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Maximum-norm a posteriori error estimates for singularly perturbed elliptic reaction-diffusion problems

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Abstract

Residual-type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polyhedral domains. Standard finite element approximations are considered. The error constants are independent of the diameters of mesh elements and the small perturbation parameter. In our analysis, we employ sharp bounds on the Green’s function of the linearized differential operator. Numerical results are presented that support our theoretical findings.

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Correspondence to Natalia Kopteva.

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The first author was partially supported by National Science Foundation Grants DMS-1016094 and DMS-1318652. The second author was partially supported by DAAD Grant A/13/05482 and Science Foundation Ireland Grant SFI/12/IA/1683.

Appendix A: Sharpness of log factors

Appendix A: Sharpness of log factors

In this section we prove that there are cases in which the logarithmic factor in the a posteriori upper bound (1.3) is necessary. Using an idea of Durán [14], we first prove a priori upper bounds and a posteriori upper and lower bounds for \(u-u_h\) in a modified BMO norm in the case that \(\Omega \) is a convex polygonal domain. These estimates are essentially the same as our \(L_\infty \) bounds, but with no logarithmic factors present. The proof is completed by employing the counterexample of Haverkamp [20] showing that a similar logarithmic factor is necessary in \(L_\infty \) a priori upper bounds for piecewise linear finite element methods. Note that our counterexample is only valid for piecewise linear elements. Logarithmic factors are not present in standard a priori \(L_\infty \) bounds for elements of degree two or higher on quasi-uniform grids, and it remains unclear whether there are cases for which they are necessary in the corresponding \(L_\infty \) a posteriori bounds. In addition, both the result of Durán [14] and ours below only consider Poisson’s problem and not the broader class of problems described in (1.1).

1.1 A.1: Adapted Hardy and BMO spaces

We begin by describing operator-adapted BMO and Hardy spaces, following [13]. Let \(-\Delta \) denote the Dirichlet Laplacian on \(\Omega \), i.e., the Laplacian with domain restricted to functions which vanish on \(\partial \Omega \). Let

$$\begin{aligned} \Vert v \Vert _{\mathrm{bmo}_\Delta (\Omega )}= & {} \sup _{B(x,r): x \in \Omega , 0<r<1} \left[ \frac{1}{|B \cap \Omega |} \int _{B \cap \Omega } | (I-(I-r^2 \Delta )^{-1})v(x)|^2 \, \mathrm{d}x \right] ^{1/2} \nonumber \\&+ \sup _{B(x,r):x \in \Omega , r \ge 1} \left[ \frac{1}{|B \cap \Omega |} \int _{B \cap \Omega } | v(x)|^2 \, \mathrm{d}x \right] ^{1/2}. \end{aligned}$$
(5.1)

The space \(\mathrm{bmo}_\Delta (\Omega )\) then consists of functions \(v \in L_2(\Omega )\) for which \(\Vert v\Vert _{\mathrm{bmo}_\Delta (\Omega )} <\infty \). Note that the resolvent \((I-r^2 \Delta )^{-1}\) replaces the usual average over B in the definition of BMO. We also define an operator-adapted atomic Hardy space \(h_\Delta ^1\) which is dual to \(\mathrm{bmo}_\Delta \). A bounded, measurable function a supported in \(\Omega \) is a local atom if there is a ball B centered in \(\Omega \) with radius \(r<2 \mathrm{diam}(\Omega )\) such that \(\Vert a\Vert _{2; \, \mathbb {R}^n} \le |B \cap \Omega |^{-1/2}\) and either \(r>1\), or \(r \le 1\) and there exists b in the domain of the Dirichlet Laplacian such that \(a = -\Delta b\), \(\mathrm{supp}(b) \cup \mathrm{supp} (-\Delta b) \subset B \cap \overline{\Omega }\), and

$$\begin{aligned} \Vert (-r^2 \Delta )^k b\Vert _{2; \, \mathbb {R}^n} \le r^2 |B \cap \Omega |^{-1/2}, ~~k=0,1. \end{aligned}$$
(5.2)

