Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains

Abstract

We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enable one to efficiently tackle the integral fractional Laplacian. In particular, we develop techniques for the treatment of the dense stiffness matrix including the computation of the entries, the efficient assembly and storage of a sparse approximation and the efficient solution of the resulting equations. The main idea consists of generalising proven techniques for the treatment of boundary integral equations to general fractional orders. Importantly, the approximation does not make any strong assumptions on the shape of the underlying domain and does not rely on any special structure of the matrix that could be exploited by fast transforms. We demonstrate the flexibility and performance of this approach in a couple of two-dimensional numerical examples.

Dedicated to Ian H. Sloan on the occasion of his 80th birthday.

Appendices

Appendix 1: Derivation of Expressions for Singular Contributions

The contributions $a^{K\times \tilde {K}}$ and a ^K×e as given in Eqs. (3) and (4) for touching elements K and $\tilde {K}$ contain removable singularities. In order to make these contributions amenable to numerical quadrature, the singularities need to be lifted. We outline the derivation for d = 2 dimensions.

The expression for $a^{K\times \tilde {K}}$ can be transformed into integrals over the reference element $\hat {K}$:

$$\displaystyle \begin{aligned} &a^{K\times\tilde{K}}(\phi_{i},\phi_{j}) \\ =& \frac{C(2,s)}{2} \int_{K} \; d \mathbf{x} \int_{\tilde{K}} \; d \mathbf{y} \frac{\left(\phi_{i}(\mathbf{x})-\phi_{i}(\mathbf{y})\right)\left(\phi_{j}(\mathbf{x})-\phi_{j}(\mathbf{y})\right)}{\left|\mathbf{x}-\mathbf{y}\right|{}^{2+2s}} \\ =& \frac{C(2,s)}{2}\frac{\left|K\right|}{\left|\hat{K}\right|} \frac{\left|\tilde{K}\right|}{\left|\hat{K}\right|} \int_{\hat{K}} \; d \hat{\mathbf{x}} \int_{\hat{K}} \; d \hat{\mathbf{y}} \frac{\left(\phi_{i}(\mathbf{x}\left(\hat{\mathbf{x}}\right))-\phi_{i}(\mathbf{y}\left(\hat{\mathbf{y}}\right))\right) \left(\phi_{j}(\mathbf{x}\left(\hat{\mathbf{x}}\right))-\phi_{j}(\mathbf{y}\left(\hat{\mathbf{y}}\right))\right)}{\left|\mathbf{x}\left(\hat{\mathbf{x}}\right)-\mathbf{y}\left(\hat{\mathbf{y}}\right)\right|{}^{2+2s}}. \end{aligned} $$

Similarly, by introducing the reference edge $\hat {e}$, we obtain

$$\displaystyle \begin{aligned} a^{K\times e}(\phi_{i},\phi_{j}) &= \frac{C(2,s)}{2s} \int_{K} \; d \mathbf{x} \int_{e} \; d \mathbf{y} \frac{\phi_{i}\left(\mathbf{x}\right) \phi_{j}\left(\mathbf{x}\right) ~ \mathbf{n}_{e}\cdot\left(\mathbf{x}-\mathbf{y}\right)}{\left|\mathbf{x}-\mathbf{y}\right|{}^{2+2s}} \\ &= \frac{C(2,s)}{2s} \frac{\left|K\right|}{\left|\hat{K}\right|} \frac{\left|e\right|}{\left|\hat{e}\right|} \int_{\hat{K}} \; d \hat{\mathbf{x}} \int_{\hat{e}} \; d \hat{\mathbf{y}} \frac{\phi_{i}\left(\mathbf{x}\left(\hat{\mathbf{x}}\right)\right) \phi_{j}\left(\mathbf{x}\left(\hat{\mathbf{x}}\right)\right) ~ \mathbf{n}_{e}\cdot\left(\mathbf{x}\left(\hat{\mathbf{x}}\right)-\mathbf{y}\left(\hat{\mathbf{y}}\right)\right)}{\left|\mathbf{x}\left(\hat{\mathbf{x}}\right)-\mathbf{y}\left(\hat{\mathbf{y}}\right)\right|{}^{2+2s}} \end{aligned} $$

for touching elements K and edges e. If K and $\tilde {K}$ or e have c ≥ 1 common vertices, and if we designate by λ _k, k = 0, …, 6 − c the barycentric coordinates of $K\cup \tilde {K}$ or K ∪ e respectively (cf. Fig. 17), we have

$$\displaystyle \begin{aligned} \lambda_{k(i)}(\hat{\mathbf{x}}) &= \phi_{i}\left(\mathbf{x}\left(\hat{\mathbf{x}}\right)\right), \end{aligned} $$

