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Numerical approximation of the integral fractional Laplacian

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Abstract

We propose a new nonconforming finite element algorithm to approximate the solution to the elliptic problem involving the fractional Laplacian. We first derive an integral representation of the bilinear form corresponding to the variational problem. The numerical approximation of the action of the corresponding stiffness matrix consists of three steps: (1) apply a sinc quadrature scheme to approximate the integral representation by a finite sum where each term involves the solution of an elliptic partial differential equation defined on the entire space, (2) truncate each elliptic problem to a bounded domain, (3) use the finite element method for the space approximation on each truncated domain. The consistency error analysis for the three steps is discussed together with the numerical implementation of the entire algorithm. The results of computations are given illustrating the error behavior in terms of the mesh size of the physical domain, the domain truncation parameter and the quadrature spacing parameter.

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Correspondence to Wenyu Lei.

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Andrea Bonito and Wenyu Lei were supported in part by the National Science Foundation through Grant DMS-1254618 while the Wenyu Lei and Joseph E. Pasciak were supported in part by the National Science Foundation through Grant DMS-1216551.

A. Proof of Lemma 7.6

A. Proof of Lemma 7.6

The proof of Lemma 7.6 requires the following auxiliary localization result. We refer to [26] for a similar result in two dimensional space.

Lemma A.1

For \(r\in (0,1/2)\), let v be in \(H^r(D)\) and \({\widetilde{v}}\) denote the extension by zero of u to \({{{\mathbb {R}}}^d}\). There exists a constant C independent of h such that

$$\begin{aligned} \Vert {\widetilde{v}}\Vert _{H^r({{{\mathbb {R}}}^d})}^2\le C \bigg (h^{-2r} \Vert v\Vert _{L^2(D)}^2 + \sum _{\tau \in {\mathcal {T}_h(D)}} |v|_{H^r(\tau )}^2\bigg ) \end{aligned}$$

with a constant C independent of h.

Proof

Let \(\widetilde{\mathcal {T}}_h(D)\) be any quasi-uniform mesh [(satisfying (63) and (64)] which extends \({\mathcal {T}_h(D)}\) beyond a unit size neighborhood of D. Fix \(\delta >0\) and for \(\tau \in {\mathcal {T}_h(D)}\) set

$$\begin{aligned} {\widetilde{\tau }}=\cup _{\{\eta \in \widetilde{\mathcal {T}}_h(D){:}\,\hbox {dist}(\eta ,\tau )<\delta h\}} \eta . \end{aligned}$$

Let

$$\begin{aligned} D_h^\delta =\mathop {\bigcup }_{\tau \in {\mathcal {T}_h(D)}} {\widetilde{\tau }}\end{aligned}$$

and let \(\widetilde{\mathcal {T}}_h(D^\delta _h)\) denote the set of \(\tau \in \widetilde{\mathcal {T}}_h(D)\) contained in \(D_h^\delta \). Finally, for \(\tau \in \widetilde{\mathcal {T}}_h(D^\delta _h){\setminus } {\mathcal {T}_h(D)}\), set

$$\begin{aligned} {\widetilde{\tau }}=\bigcup _{\{\eta \in \widetilde{\mathcal {T}}_h(D^\delta _h){:}\,\hbox {dist}(\eta ,\tau )<\delta h\}}\eta . \end{aligned}$$

Fix \(v\in H^r(D)\). Since \({\widetilde{v}}\) vanishes outside of \(D_h^\delta \),

$$\begin{aligned} \begin{aligned} |{\widetilde{v}}|_{H^r({{{\mathbb {R}}}^d})}^2&=\int _{D_h^\delta }\int _{D_h^\delta } \frac{({\widetilde{v}}(x)-{\widetilde{v}}(y))^2}{|x-y|^{d+2r}} \, dx \, dy \\&\quad +\,2\int _{D}\int _{(D_h^\delta )^c} \frac{v(y)^2}{|x-y|^{d+2r}} \, dx \, dy=:J_1+J_2. \end{aligned} \end{aligned}$$

The second integral above is bounded by

$$\begin{aligned} \begin{aligned} J_2&\le 2 \int _{D}\int _{|x-y|\ge \delta h} \frac{v(y)^2}{|x-y|^{d+2r}} \, dx \, dy\\&=2 (\delta h)^{-2r} \int _{D} v(y)^2 \, dy \int _{|z|\ge 1} |z|^{-d-2r} \, dz= C h^{-2r}\Vert v\Vert _{L^2(D)}^2. \end{aligned} \end{aligned}$$
(97)