An atomic representation of w is a series \(w = \sum _{j} \lambda _j a_j\), where \(\{\lambda _j\}_{j=0}^\infty \in \ell ^1\), each \(a_j\) is a local atom, and the series converges in \(L_2(\Omega )\). We then define the norm

$$\begin{aligned} \Vert w\Vert _{h_\Delta ^1(\Omega )} = \inf \left\{ \sum _{j=0}^\infty |\lambda _j| : w = \sum _{j=0}^\infty \lambda _j a_j \hbox { is an atomic representation of } w \right\} . \end{aligned}$$
(5.3)

The Hardy space \(h_\Delta ^1(\Omega )\) is the completion in \((\mathrm{bmo}_\Delta (\Omega ))^*\) of the set of functions having an atomic representation with respect to the metric induced by the above norm. In addition, \(\mathrm{bmo}_\Delta \) is the dual space of \(h_\Delta ^1\) in the sense that if \(w= \sum _{j=0}^\infty \lambda _j a_j \in h_\Delta ^1(\Omega )\), then

$$\begin{aligned} w \mapsto v(w) := \lim _{k \rightarrow \infty } \sum _{j=0}^k \lambda _j \int _\Omega a_j v \, \mathrm{d}x \end{aligned}$$
(5.4)

is a well-defined and continuous linear functional for each \(v \in \mathrm{bmo}_\Delta (\Omega )\) whose norm is equivalent to \(\Vert v\Vert _{\mathrm{bmo}_\Delta (\Omega )}\). In addition, each continuous linear functional on \(h_\Delta ^1(\Omega )\) has this form (cf. Theorem 3.11 of [13]).

We finally list an essential regularity result; cf. Theorem 4.1 of [13].

Lemma 9

Let \(\Omega \) be a bounded, simply connected, semiconvex domain in \(\mathbb {R}^n\), and let G be the Dirichlet Green’s function for \(-\Delta \). Let \(\mathbb {G}_\Delta \) be the corresponding Green operator given by \(\mathbb {G}(v)(x) = \int _\Omega G(x,y) v(y) \, \mathrm{d}y\). Then the operators \(\frac{\partial \mathbb {G}}{\partial x_i \partial x_j}\) are bounded from \(h_\Delta ^1(\Omega )\) to \(L_1(\Omega )\). In other terms, given \(u \in H_0^1(\Omega )\) with \(-\Delta u \in h_\Delta ^1(\Omega )\), we have \(u \in W_1^2(\Omega )\) with

$$\begin{aligned} |u|_{2, 1 ; \, \Omega } \lesssim \Vert \Delta u \Vert _{h_\Delta ^1(\Omega )}. \end{aligned}$$
(5.5)

We remark that the above regularity result does not in general hold on nonconvex Lipschitz (or even \(C^1\)) domains; cf. Theorem 1.2.b of [21]. It is not clear whether (5.5) holds on nonconvex polyhedral domains, but a different approach to the analysis than that taken in [13] would be in any case needed to establish this. Such a result would allow us to extend a posteriori estimates in \(bmo_{\Delta }\) that we obtain below for convex polyhedral domains to general polyhedral domains, which would be desirable since the corresponding \(L_\infty \) estimates also hold on general polyhedral domains. However, for our immediate purpose of providing a counterexample it suffices to consider convex domains.

1.2 A.2: A priori and a posteriori estimates in \(bmo_{\Delta }\)

In [14], Durán proved that given a smooth convex domain \(\Omega \subset \mathbb {R}^2\) and piecewise linear finite element solution \(u_h\) on a quasi-uniform mesh of diameter h, \(\Vert u-u_h\Vert _{\mathrm{BMO}(\Omega )} \lesssim h^2 |u|_{W_\infty ^2(\Omega )}\). Here BMO\((\Omega )\) is the classical BMO space; cf. [14] for a definition. We prove the same on convex polyhedral domains in arbitrary space dimension, but with BMO replaced by its operator-adapted counterpart. For notational simplicity we also consider only piecewise linear finite element spaces below, but our a priori and a posteriori bounds easily generalize to arbitrary polynomial degree.

