word, let $A$ be the earlier endpoint of its labeled reference edge in its stored vertex list, let $B$ be the other endpoint, let $C$ be the opposite vertex, and let $M=(A+B)\mathrm{/}2$. Its children with vertex sets $\{A,M,C\}$ and $\{B,M,C\}$ receive identifiers $(r,w0)$ and $(r,w1)$, respectively. Each child's stored vertex list is the parent list restricted to its two retained vertices, in their previous order, followed by $M$. These lists are bookkeeping orders, independent of the orientation convention for affine reference maps. Compare identifiers first by root rank, then lexicographically by binary word, with $0<1$ and a proper prefix before its extensions. This is a total order, so every finite nonempty pending set has a unique least identifier. Distinct roots or distinct binary paths have distinct identifiers, and a retained cell keeps its identifier and vertex list through all subsequent calls. This operational decoration changes no geometric bisection, reference edge, admissible triangulation or finite-element space.

The compatible initial labeling is a hypothesis, not a consequence of an arbitrary initial triangulation. This definition is two-dimensional and makes no claim about three-dimensional closure or complexity.

The geometry and closure properties of this family are separate theorems on the same page.

Assume Countable Choice for the $L^2$ and Sobolev conventions of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

All functions and data are real-valued. Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be a bounded polygon and let $\mathcal{T}$ be a finite conforming matching triangulation by nondegenerate affine triangles, with maximum shape-regularity constant $\mathrm{\Gamma }$. Write $h_K=\mathrm{diam}K$ and $h_E=\mathrm{diam}E$. Let $V(\mathcal{T})$ be the continuous, piecewise-affine zero-boundary subspace of Broken and conforming polynomial spaces on a matching mesh, which lies in $H_0^1(\mathrm{\Omega };\mathbb{R})$ by Continuous matching piecewise polynomials belong to W1p and Zero-boundary conforming polynomials lie in W0 1p and satisfy Poincare. The exact weak model is $-\mathrm{\Delta }u+b\cdot \mathrm{\nabla }u=f,u\in H_0^1(\mathrm{\Omega };\mathbb{R}),f\in L^2(\mathrm{\Omega };\mathbb{R}),b\in {\mathbb{R}}^2\text{ constant}\mathrm{.}$ The weak form is ${\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\cdot \mathrm{\nabla }v+(b\cdot \mathrm{\nabla }u)vdx={\int }_{\mathrm{\Omega }}fvdx$ for every $v\in H_0^1(\mathrm{\Omega };\mathbb{R})$. The case $b=0$ is the symmetric Laplace model. Given $U\in V(\mathcal{T})$, its cell residual is $R_K(U)=f{\mathrm{\mid }}_K-b\cdot \mathrm{\nabla }(U{\mathrm{\mid }}_K)$; this is $f+\mathrm{\Delta }U-b\cdot \mathrm{\nabla }U$ because $\mathrm{\Delta }U=0$ on each cell. For an interior edge $E=K^{+}\cap K^{-}$ define the outward flux jump $J_E(U)=\mathrm{\nabla }U{\mathrm{\mid }}_{K^{+}}\cdot n_{K^{+}}+\mathrm{\nabla }U{\mathrm{\mid }}_{K^{-}}\cdot n_{K^{-}},$ where $n_{K^{\pm }}$ are the unit outward normals. Set ${\eta }_K(U{)}^2=h_K^2\mathrm{\parallel }{R}_K(U){\mathrm{\parallel }}_{L^2(K)}^2+\frac{1}{2}{\sum }_{E\subset \mathrm{\partial }K,E\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}}{h}_E\mathrm{\parallel }{J}_E(U){\mathrm{\parallel }}_{L^2(E)}^2,$ and $\eta (U,\mathcal{S}{)}^2={\sum }_{K\in \mathcal{S}}{\eta }_K(U{)}^2$ for $\mathcal{S}\subset \mathcal{T}$. The factor $1\mathrm{/}2$ assigns each interior-edge contribution once globally. Let ${\mathrm{\Pi }}_K^0$ be the $L^2(K)$ projection onto constants. The cell data oscillation is ${\mathrm{osc}}_K(U{)}^2=h_K^2\mathrm{\parallel }{R}_K(U)-{\mathrm{\Pi }}_K^0{R}_K(U){\mathrm{\parallel }}_{L^2(K)}^2,\mathrm{osc}(U,\mathcal{S}{)}^2={\sum }_{K\in \mathcal{S}}{\mathrm{osc}}_K(U{)}^2\mathrm{.}$ Since $b\cdot \mathrm{\nabla }U$ is constant on $K$, this oscillation equals $h_K^2\mathrm{\parallel }f-{\mathrm{\Pi }}_K^{0}f{\mathrm{\parallel }}_2^2$.