Fig. 17

Numbering of local nodes for touching triangular elements K and $\tilde {K}$ or element K and edge e. (a) $K\cap \tilde {K}=K$. (b) $K\cap \tilde {K}=$edge. (c) $K\cap \tilde {K}=$vertex. (d) K ∩ e = e. (e) K ∩ e = vertex

where k(i) is the local index on $K\cup \tilde {K}$ or K ∪ e of the global degree of freedom i. Moreover, we have

$$\displaystyle \begin{aligned} \mathbf{x}\left(\hat{\mathbf{x}}\right) - \mathbf{y}\left(\hat{\mathbf{y}}\right) &= \sum_{k=0}^{6-c} \lambda_{k}\left(\hat{\mathbf{x}}\right)\mathbf{x}_{k}-\sum_{k=0}^{6-c} \lambda_{k}\left(\hat{\mathbf{y}}\right)\mathbf{x}_{k} \\ &= \sum_{k=0}^{6-c} \left[\lambda_{k}\left(\hat{\mathbf{x}}\right) - \lambda_{k}\left(\hat{\mathbf{y}}\right)\right]\mathbf{x}_{k}. \end{aligned} $$

Here, x _k, k = 0, …, 6 − c are the vertices that span $K\cup \tilde {K}$ or K ∪ e respectively.

By setting

$$\displaystyle \begin{aligned} \psi_{k}\left(\hat{\mathbf{x}},\hat{\mathbf{y}}\right) := \lambda_{k}\left(\hat{\mathbf{x}}\right) - \lambda_{k}\left(\hat{\mathbf{y}}\right), \end{aligned} $$

we can therefore write

$$\displaystyle \begin{aligned} a^{K\times\tilde{K}}(\phi_{i},\phi_{j}) &= \frac{C(2,s)}{2}\frac{\left|K\right|}{\left|\hat{K}\right|} \frac{\left|\tilde{K}\right|}{\left|\hat{K}\right|} \int_{\hat{K}} \; d \hat{\mathbf{x}} \int_{\hat{K}} \; d \hat{\mathbf{y}} \frac{\psi_{k(i)}\left(\hat{\mathbf{x}},\hat{\mathbf{y}}\right) \psi_{k(j)}\left(\hat{\mathbf{x}},\hat{\mathbf{y}}\right)}{\left|\sum_{k=0}^{6-c}\psi_{k}\left(\hat{\mathbf{x}},\hat{\mathbf{y}}\right) \mathbf{x}_{k}\right|{}^{2+2s}}.\end{aligned} $$

By carefully splitting the integration domain $\hat {K}\times \hat {K}$ into L _c parts and applying a Duffy transformation to each part, the contributions can be rewritten into integrals over a unit hyper-cube, where the singularities are lifted.

$$\displaystyle \begin{aligned} \begin{array}{rcl} a^{K\times\tilde{K}}(\phi_{i},\phi_{j}) &\displaystyle =&\displaystyle \frac{C(2,s)}{2} \frac{\left|K\right|}{\left|\hat{K}\right|} \frac{\left|\tilde{K}\right|}{\left|\hat{K}\right|} \\ &\displaystyle &\displaystyle \quad \sum_{\ell=1}^{L_{c}} \int_{[0,1]^{4}} \; d \boldsymbol{\eta} ~ \Bar J^{(\ell,c)}\frac{\Bar \psi_{k(i)}^{(\ell,c)}\left(\boldsymbol{\eta}\right) \Bar \psi_{k(j)}^{(\ell,c)}\left(\boldsymbol{\eta}\right)}{\left|\sum_{k=0}^{2d-c}\Bar\psi_{k}^{(\ell,c)}\left(\boldsymbol{\eta}\right) \mathbf{x}_{k}\right|{}^{2+2s}}. {}\vspace{-3pt} \end{array} \end{aligned} $$

(16)

The details of this approach can be found in Chapter 5 of [22] for the interactions between K and $\tilde {K}$. We record the obtained expressions in this case.