Lemma 10

Assume that \(\Omega \subset \mathbb {R}^n\) is convex and polyhedral, and \(u \in W_\infty ^2(\Omega )\). Let also \(u_h\) be the piecewise linear finite element approximation to u with respect to a quasi-uniform simplicial mesh of diameter h. Then

$$\begin{aligned} \Vert u-u_h\Vert _{\mathrm{bmo}_\Delta (\Omega )} \lesssim h^2 |u|_{2, \infty ; \, \Omega }. \end{aligned}$$
(5.6)

Proof

Let \(\sum _{j=0}^k \lambda _j a_j = z \in h_\Delta ^1(\Omega )\) with k arbitrary but finite. Such functions are dense in \(h_\Delta ^1\), so to prove our claim it suffices by the duality of \(\mathrm{bmo}_\Delta \) and \(h_\Delta ^1\) to show that \(\int _\Omega (u-u_h) z \, \mathrm{d}x \lesssim h^2 |u|_{2, \infty ; \, \Omega } \Vert z\Vert _{h_\Delta ^1(\Omega )}\). Let \(-\Delta v=z\) with \(v=0\) on \(\partial \Omega \). Letting \(I_h v\) be a Scott-Zhang interpolant of v, we have

$$\begin{aligned} (u-u_h, z)&= (u-u_h, -\Delta v) = (\nabla (u-u_h), \nabla (v-I_h v)) \nonumber \\&\lesssim h \Vert u-u_h\Vert _{W_\infty ^1(\Omega )} |v|_{2, 1; \, \Omega } \lesssim h\Vert u-u_h\Vert _{1, \infty ; \, \Omega } \Vert z\Vert _{h_\Delta ^1(\Omega )}. \end{aligned}$$
(5.7)

The proof is completed by recalling the \(W_\infty ^1\) error estimate \(\Vert u-u_h\Vert _{1, \infty ; \, \Omega } \lesssim h |u|_{2, \infty ; \, \Omega }\); cf. [12, 19, 36]. \(\square \)

We next prove a posteriori upper and lower bounds for \(\Vert u-u_h\Vert _{\mathrm{bmo}_\Delta (\Omega )}\). Note that the a posteriori lower bound for the error is critical in establishing that the logarithmic factor in (1.3) is necessary.

Lemma 11

Assume that \(\Omega \subset \mathbb {R}^n\) is convex and polyhedral. Let also \(u_h\) be the piecewise linear finite element approximation to u with respect to a shape-regular simplicial mesh, where \(u \in H_0^1(\Omega )\) with \(-\Delta u = f \in L_\infty (\Omega )\). Then

$$\begin{aligned}&\Vert u -u_h\Vert _{\mathrm{bmo}_\Delta (\Omega )} + \max _{T \in \mathcal {T}_h} h_T^2 \Vert f-f_T\Vert _{\infty ; \, T} \nonumber \\&\quad \simeq \max _{T \in \mathcal {T}_h} [ h_T^2 \Vert f + \Delta u_h\Vert _{\infty ; \, T} + h_T \Vert \llbracket \nabla u_h \rrbracket \Vert _{\infty ; \, \partial T}]. \end{aligned}$$
(5.8)

Here \(f_T = \frac{1}{|T|} \int _T f \,\mathrm{d}x\).

Proof

The upper bound for \(\Vert u-u_h\Vert _{\mathrm{bmo}_\Delta (\Omega )}\) follows by first noting that \(h_T^2 \Vert f-f_T \Vert _{\infty ; \, T} \le h_T^2 \Vert f+ \Delta u_h\Vert _{\infty ; \, T}\) and then employing a duality argument precisely as in the preceding lemma; one must only substitute standard residual error estimation techniques for the a priori error analysis techniques above. In order to prove the lower bound we employ a discrete \(\delta \)-function; cf. (A.5) of [38]. Given \(x_0 \in T \in \mathcal {T}_h\), let \(\delta _{x_0}\) be a smooth, fixed function compactly supported in T such that \((v_h, \delta _{x_0})=v_h(x_0)\) for all \(v_h \in S_h\). \(\delta _{x_0}\) may be constructed to satisfy \(\Vert \delta _{x_0}\Vert _{m, p ; \, T} \lesssim h_T^{-m-n(1-\frac{1}{p})}\) with constant independent of \(x_0\). A short computation shows that \(-c h_T^2 \Delta \delta _{x_0}\) is an atom satisfying (5.2) with the required value of c and the constant in \(r \simeq h_T\) independent of essential quantities. Thus