For a separate definitional interface, consider the symmetric diffusion model $-\mathrm{div}(A\mathrm{\nabla }u)=f$, where $A$ is a symmetric uniformly elliptic matrix field in the sense of Uniformly elliptic divergence-form operators and their sesquilinear forms and is Lipschitz on each cell. Its cell residual for a piecewise-affine $U$ is $R_K^A(U)=f+\mathrm{div}(A\mathrm{\nabla }U)$, and its outward conormal-flux jump is the sum of $A\mathrm{\nabla }U\cdot n_K$ from the two incident cells. Let ${\mathrm{\Pi }}_E^0$ be the $L^2(E)$ projection onto constants. The projected edge term $h_E\mathrm{\parallel }{J}_E^A-{\mathrm{\Pi }}_E^0{J}_E^A{\mathrm{\parallel }}_{L^2(E)}^2$ records nonconstant conormal flux. The reliability, contraction, and complexity results authored here concern $A=I$ only; this extension supplies notation and no coefficient-dependent theorem.

The matching affine mesh and continuous broken-polynomial conventions are those of Matching affine simplicial meshes and shape regularity and Broken and conforming polynomial spaces on a matching mesh. The proof of the weak representation uses Sobolev face traces and Gauss–Green on a reference simplex and Interior facet orientations cancel in a matching mesh.

Assume Countable Choice, as in the reference-simplex trace and Poincare suppliers.

Let $\mathcal{T}$ be a conforming shape-regular triangular mesh. For an interior vertex $z$, let ${\omega }_z$ be the union of its incident triangles; for a boundary vertex use the same patch. There is a constant $C$ depending only on the shape-regularity bound such that, at an interior vertex, every $v\in H^1({\omega }_z)$ satisfies $\mathrm{\parallel }v-v_{{\omega }_z}{\mathrm{\parallel }}_{L^2({\omega }_z)}\le C{h}_z\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2({\omega }_z)},$ where $v_{{\omega }_z}$ is the patch mean and $h_z={\max }_{K\subset {\omega }_z}{h}_K$. On a boundary vertex patch, every $v\in H_0^1(\mathrm{\Omega })$ satisfies $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2({\omega }_z)}\le C{h}_z\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2({\omega }_z)}\mathrm{.}$ For a cell $K$, define ${\omega }_K={\bigcup }_{z\in \mathcal{V}(K)}{\omega }_z$, where $\mathcal{V}(K)$ is its set of three vertices. If $K$ does not meet $\mathrm{\partial }\mathrm{\Omega }$, the corresponding mean-zero estimate holds on ${\omega }_K$ with scale $h_K$; if $K$ meets $\mathrm{\partial }\mathrm{\Omega }$ and $v\in H_0^1(\mathrm{\Omega })$, then $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2({\omega }_K)}\le C{h}_K\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2({\omega }_K)}\mathrm{.}$

Input: a finite labeled conforming mesh $\mathcal{T}$, exact nonnegative arrays $({\eta }_K^2{)}_{K\in \mathcal{T}}$ and $(o_K^2{)}_{K\in \mathcal{T}}$, where ${\eta }_K^2$ is the cell estimator contribution and $o_K^2$ the cell data-oscillation contribution, bulk-fraction parameters ${\theta }_E,{\theta }_O\in (0,1]$, a per-call total bisection budget $B\in {\mathbb{N}}_0$, and a stable total order of mesh identifiers. On the first call on a mesh not yet carrying operational forest identifiers, rank its cells by the supplied order and initialize the operational roots and vertex lists of Compatible 2D newest-vertex-bisection meshes. This initialization preserves the supplied order on those cells and does not change their geometric generations or labels. On subsequent calls retain that forest decoration and use its identifier order. Thus each current input leaf, including every generated descendant, has a specified identifier. For an array $q_K\ge 0$ and $\theta \in (0,1]$, sort by decreasing $q_K$ and then increasing identifier. If ${\sum }_K{q}_K=0$, return the empty set. Otherwise return the shortest initial segment whose sum is at least $\theta {\sum }_K{q}_K$. Thus these parameters are fractions of the squared contributions (the square-root convention for the usual estimator-norm parameter). Call the resulting sets $M_E,M_O$ and set $M=M_E\cup M_O$.