Expanding the first integral gives

$$\begin{aligned} \begin{aligned} J_1&= \sum _{\tau \in \widetilde{\mathcal {T}}_h(D^\delta _h)} \int _\tau \int _{\widetilde{\tau }}\frac{(v(x)-v(y))^2}{|x-y|^{d+2r}} \, dx \, dy\\&\quad +\,\sum _{\tau \in \widetilde{\mathcal {T}}_h(D^\delta _h)} \int _\tau \int _{{\widetilde{\tau }}^c} \frac{(v(x)-v(y))^2}{|x-y|^{d+2r}}\, dx \, dy=:J_3+J_4. \end{aligned} \end{aligned}$$
(98)

Applying the arithmetic-geometric mean inequality gives

$$\begin{aligned} J_4\le 2 \sum _{\tau \in \widetilde{\mathcal {T}}_h(D^\delta _h)} \int _\tau \int _{{\widetilde{\tau }}^c} \frac{v(x)^2+v(y)^2}{|x-y|^{d+2r}}\, dx \, dy. \end{aligned}$$
(99)

As in (97),

$$\begin{aligned} J_5:= \sum _{\tau \in \widetilde{\mathcal {T}}_h(D^\delta _h)} \int _\tau \int _{{\widetilde{\tau }}^c} \frac{v(y)^2}{|x-y|^{d+2r}}\ \, dx \, dy \le C h^{-2r} \Vert v\Vert _{L^2(D)}^2. \end{aligned}$$
(100)

Now,

$$\begin{aligned} \begin{aligned}&\{(\tau ,\tau _1){:}\, \tau \in \widetilde{\mathcal {T}}_h(D^\delta _h)\hbox { and } \tau _1\in {\widetilde{\tau }}^c\}\\&\quad =\left\{ (\tau ,\tau _1)\in \widetilde{\mathcal {T}}_h(D^\delta _h)\times \widetilde{\mathcal {T}}_h(D^\delta _h){:}\, \hbox {dist}(\tau ,\tau _1)>\delta h\right\} \\&\quad =\left\{ (\tau ,\tau _1){:}\, \tau _1\in \widetilde{\mathcal {T}}_h(D^\delta _h)\hbox { and } \tau \in {\widetilde{\tau }}_1^c\right\} . \end{aligned} \end{aligned}$$

Using this and Fubini’s Theorem gives

$$\begin{aligned} \begin{aligned} \sum _{\tau \in \widetilde{\mathcal {T}}_h(D^\delta _h)}&\int _\tau \int _{{\widetilde{\tau }}^c} \frac{v(x)^2}{|x-y|^{d+2r}}\ \, dx \, dy\\&\quad = \sum _{\tau _1\in \widetilde{\mathcal {T}}_h(D^\delta _h)} \int _{\tau _1} \int _{{\widetilde{\tau }}_1^c} \frac{v(x)^2}{|x-y|^{d+2r}}\ \, dy \, dx= J_5. \end{aligned} \end{aligned}$$

Thus \(J_4\le 4 J_5\) and is bounded by the right hand side of (100).

For \(J_3\), we clearly have

$$\begin{aligned} J_3\le \sum _{\tau \in \widetilde{\mathcal {T}}_h(D^\delta _h)} |v|_{H^r({\widetilde{\tau }})}^2. \end{aligned}$$

For any element \(\tau ' \in \widetilde{\mathcal {T}}_h(D^\delta _h)\), let \(v_{\tau '}\) denote \({\widetilde{v}}\) restricted to \(\tau '\) and extended by zero outside. As \(r\in (0,1/2)\), \(v_{\tau '} \in H^r({{{\mathbb {R}}}^d})\) and satisfies

$$\begin{aligned} \Vert v_{\tau '}\Vert _{H^r({{{\mathbb {R}}}^d})}\le C \Vert v\Vert _{H^r(\tau ')}. \end{aligned}$$

The constant C above only depends on Lipschitz constants associated with \(\tau '\) (see [22, 28]), which in turn only depend on the constants appearing in (63). We use the triangle inequality to get

$$\begin{aligned} |v|_{H^r({\widetilde{\tau }})} \le \sum _{\tau ^\prime \subset {\widetilde{\tau }}} |v_{\tau '}|_{H^r({\widetilde{\tau }})} \end{aligned}$$

and hence a Cauchy–Schwarz inequality implies that

$$\begin{aligned} |v|_{H^r({\widetilde{\tau }})}^2 \le N_\tau \sum _{\tau ' \subset {\widetilde{\tau }}} |v_{\tau '}|^2_{H^r({\widetilde{\tau }})}\le C \sum _{\tau ' \subset {\widetilde{\tau }}} |v_{\tau '}|^2_{H^r(\tau ^\prime )} \end{aligned}$$

with \(N_\tau \) denoting the number of elements in \({\widetilde{\tau }}\). As the mesh is quasi-uniform, \(N_\tau \) can be bounded independently of h. In addition, the mesh quasi-uniformity condition also implies that each \(\tau ' \in {\mathcal {T}_h(D)}\) is contained in a most a fixed number (independent of h) of \({\widetilde{\tau }}\) (with \(\tau \in \widetilde{\mathcal {T}}_h(D^\delta _h)\)). Thus,

$$\begin{aligned} J_3 \le C\sum _{\tau ^\prime \in {\mathcal {T}_h(D)}} \Vert v\Vert ^2_{H^r(\tau ^\prime )}. \end{aligned}$$