• K and $\tilde {K}$ are identical, i.e. c = 3

  $$\displaystyle \begin{aligned} L_{3}=3,&& \Bar J^{(1,3)}=\Bar J^{(2,3)}=\Bar J^{(3,3)}= \eta_{0}^{3-2s}\eta_{1}^{2-2s}\eta_{2}^{1-2s},\end{aligned} $$
  $$\displaystyle \begin{aligned} \Bar\psi_{k}^{(1,3)} &= \begin{cases} -\eta_{3} \\ \eta_{3}-1 \\ 1 \end{cases} & \Bar\psi_{k}^{(2,3)} &= \begin{cases} -1 \\ 1-\eta_{3} \\ \eta_{3} \end{cases} & \Bar\psi_{k}^{(3,3)} &= \begin{cases} \eta_{3} \\ -1 \\ 1-\eta_{3} \end{cases}\end{aligned} $$

• K and $\tilde {K}$ share an edge, i.e. c = 2

  $$\displaystyle \begin{aligned} L_{2}=5, && \Bar J^{(1,2)}=\eta_{0}^{3-2s}\eta_{1}^{2-2s}, \\ && \Bar J^{(2,2)}=\Bar J^{(3,2)}=\Bar J^{(4,2)}=\Bar J^{(5,2)} = \eta_{0}^{3-2s}\eta_{1}^{2-2s}\eta_{2}\end{aligned} $$
  $$\displaystyle \begin{aligned} \Bar\psi_{k}^{(1,2)} &= \begin{cases} -\eta_{2} \\ 1-\eta_{3} \\ \eta_{3} \\ \eta_{2}-1 \end{cases} & \Bar\psi_{k}^{(2,2)} &= \begin{cases} -\eta_{2}\eta_{3} \\ \eta_{2}-1 \\ 1 \\ \eta_{2}\eta_{3}-\eta_{2} \end{cases} & \Bar\psi_{k}^{(3,2)} &= \begin{cases} \eta_{2} \\ \eta_{2}\eta_{3}-1 \\ 1-\eta_{2} \\ -\eta_{2}\eta_{3} \end{cases} \\ \Bar\psi_{k}^{(4,2)} &= \begin{cases} \eta_{2} \eta_{3} \\ 1-\eta_{2} \\ \eta_{2}-\eta_{2} \eta_{3} \\ -1 \end{cases} & \Bar\psi_{k}^{(5,2)} &= \begin{cases} \eta_{2}\eta_{3} \\ \eta_{2}-1 \\ 1-\eta_{2}\eta_{3} \\ -\eta_{2} \end{cases} \end{aligned} $$

• K and $\tilde {K}$ share a vertex, i.e. c = 1

  $$\displaystyle \begin{aligned} L_{1}=2, && \Bar J^{(1,1)}= \Bar J^{(2,1)}= \eta_{0}^{3-2s}\eta_{2}\end{aligned} $$
  $$\displaystyle \begin{aligned} \Bar\psi_{k}^{(1,1)} &= \begin{cases} \eta_{2}-1 \\ 1-\eta_{1} \\ \eta_{1} \\ \eta_{2}\eta_{3}-\eta_{2} \\ -\eta_{2}\eta_{3} \end{cases} & \Bar\psi_{k}^{(2,1)} &= \begin{cases} 1-\eta_{2} \\ \eta_{2}-\eta_{2} \eta_{3} \\ \eta_{2}\eta_{3} \\ \eta_{1}-1 \\ -\eta_{1} \end{cases}\end{aligned} $$

We notice that the contributions for identical elements only depend on η ₃, so that in fact only one-dimensional integrals need to be computed. Similarly, the cases of common edges or common vertices only require two and three dimensional integration.

