27. The residual error estimator

The idea is to compute the residual of the Poisson equation

\[ f + \Delta \, u_h, \]

in the natural norm \(H^{-1}\). The classical \(\Delta\)-operator cannot be applied to \(u_h\), since the first derivatives, \(\nabla u_h\), are non-continuous across element boundaries. One can compute the residuals on the elements

\[ f_{|T} + \Delta \, u_{h|T} \qquad \forall \, T \in {\cal T}, \]

and one can also compute the violation of the continuity of the gradients on the edge \(E = T_1 \cap T_2\). We define the normal-jump term

\[ \left[\frac{\partial u_h}{\partial n} \right] := \frac{\partial u_h}{\partial n_1}|_{T_1} + \frac{\partial u_h}{\partial n_2}|_{T_2}. \]

The residual error estimator is

\[ \eta^{res}(u_h,f)^2 := \sum_T \eta_T^{res}(u_h,f)^2 \]

with the element contributions

\[ \eta_T^{res}(u_h, f)^2 := h_T^2 \| f + \Delta u_h \|_{L_2(T)}^2 + \sum_{E : E \subset T \atop E \subset \Omega} h_E \left\| \left[ \frac{\partial u_h}{\partial n} \right] \right\|_{L_2(E)} ^2. \]

The scaling with \(h_T\) corresponds to the natural \(H^{-1}\) norm of the residual.

27.1. The Clément- operator

To show the reliability of the residual error estimator, we need a new quasi-interpolation operator, the Clément- operator \(\Pi_h\). In contrast to the interpolation operator, this operator is well defined for functions in \(L_2\).

We define the vertex patch of all elements connected with the vertex \(x\)

\[ \omega_x = \bigcup_{T : x \in T} T, \]

the edge patch consisting of all elements connected with the edge \(E\)

\[ \omega_E = \bigcup_{T : E \cap T \neq \emptyset} T, \]

and the element patch consisting of the element \(T\) and all its neighbors

\[ \omega_T = \bigcup_{T^\prime : T \cap T^\prime \neq \emptyset} T^\prime. \]

The nodal interpolation operator \(I_h\) was defined as

\[ I_h v = \sum_{x_i \in {\cal V}} v(x_i) \varphi_i, \]

where \(\varphi_i\) are the nodal basis functions. Now, we replace the nodal value \(v(x_i)\) by a local mean value.

Definition: Clément quasi-interpolation operator

For each vertex \(x\),let \(\overline{v}^{\omega_x}\) be the mean value of \(v\) on the patch \(\omega_x\), i.e.,

\[ \overline{v}^{\omega_x} = \frac{1}{|\omega_x|} \int_{\omega_x} v \, dx. \]

The Clément operator is

\[ \Pi_h v := \sum_{x_i \in {\cal V}} \overline{v}^{\omega_{x_i}} \varphi_i. \]

In the case of homogeneous Dirichlet boundary values, the sum contains only inner vertices.

Theorem:

The Clément operator satisfies the following continuity and approximation estimates:

\[\begin{align*} \| \nabla \Pi_h v \|_{L_2(T)} & \preceq \| \nabla v \|_{L_2(\omega_T)} \\ \| v - \Pi_h v \|_{L_2(T)} & \preceq h_T \| \nabla v \|_{L_2(\omega_T)} \\ \| v - \Pi_h v \|_{L_2(E)} & \preceq h_E^{1/2} \| \nabla v \|_{L_2(\omega_E)} \\ \end{align*}\]

Proof: First, choose a reference patch \(\widehat \omega_T\) of dimension \(\simeq 1\). The quasi-interpolation operator is bounded on \(H^1(\omega_T)\):

\[ % \label{equ_clement_bh} \| v - \Pi_h v \|_{L_2(\widehat T)} + \| \nabla (v - \Pi_h v) \|_{L_2(\widehat T)} \preceq \| v \|_{H^1(\widehat \omega_T)} \]

If \(v\) is constant on \(\omega_T\), then the mean values in the vertices take the same values, and also \((\Pi_h v)_{|T}\) is the same constant. The constant function (on \(\omega_T\)) is in the kernel of \(\| v - \Pi_h v \|_{H^1(T)}\). Due to the Bramble-Hilbert lemma, we can replace the norm on the right hand side by the semi-norm:

\[ % \label{equ_clement_bh2} \| v - \Pi_h v \|_{L_2(\widehat T)} + \| \nabla (v - \Pi_h v) \|_{L_2(\widehat T)} \preceq \| \nabla v \|_{L_2(\widehat \omega_T)} \]

The rest follows from scaling. Let \(F : x \rightarrow h x\) scale the reference patch \(\widehat \omega_T\) to the actual patch \(\omega_T\). Then

\[ \| v - \Pi_h v \|_{L_2(T)} + h \, \| \nabla (v-\Pi_h v) \|_{L_2(T)} \preceq h \, \| \nabla v \|_{L_2(\omega_T)} \]

The estimate for the edge term is similar. One needs the scaling of integrals from the reference edge \(\widehat E\) to \(E\):

\[ \| v \|_{L_2(E)} = h_E^{1/2} \| v \circ F \|_{L_2(\hat E)} \]

27.2. Reliability of the residual error estimator

Theorem:

The residual error estimator is reliable:

\[ \| u - u_h \| \preceq \eta^{res} (u_h, f) \]

Proof: From the coercivity of \(A(.,.)\) we get

\[ \| u - u_h \|_{H^1} \preceq \frac{A(u-u_h, u-u_h)}{ \| u-u_h \|_{H^1} } \leq \sup_{0 \neq v \in H^1} \frac{A(u-u_h, v)}{ \| v \|_{H^1} }. \]