$$\begin{aligned} h_T^2 \Vert f + \Delta u_h\Vert _{\infty ; \, T}&\le h_T^2 \Vert f-f_T\Vert _{\infty ; \, T} + h_T^2 (f_T + \Delta u_h, \delta _{x_0}) \nonumber \\&\lesssim h_T^2 \Vert f-f_T\Vert _{\infty ; \, T} - h_T^2 (\Delta (u-u_h), \delta _{x_0}) \nonumber \\&= h_T^2 \Vert f-f_T\Vert _{\infty ; \,T} - h_T^2 (u-u_h, \Delta \delta _{x_0}) \nonumber \\&\lesssim h_T^2 \Vert f-f_T\Vert _{\infty ; \, T} + \Vert u-u_h\Vert _{\mathrm{bmo}_\Delta (\Omega )}. \end{aligned}$$
(5.9)

To bound \(h_T \Vert \llbracket \nabla u_h \rrbracket \Vert _{\infty ; \, e}\) on a face e of the triangulation, let \(e = T_1 \cap T_2\) with \(T_1,T_2 \in \mathcal {T}_h\). Modest modification of the arguments in (A.5) of [38] yields that for \(x_0 \in e\) and fixed polynomial degree \(r-1\), there is a function \(\widetilde{\delta }_{x_0}\) compactly supported in \(\mathcal {T}_1 \cup T_2\) such that \(v_h (x_0) = \int _e \widetilde{\delta }_{x_0} v_h \, \mathrm{d}s\) for \(v_h \in \mathbb {P}_{r-1}\), and in addition, \(\Vert \widetilde{\delta }_{x_0} \Vert _{m,p; \,T_1 \cup T_2} \lesssim h_T^{-m+1+n (1-\frac{1}{p})}\). Similar to above, \(-c h_T \Delta \widetilde{\delta }_{x_0}\) is an atom with \(r \simeq h_T\). Thus

$$\begin{aligned} h_T \Vert \llbracket \nabla u_h \rrbracket \Vert _{\infty ; \, e}&= \int _e \llbracket \nabla u_h \rrbracket \widetilde{\delta }_{x_0} \, \mathrm{d}s \nonumber \\&= h_T \int _{T_1 \cup T_2} \nabla (u-u_h) \nabla \widetilde{\delta }_{x_0} \, \mathrm{d}x - h_T \int _{T_1 \cup T_2} ( \Delta u_h+f) \widetilde{\delta }_{x_0} \, \mathrm{d}x \nonumber \\&\lesssim \int _{T_1 \cup T_2} (u-u_h) (-h_T \Delta \widetilde{\delta }_{x_0}) \, \mathrm{d}x + h_T \Vert \Delta u_h +f\Vert _{\infty ; \, T_1 \cup T_2} \Vert \widetilde{\delta }_{x_0} \Vert _{1; \, T_1 \cup T_2} \nonumber \\&\lesssim \Vert u-u_h\Vert _{\mathrm{bmo}_\Delta (\Omega )} + h_T^2 \Vert f + \Delta u_h \Vert _{\infty ; \, \mathcal {T}_1 \cup T_2}. \end{aligned}$$
(5.10)

Combining (5.9) and (5.10) completes the proof. \(\square \)