In a similar fashion, the integration domain of a ^K×e can be split into several parts, so that the singularity can be lifted:

$$\displaystyle \begin{aligned} &a^{K\times e}(\phi_{i},\phi_{j}) \\ &= \frac{C(2,s)}{2s} \frac{\left|K\right|}{\left|\hat{K}\right|} \frac{\left|e\right|}{\left|\hat{e}\right|} \int_{[0,1]^{3}} \; d \boldsymbol{\eta} \Bar J^{(\ell,c)} \frac{\phi_{k(i)}^{(\ell,c)}\left(\boldsymbol{\eta}\right) \phi_{k(j)}^{(\ell,c)}\left(\boldsymbol{\eta}\right) ~ \sum_{k=0}^{5-c}\Bar\psi_{k}^{(\ell,c)}\left(\boldsymbol{\eta}\right) \mathbf{n}_{e}\cdot\mathbf{x}_{k}}{\left|\sum_{k=0}^{5-c}\Bar\psi_{k}^{(\ell,c)}\left(\boldsymbol{\eta}\right) \mathbf{x}_{k}\right|{}^{2+2s}}.\end{aligned} $$

Here, $\phi _{k}^{(\ell ,c)}$ are the expressions for the local shape functions under the Duffy transformations. The obtained expressions are

• e is an edge of K, i.e. c = 2

  $$\displaystyle \begin{aligned} L_{2}=3, && \Bar J^{(1,2)}=\Bar J^{(2,2)}=\Bar J^{(3,2)}=\eta_{0}^{-2s}\left(1-\eta_{0}\right),\end{aligned} $$
  $$\displaystyle \begin{aligned} \phi_{k}^{(1,2)} &= \begin{cases} 1-\eta_{0}-\eta_{2}+\eta_{0}\eta_{2} \\ \eta_{0}+\eta_{2}-\eta_{0}\eta_{1}-\eta_{0}\eta_{1} \\ \eta_{0}\eta_{1} \end{cases} & \phi_{k}^{(2,2)} &= \begin{cases} 1-\eta_{0}-\eta_{2}+\eta_{0}\eta_{2} \\ \eta_{2}-\eta_{0}\eta_{2} \\ \eta_{0} \end{cases}\\ \phi_{k}^{(3,2)} &= \begin{cases} 1-\eta_{2}+\eta_{0}\eta_{2}-\eta_{0}\eta_{1} \\ \eta_{2}-\eta_{0}\eta_{2} \\ \eta_{0}\eta_{1} \end{cases}\end{aligned} $$
  $$\displaystyle \begin{aligned} \Bar\psi_{k}^{(1,2)} &= \begin{cases} -1 \\ 1-\eta_{1} \\ \eta_{1} \end{cases} & \Bar\psi_{k}^{(2,2)} &= \begin{cases} -\eta_{1} \\ \eta_{1}-1 \\ 1 \end{cases} & \Bar\psi_{k}^{(3,2)} &= \begin{cases} 1-\eta_{1} \\ -1 \\ \eta_{1} \end{cases} \end{aligned} $$

  We notice that for s ≥ 1∕2, the integrand still contains a singularity. In this case, the finite element space V _h does not include the degrees of freedom on the boundary. For the interaction of the single degree of freedom that is not on the boundary (k = 2), we obtain

  $$\displaystyle \begin{aligned} \Bar J^{(1,2)}=\Bar J^{(2,2)}=\Bar J^{(3,2)}=\eta_{0}^{2-2s}\left(1-\eta_{0}\right), \end{aligned} $$
  $$\displaystyle \begin{aligned} \phi_{2}^{(1,2)} &= \eta_{1} & \phi_{2}^{(2,2)} &= 1 & \phi_{2}^{(3,2)} &= \eta_{1} \end{aligned} $$

  and $\Bar \psi _{2}^{\ell ,c}$ as above.