Before modifying the mesh, form the original edge-neighbor request set $Q=\{L\in \mathcal{T}:L=K\text{ or }L\text{ shares an edge with some }K\in M\}\mathrm{.}$ Thus $M_E,M_O$ and their union $M$ remain the bulk marks; $Q$ is the larger refinement request, with $M\subseteq Q$ and $\mathrm{\#}Q\le 4\mathrm{\#}M$ because a triangle has at most three edge neighbors. For $M=\mathrm{\varnothing }$, also $Q=\mathrm{\varnothing }$. Set a remaining-generation flag $q_K=3$ on each cell in $Q$ and $q_K=0$ on every other cell. Repeatedly take the pending leaf with least stable identifier and process its reference edge. If the reference edge is on the polygon boundary, bisect that leaf. If its interior reference neighbor has the same generation, bisect the compatible pair simultaneously. If the neighbor has generation one lower, recursively process that neighbor first and then re-inspect the original leaf; this follows the finite reference-edge chain until the edge can be split conformingly. Every split leaf $T$ is replaced by its two children with the binary-path identifiers and stored vertex lists prescribed by Compatible 2D newest-vertex-bisection meshes, each inheriting $q_T-1$ when $q_T>0$ and otherwise $0$. Thus conformity splits may propagate without scheduling extra uniform generations, while every originally requested cell in $Q$ and its descendants complete all three requested generations. Process simultaneous pairs in the same transaction and use the fixed identifier order for all pending leaves. Stop when all remaining-generation flags vanish. Count every bisection against $B$. If closure or a pending request would exceed the remaining budget, discard the tentative splits and their new identifiers, retaining the original identifiers and vertex lists, and return the original mesh with status budget-exhausted; otherwise return the refined matching mesh with status complete. Invalid arrays, nonfinite or negative entries, invalid parameters, or incompatible labels return invalid-input without changing the mesh. All arithmetic and comparisons in this mathematical algorithm are exact; implementation error, linear algebra, and quadrature are outside its scope.

A complete output is a conforming descendant, contains the uniform three-generation subdivision of every original cell in $Q$, and hence has an interior vertex in every bulk-marked cell and all its original edge neighbors. Every originally marked cell is refined; the sets $M_E,M_O$ themselves are unchanged by the patch expansion. Termination, shape regularity and cumulative element counts are separate result claims on this page. This definition makes no operation bound claim for a single refinement call.

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.

Assume Countable Choice, as in the zero-boundary Sobolev convention.

Let ${\mathcal{T}}_H\le {\mathcal{T}}_h$ be nested conforming meshes and let $U_H,U_h$ be exact P1 Galerkin solutions for the constant-convection model of Residual indicators and data oscillation for P1 AFEM. Write $e_j=u-U_j$ and $d=U_h-U_H$. For $b=0$, $\mathrm{\parallel }\mathrm{\nabla }{e}_h{\mathrm{\parallel }}_2^2=\mathrm{\parallel }\mathrm{\nabla }{e}_H{\mathrm{\parallel }}_2^2-\mathrm{\parallel }\mathrm{\nabla }d{\mathrm{\parallel }}_2^2\mathrm{.}$ For constant $b$, assume the explicit dual-pickup bound $\mathrm{\parallel }{e}_h{\mathrm{\parallel }}_{L^2}\le C_D{h}_0^s\mathrm{\parallel }\mathrm{\nabla }{e}_h{\mathrm{\parallel }}_{L^2}$ for the nested sequence, with $s>0$, where $h_0={\max }_{K\in {\mathcal{T}}_0}\mathrm{diam}K$, and put $\mu =C_D{h}_0^s\mathrm{\mid }b\mathrm{\mid }<1$. Then $\mathrm{\parallel }\mathrm{\nabla }{e}_h{\mathrm{\parallel }}_2^2\le (1-\mu {)}^{-1}\mathrm{\parallel }\mathrm{\nabla }{e}_H{\mathrm{\parallel }}_2^2-\mathrm{\parallel }\mathrm{\nabla }d{\mathrm{\parallel }}_2^2\mathrm{.}$ The dual-pickup bound is an additional hypothesis. It is not asserted for every polygon or mesh.

For any compatible 2D initial labeling from Compatible 2D newest-vertex-bisection meshes, every conforming NVB descendant has shape regularity bounded by a constant depending only on the initial mesh. Three complete generations of uniform bisection of a triangle split each original edge at its midpoint, create a new interior midpoint on the median from the opposite vertex, and give child diameters at most half the parent diameter.