Combining the estimates for \(J_2,J_3\) and \(J_4\) completes the proof of the lemma. \(\square \)

Proof of Lemma 7.6

In this proof, C denotes a generic constant independent of h and j defined later. The inequality (4.1) of [43] guarantees that for \(\tau \in {{\mathcal {T}}}_h\), we have

$$\begin{aligned} \Vert v-\pi ^{sz}_hv\Vert _{H^{m}(\tau )} \le C \sum _{k=0}^m h^{k-m} \Vert v-p\Vert _{H^k(S_\tau )},\quad \hbox { for }m=0,1, \end{aligned}$$
(101)

for any linear polynomial p and \(v\in H^{1}(S_\tau )\). Here \(S_\tau \) denotes the union of \(\tau ^\prime \in {{\mathcal {T}}}_h\) with \(\tau \cap \tau ^\prime \ne \emptyset \).

Now, we map \(\tau \) to the reference element using an affine transformation. The mapping takes \(S_\tau \) to \({\widehat{S}}_\tau \). Our aim is to take advantage of the averaged Taylor polynomial constructed in [25], which requires the domain to be star-shaped with respect to a ball (of uniform diameter). The patch \({\widehat{S}}_\tau \) may not satisfy this property. However, it can be written as the (overlapping) union of domains \({\widehat{D}}_j\) with each \( {\widehat{D}}_j\) consisting of the union of pairs of elements of \({\widehat{S}}_\tau \) sharing a common face. These \({\widehat{D}}_j\) are star-shaped with respect to balls of diameter depending on the shape regularity constant of the subdivision, which is uniform thanks to (63). Hence, the averaged Taylor polynomial \({{\mathcal {Q}}}_j\) constructed in [25] satisfies (see Theorem 6.1 of [25]), for all \(v\in H^{\beta }({\widehat{S}}_j)\),

$$\begin{aligned} \Vert v-{{\mathcal {Q}}}_jv\Vert _{H^1({\widehat{D}}_j)} \le C |v|_{H^{\beta }({\widehat{D}}_j)}. \end{aligned}$$
(102)

Taking \(\Vert \cdot \Vert _{{\widehat{D}}_j}\) to be \(\Vert \cdot \Vert _{L^2({\widehat{D}}_j)} \) or \(\Vert \cdot \Vert _{H^1({\widehat{D}}_j)}\) and \(|\cdot |_{{\widehat{D}}_j} = |\cdot |_{H^\beta ({\widehat{D}}_j)}\) in Theorem 7.1 of [25] implies that (102) holds with \({\widehat{D}}_j\) replaced by \({\widehat{S}}_j\). This, (101) and a Bramble-Hilbert argument implies that for \(v\in H^\beta (D)\cap H^1_0(D)\),

$$\begin{aligned} \Vert v-\pi _h^{sz} v\Vert _{L^2(D)} + h \Vert v-\pi _h^{sz}v\Vert _{H^1(D)} \le C h^\beta |v|_{H^{\beta }(D)}. \end{aligned}$$
(103)

Inequality (88) follows from (103) and interpolation.

We cannot use Theorem 7.1 of [25] to derive (87) because of the non-locality of the norm \(|\cdot |_{H^\beta (D)}\). Instead, we apply Lemma A.1, (103), and the fact that \(|\pi _h^{sz}v|_{H^\beta (\tau )}=0\) to obtain, for \(v\in H^\beta (D)\cap H^1_0(D)\),

$$\begin{aligned} |v-\pi _h^{sz}v |_{H^\beta (D)}^2&\le C \bigg (h^{2-2\beta } \Vert \nabla (v-\pi _h^{sz}v)\Vert _{L^2(D)}^2 + \sum _{\tau \in {\mathcal {T}_h(D)}} |v|_{H^r(\tau )}^2\bigg )\nonumber \\&\le |v|^2_{H^\beta (D)}. \end{aligned}$$
(104)

The norms in (87) can be replaced by \(\Vert \cdot \Vert _{H^\beta (D)}\) and hence (87) follows from (103) and (104). \(\square \)

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Bonito, A., Lei, W. & Pasciak, J.E. Numerical approximation of the integral fractional Laplacian. Numer. Math. 142, 235–278 (2019). https://doi.org/10.1007/s00211-019-01025-x

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