• K and e share a vertex, i.e. c = 1

  $$\displaystyle \begin{aligned} L_{1}=2, && \Bar J^{(1,1)} = \eta_{0}^{1-2s}, \Bar J^{(2,1)}= \eta_{0}^{1-2s}\eta_{1} \end{aligned} $$
  $$\displaystyle \begin{aligned} \Bar\psi_{k}^{(1,1)} &= \begin{cases} \eta_{2}-1 \\ 1-\eta_{1} \\ \eta_{1} \\ -\eta_{2} \end{cases} & \Bar\psi_{k}^{(2,1)} &= \begin{cases} 1-\eta_{1} \\ \eta_{1}-\eta_{1} \eta_{2} \\ \eta_{1}\eta_{2} \\ -1 \end{cases} \end{aligned} $$

Appendix 2: Proof of Consistency Error Due to Quadrature

Next, we give the proof for the consistency error of the quadrature approximation first stated in Sect. 4.2.

Theorem 3

For d = 2, let $\mathcal {I}_{K}$ index the degrees of freedom on $K\in \mathcal {P}_{h}$ , and define $\mathcal {I}_{K\times \tilde {K}}:=\mathcal {I}_{K}\cup \mathcal {I}_{\tilde {K}}$ . Let k _T (respectively k _T,∂ ) be the quadrature order used for touching pairs $K\times \tilde {K}$ (respectively K × e), and let $k_{NT}\left (K,\tilde {K}\right )$ (respectively $k_{NT,\partial }\left (K,e\right )$ ) be the quadrature order used for pairs that have empty intersection. Denote the resulting approximation to the bilinear form $a\left (\cdot ,\cdot \right )$ by $a_{Q}\left (\cdot ,\cdot \right )$ . Then the consistency error due to quadrature is bounded by

where the errors are given by

$$\displaystyle \begin{aligned} E_{T}&=h^{-2-2s}\rho_{1}^{-2k_{T}},\\ E_{NT} &= \max_{K,\tilde{K}\in\mathcal{P}_{h}, \overline{K}\cap\overline{\tilde{K}}=\emptyset} h^{-2}d_{K,\tilde{K}}^{-2s}\left(\rho_{2}\frac{d_{K,\tilde{K}}}{h}\right)^{-2k_{NT}\left(K,\tilde{K}\right)}, \\ E_{T,\partial} &= h^{-1-2s}\rho_{3}^{-2k_{T,\partial}},\\ E_{NT,\partial} &= \max_{K\in\mathcal{P}_{h}, e\in\mathcal{P}_{h,\partial}, \overline{K}\cap\overline{e}=\emptyset} h^{-1}d_{K,e}^{-2s}\left(\rho_{4}\frac{d_{K,e}}{h}\right)^{-2k_{NT,\partial}\left(K,e\right)}, \end{aligned} $$

$d_{K,\tilde {K}}:=\inf _{\mathbf {x}\in K, \mathbf {y}\in \tilde {K}}\left |\mathbf {x}-\mathbf {y}\right |$ , $d_{K,e}:=\inf _{\mathbf {x}\in K, \mathbf {y}\in e}\left |\mathbf {x}-\mathbf {y}\right |$ , and ρ _j > 1, j = 1, 2, 3, 4, are constants.

Proof

Let the quadrature rules for the pairs $K\times \tilde {K}$ and K × e be denoted by $a^{K\times \tilde {K}}_{Q}\left (\cdot ,\cdot \right )$ and $a^{K\times e}_{Q}\left (\cdot ,\cdot \right )$. Set

$$\displaystyle \begin{aligned} E_{K\times\tilde{K}}^{i,j} &= a^{K\times\tilde{K}}\left(\phi_{i},\phi_{j}\right)-a_{Q}^{K\times\tilde{K}}\left(\phi_{i},\phi_{j}\right),\\ E_{K\times e}^{i,j} &= a^{K\times e}\left(\phi_{i},\phi_{j}\right)-a_{Q}^{K\times e}\left(\phi_{i},\phi_{j}\right). \end{aligned} $$