The drawing shows three uniform NVB generations of $ABC$ with reference edge $AB$. $M,N,P$ are the midpoints of the three original sides; $S$ is the midpoint of median $CM$ and is interior; $Q,R$ are the midpoints of $AM,BM$. The eight leaves and their coordinates are verified in proof step 2.1. The dashed original edge $AB$ is the reference edge. The picture fixes the point conventions only; no proof relies on it.

Assume Countable Choice, as in the Sobolev and simplex trace suppliers.

Let $\mathcal{T}$ be a matching triangular mesh, $U\in V(\mathcal{T})$, and $u$ solve the weak model from Residual indicators and data oscillation for P1 AFEM. For every $v\in H_0^1(\mathrm{\Omega })$, ${\int }_{\mathrm{\Omega }}\mathrm{\nabla }(u-U)\cdot \mathrm{\nabla }v+(b\cdot \mathrm{\nabla }(u-U))v={\sum }_{K\in \mathcal{T}}{\int }_K{R}_K(U)v-{\sum }_{E\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}}{\int }_E{J}_E(U)v\mathrm{.}$

Assume Countable Choice, as in the Sobolev, affine-scaling and trace suppliers.

Let $\mathcal{T}$ be a shape-regular conforming triangular mesh and let $I_{\mathcal{T}}$ be the moment interpolant from A local moment interpolant for conforming P1 meshes. For each cell $K$, let ${\omega }_K$ be the union of vertex stars of its vertices. There is a constant $C$ depending only on shape regularity such that for every $v\in H_0^1(\mathrm{\Omega })$, $\mathrm{\parallel }v-I_{\mathcal{T}}v{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2({\omega }_K)},\mathrm{\parallel }\mathrm{\nabla }{I}_{\mathcal{T}}v{\mathrm{\parallel }}_{L^2(K)}\le C\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2({\omega }_K)},$ and, for every interior edge $E\subset \mathrm{\partial }K$, $\mathrm{\parallel }v-I_{\mathcal{T}}v{\mathrm{\parallel }}_{L^2(E)}\le C{h}_E^{1\mathrm{/}2}\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2({\omega }_K\cup {\omega }_{K^{\mathrm{\prime }}})},$ where $K^{\mathrm{\prime }}$ is the other incident cell. The operator maps into $V(\mathcal{T})$ and reproduces every member of $V(\mathcal{T})$.

For two compatible conforming NVB descendants ${\mathcal{T}}_1,{\mathcal{T}}_2$ of the same initial mesh ${\mathcal{T}}_0$, the union of their bisection forests defines a smallest common conforming overlay ${\mathcal{T}}_1\oplus {\mathcal{T}}_2$ and $\mathrm{\#}({\mathcal{T}}_1\oplus {\mathcal{T}}_2)\le \mathrm{\#}{\mathcal{T}}_1+\mathrm{\#}{\mathcal{T}}_2-\mathrm{\#}{\mathcal{T}}_0\mathrm{.}$

Assume Countable Choice for the $L^2$ projection convention.

Suppose the three-generation marked refinement call completes and returns ${\mathcal{T}}_{\ast }\ge \mathcal{T}$, and let ${\mathrm{osc}}_{\mathcal{T}}(K{)}^2=h_K^2\mathrm{\parallel }f-{\mathrm{\Pi }}_K^{0}f{\mathrm{\parallel }}_2^2$. For a Dörfler set $M_O$ with parameter ${\theta }_O\in (0,1]$, ${\mathrm{osc}}_{{\mathcal{T}}_{\ast }}^2\le {\gamma }_O{\mathrm{osc}}_{\mathcal{T}}^2,{\gamma }_O=1-\frac{3}{4}{\theta }_O<1.$

Assume Countable Choice, as in the Sobolev interpolation suppliers.

Let $\mathcal{T}\le {\mathcal{T}}_{\ast }$ be conforming P1 meshes with exact Galerkin solutions $U,U_{\ast }$. Let $\mathcal{R}=\{K\in \mathcal{T}:K\text{ is subdivided in }{\mathcal{T}}_{\ast }\}$ and let $\omega (\mathcal{R})$ be the union of the vertex stars of cells in $\mathcal{R}$. Then $\mathrm{\parallel }\mathrm{\nabla }(U_{\ast }-U){\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2\le C{\sum }_{K\in \mathcal{R}}{\eta }_K(U{)}^2,$ where $C$ depends on shape regularity, $\mathrm{\mid }b\mathrm{\mid }{h}_{\max }$, and the local interpolant constants.