The Galerkin orthogonality \(A(u-u_h,v_h) = 0\) for all \(v_h \in V_h\) allows to insert the Cl’ement interpolant in the numerator. It is well defined for \(v \in H^1\):

\[ \| u - u_h \|_{H^1} \leq \sup_{0 \neq v \in H^1} \frac{A(u-u_h, v - \Pi_h v)}{ \| v \|_{H^1} }. \]

We use that the true solution \(u\) fulfills \(A(u,v) = f(v)\), and insert the definitions of \(A(.,.)\) and \(f(.)\):

\[\begin{align*} A(u-u_h, v-\Pi_h v) & = f(v-\Pi_h v) - A(u_h, v - \Pi_h v) \\ & = \int_{\Omega} f (v-\Pi_h v) \, dx - \int_\Omega \nabla u_h \nabla (v - \Pi_h v) \, dx \\ & = \sum_{T \in \cal T} \int_{T} f (v-\Pi_h v) \, dx - \sum_{T \in \cal T} \int_T \nabla u_h \nabla (v - \Pi_h v) \, dx \end{align*}\]

On each \(T\), the finite element function \(u_h\) is a polynomial. This allows integration by parts on each element:

\[\begin{eqnarray*} A(u-u_h, v-\Pi_h v) & = \sum_{T \in \cal T} \int_{T} f (v-\Pi_h v) \, dx - \sum_{T \in \cal T} \left\{ -\int_T \Delta u_h (v - \Pi_h v) \, dx + \int_{\partial T} \frac{\partial u_h}{\partial n} (v - \Pi_h v) \, ds \right\} \end{eqnarray*}\]

All inner edges \(E\) have contributions from normal derivatives from their two adjacent triangles \(T_{E,1}\) and \(T_{E,2}\). On boundary edges, \(v-\Pi_h v\) vanishes.

\[\begin{align*} A(u-u_h, v-\Pi_h v) & = \sum_T \int_T (f + \Delta u_h) (v - \Pi_h v) \, dx + \sum_E \int_E \left\{ \frac{\partial u_h}{\partial n}|_{T_{E,1}} + \frac{\partial u_h}{\partial n}|_{T_{E,2}} \right\} (v - \Pi_h v) \, ds \\ & = \sum_T \int_T (f + \Delta u_h) (v - \Pi_h v) \, dx + \sum_E \int_E \left[ \frac{\partial u_h}{\partial n} \right] (v - \Pi_h v) \, ds \end{align*}\]

Applying Cauchy-Schwarz first on \(L_2(T)\) and \(L_2(E)\), and then in \({\mathbb R}^n\):

\[\begin{align*} A(u-u_h, v-\Pi_h v) & \leq \sum_T \| f + \Delta u_h \|_{L_2(T)} \| v - \Pi_h v \|_{L_2(T)} + \sum_E \left\| \left[ \frac{\partial u_h}{\partial n} \right] \right\|_{L_2(E)} \| v - \Pi_h v \|_{L_2(E)} \\ & = \sum_T h_T \| f + \Delta u_h \|_{L_2(T)} h_T^{-1} \| v - \Pi_h v \|_{L_2(T)} + \sum_E h_E^{1/2} \left\| \left[ \frac{\partial u_h}{\partial n} \right] \right\|_{L_2(E)} h_E^{-1/2} \| v - \Pi_h v \|_{L_2(E)} \\ & \leq \left\{ \sum_T h_T^2 \| f + \Delta u_h \|_{L_2(T)}^2 \right\}^{1/2} \left\{ \sum_T h_T^{-2} \| v - \Pi_h v \|_{L_2(T)}^2 \right\}^{1/2} + \\ & + \left\{ \sum_E h_E \left\| \left[ \frac{\partial u_h}{\partial n} \right] \right\|_{L_2(E)}^2 \right\}^{1/2} \left\{ \sum_E h_E^{-1} \| v - \Pi_h v \|_{L_2(E)}^2 \right\}^{1/2} \end{align*}\]

We apply the approximation estimates of the Cl’ement operator, and use that only a bounded number of patches are overlapping:

\[ \sum_T h_T^{-2} \| v - \Pi_h v \|_{L_2(T)}^2 \preceq \sum_T \| \nabla v \|_{L_2(\omega_T)}^2 \preceq \| \nabla v \|_{L_2(\Omega)}^2, \]

and similar for the edges

\[ \sum_E h_E^{-1} \| v - \Pi_h v \|_{L_2(E)}^2 \leq \| \nabla v \|_{L_2(\Omega)}^2. \]

Combining the steps above we observe

\[\begin{eqnarray*} \| u - u_h \|_V & \preceq & \sup_{v \in H^1} \frac{A(u-u_h, v -\Pi_h v)}{\| v \|_H^1} \\ & \preceq & \sup_{V \in H^1} \frac{ \left\{ \sum_T h_T^2 \| f + \Delta u_h \|_{L_2(T)}^2 + \sum_E h_E \left\| \left[ \frac{\partial u_h}{\partial n} \right] \right\|_{L_2(E)}^2 \right\}^{1/2} \; \| \nabla v \|_{L_2(\Omega)} } { \| v \|_{H^1} } \\ & \leq & \left\{ \sum_T h_T^2 \| f + \Delta u_h \|_{L_2(T)}^2 + \sum_E h_E \left\| \left[ \frac{\partial u_h}{\partial n} \right] \right\|_{L_2(E)}^2 \right\}^{1/2}, \end{eqnarray*}\]

what is the reliability of the error estimator \(\eta^{res}(u_h,f)\)

27.2.1. Efficiency of the residual error estimator

If the source term \(f\) is piecewise polynomial on the mesh, then the error estimator \(\eta^{res}\) is efficient:

\[ \| u - u_h \|_V \succeq \eta^{res} (u_h, f) \]

Proof: Literature