For u, v ∈ V _h, we set

$$\displaystyle \begin{aligned} E_{K\times\tilde{K}}(u,v)&= \sum_{i\in\mathcal{I}_{K\times\tilde{K}}} \sum_{j\in\mathcal{I}_{K\times\tilde{K}}} u_{i}v_{j}E_{K\times\tilde{K}}^{i,j}, \\ E_{K\times e}(u,v)&= \sum_{i\in\mathcal{I}_{K}} \sum_{j\in\mathcal{I}_{K}} u_{i}v_{j}E_{K\times e}^{i,j} \end{aligned} $$

so that

$$\displaystyle \begin{aligned} \left|E_{K\times\tilde{K}}(u,v)\right| &\leq \left(\max_{i,j}\left|E_{K\times\tilde{K}}^{i,j}\right|\right) \sum_{i\in\mathcal{I}_{K\times\tilde{K}}} \left|u_{i}\right| \sum_{j\in\mathcal{I}_{K\times\tilde{K}}} \left|v_{j}\right| \\ &\leq \left(\max_{i,j}\left|E_{K\times\tilde{K}}^{i,j}\right|\right) \left|\mathcal{I}_{K\times\tilde{K}}\right| \sqrt{\sum_{i\in\mathcal{I}_{K\times\tilde{K}}} \left|u_{i}\right|{}^{2}} \sqrt{\sum_{j\in\mathcal{I}_{K\times\tilde{K}}} \left|v_{j}\right|{}^{2}}, \\ \left|E_{K\times e}(u,v)\right| &\leq \left(\max_{i,j}\left|E_{K, e}^{i,j}\right|\right) \sum_{i\in\mathcal{I}_{K}} \left|u_{i}\right| \sum_{j\in\mathcal{I}_{K}} \left|v_{j}\right| \\ &\leq \left(\max_{i,j}\left|E_{K, e}^{i,j}\right|\right) \left|\mathcal{I}_{K}\right| \sqrt{\sum_{i\in\mathcal{I}_{K}} \left|u_{i}\right|{}^{2}} \sqrt{\sum_{j\in\mathcal{I}_{K}} \left|v_{j}\right|{}^{2}} \end{aligned} $$

Since

$$\displaystyle \begin{aligned} \sum_{i\in\mathcal{I}_{K\times\tilde{K}}} \left|u_{i}\right|{}^{2} &\leq C\left[h_{K}^{-d}\int_{K}u^{2} + h_{\tilde{K}}^{-d}\int_{\tilde{K}}u^{2}\right], \\ \sum_{i\in\mathcal{I}_{K}} \left|u_{i}\right|{}^{2} &\leq C h_{K}^{-d}\int_{K}u^{2}, \end{aligned} $$

we find

Because

and

we obtain

For d = 2, using Theorem 6 stated below permits to conclude. □

Theorem 6 ([22], Theorems 5.3.23 and 5.3.24)

If K and $\tilde {K}$ (K and e) are touching elements, then

$$\displaystyle \begin{aligned} \left|E_{K\times\tilde{K}}^{i,j}\right|&\leq Ch^{2-2s}\rho_{1}^{-2k_{T}},\\ \left|E_{K\times e}^{i,j}\right| &\leq Ch^{2-2s}\rho_{3}^{-2k_{T,\partial}}, \end{aligned} $$

where ρ ₁, ρ ₃ > 1 and k _T , k _T,∂ are the quadrature orders in every dimension of Eqs.(5) and (6).

If K and $\tilde {K}$ (K and e) are not touching, then

$$\displaystyle \begin{aligned} \left|E_{K\times\tilde{K}}^{i,j}\right| &\leq C h^{2} d_{K,\tilde{K}}^{-2s}\tilde{\rho}_{2}\left(K,\tilde{K}\right)^{-2k_{NT}},\\ \left|E_{K\times e}^{i,j}\right| &\leq C h^{2} d_{K,e}^{-2s}\tilde{\rho}_{4}\left(K,e\right)^{-2k_{NT.\partial}}, \end{aligned} $$

where $d_{K,\tilde {K}}:=dist(K,\tilde {K})$ , d _K,e := dist(K, e), $\tilde {\rho }_{2}(K,\tilde {K}):=\rho _{2}\max \left \{\frac {d_{K,\tilde {K}}}{h},1\right \}$ , and $\tilde {\rho }_{4}(K,\tilde {K}):=\rho _{4}\max \left \{\frac {d_{K,e}}{h},1\right \}$ , with ρ ₂, ρ ₄ > 1, and k _NT , k _NT,∂ are the quadrature order in every dimension of Eqs.(3) and (4).