1.3 A.3: Necessity of logarithmic factors

In this section we show that logarithmic factors are necessary in maximum-norm a posteriori upper bounds at least in the case of piecewise linear function spaces in two space dimensions. In [20], Haverkamp showed that given a convex polygonal domain \(\Omega \) and quasi-uniform mesh of size h, there exists u (which depends on h) such that \(\Vert u-u_h\Vert _{ \infty ; \, \Omega } \gtrsim h^2 \log h^{-1} |u|_{2, \infty ; \, \Omega }\). Given such a u, employing this result, (1.3), and the preceding two lemmas yields

$$\begin{aligned} h^2 \log h^{-1} |u|_{2, \infty ; \, \Omega }&\lesssim \Vert u-u_h\Vert _{\infty ; \, \Omega } \nonumber \\&\lesssim \log h^{-1}\max _{T \in \mathcal {T}_h} [ h^2 \Vert f + \Delta u_h\Vert _{\infty ; \, T} + h \Vert \llbracket \nabla u_h \rrbracket \Vert _{\infty ; \, \partial T}] \nonumber \\&\lesssim \log h^{-1} [\Vert u -u_h\Vert _{\mathrm{bmo}_\Delta (\Omega )} + \max _{T \in \mathcal {T}_h} h^2 \Vert f-f_T\Vert _{\infty ; \, T}] \nonumber \\&\lesssim h^2 \log h^{-1}|u|_{2, \infty ; \, \Omega }. \end{aligned}$$
(5.11)

We have thus proved the following lemma.

Lemma 12

The bound

$$\begin{aligned} \Vert u-u_h\Vert _{\infty ; \, \Omega } \lesssim \log \underline{h}^{-1} \max _{T \in \mathcal {T}_h} [h_T^2 \Vert f+ \Delta u_h\Vert _{\infty ; \, T} + h_T \Vert \llbracket \nabla u_h \rrbracket \Vert _{\infty ; \, \partial T}] \end{aligned}$$
(5.12)

does not in general hold if the term \(\log \underline{h}^{-1}\) is omitted.

We now also remark on two further important consequences of Lemma 11. First, the standard a priori and a posteriori upper bounds for \(L_\infty \) are

$$\begin{aligned}&\Vert u -u_h\Vert _{\infty ; \, \Omega } \nonumber \\&\quad \lesssim \log \underline{h}^{-1} \max _{T \in \mathcal {T}_h} [ h_T^2 \Vert f + \Delta u_h\Vert _{\infty ; \, T} + h_T \Vert \llbracket \nabla u_h \rrbracket \Vert _{\infty ; \, \partial T} ] \nonumber \\&\quad \lesssim \log \underline{h}^{-1} \left( \Vert u-u_h\Vert _{\infty ; \, \Omega } + \max _{T \in \mathcal {T}_h} h_T^2 \Vert f-f_T\Vert _{\infty ; \, T} \right) . \end{aligned}$$
(5.13)

Lemma 12 establishes that the logarithmic factor in the first inequality above is necessary. Our estimates also show that the logarithmic factor in the second inequality (efficiency estimate) sometimes is not sharp, since \(\Vert u-u_h\Vert _{\infty ; \, \Omega }\) in the third line above may be replaced by \(\Vert u-u_h\Vert _{\mathrm{bmo}_\Delta (\Omega )}\) and the latter may grow strictly (logarithmically) slower than the former.

Secondly, an interesting question that has yet to be successfully approached in the literature is proof of convergence of adaptive FEM for controlling maximum errors. Among other difficulties, the presence of the logarithmic factor in the a posteriori bounds for the maximum error makes adaptation of standard AFEM convergence and optimality proofs much more challenging. Because logarithmic factors are global, they play no role in AFEM marking schemes, so the natural AFEM for controlling \(\Vert u-u_h\Vert _{\infty ; \, \Omega }\) is precisely the same as that for controlling \(\Vert u-u_h\Vert _{\mathrm{bmo}_\Delta (\Omega )}\). Lemma 11 indicates that at least for convex domains the BMO norm of the error is more directly controlled by the standard \(L_\infty \) AFEM since the bounds involve no logarithmic factors.

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Demlow, A., Kopteva, N. Maximum-norm a posteriori error estimates for singularly perturbed elliptic reaction-diffusion problems. Numer. Math. 133, 707–742 (2016). https://doi.org/10.1007/s00211-015-0763-0

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