Error analysis of finite element method for nonlocal diffusion model

We consider the Poisson equation with Neumann boundary condition.

	$\displaystyle-\Delta u(\bm{x})+u(\bm{x})$	$\displaystyle=f(\bm{x}),\quad\bm{x}\in\Omega\subset\mathbb{R}^{d},$		(1.1)
	$\displaystyle\frac{\partial u}{\partial\mathbf{n}}(\bm{x})$	$\displaystyle=g(\bm{x}),\quad\bm{x}\in\partial\Omega.$

A nonlocal counterpart of Poisson equation is given as follows

$\displaystyle\frac{1}{\delta^{2}}\int_{\Omega}R_{\delta}(\bm{x},\bm{y})(u(\bm{x})-u(\bm{y}))\mathrm{d}\bm{y}+\int_{\Omega}\bar{R}_{\delta}(\bm{x},\bm{y})u(\bm{y})\mathrm{d}\bm{y}=\int_{\Omega}\bar{R}_{\delta}(\bm{x},\bm{y})f(\bm{y})\mathrm{d}\bm{y}+2\int_{\partial\Omega}\bar{R}_{\delta}(\bm{x},\bm{y})g(\bm{y})\mathrm{d}\tau_{\bm{y}},\quad$

(1.2)

$\displaystyle R_{\delta}(\bm{x},\bm{y})=C_{\delta}R\left(\frac{\|\bm{x}-\bm{y}\|^{2}}{4\delta^{2}}\right),\quad\bar{R}_{\delta}(\bm{x},\bm{y})=C_{\delta}\bar{R}\left(\frac{\|\bm{x}-\bm{y}\|^{2}}{4\delta^{2}}\right){,}$

(1.3)

where

$\displaystyle\bar{R}(r)=\int_{r}^{+\infty}R(s)\mathrm{d}s=\int_{r}^{1}R(s)\mathrm{d}s,$

(1.4)

which satisfies obviously

$$\bar{R}^{\prime}(r)=\frac{d}{dr}\bar{R}(r)=-R(r),\;\forall r\in\mathbb{R}^{+},\quad\text{and}\quad\bar{R}(r)=0,\;\forall r>1.$$

The constant $C_{\delta}=\alpha_{d}\delta^{-d}$ in (1.3) is a normalization factor so that

$\displaystyle\int_{\mathbb{R}^{d}}\bar{R}_{\delta}(\bm{x},\bm{y})\mathrm{d}\bm{y}=\alpha_{d}S_{d}\int_{0}^{1}\bar{R}(\frac{r^{2}}{4})r^{d-1}\mathrm{d}r=1,$

(1.5)

with $S_{d}$ denotes area of the unit sphere in $\mathbb{R}^{d}$.

$R$ is a kernel function which satisfies following conditions:

• (a)

  (regularity) $R\in C^{1}[0,1]$;

• (b)

  (positivity and compact support) $R(r)\geq 0$ and $R(r)=0$ for $\forall r>1$;

• (c)

  (nondegeneracy) $\exists\gamma_{0}>0$ so that $R(r)\geq\gamma_{0}$ for $0\leq r\leq\frac{1}{2}$.

For the truncation error of the nonlocal model (1.2), we have following theorem [SS17].

Let $\Omega_{h}$ be a polyhedral approximation of $\Omega$, and $\mathcal{T}_{h}$ be the mesh associated with $\Omega_{h}$, where $h=\max_{T\in\mathcal{T}_{h}}\mbox{diam}(T)$ is the maximum diameter, $\rho=\min_{T\in\mathcal{T}_{h}}\rho(T)$ where $\rho(T)$ denotes the radius of the inscribed ball of $T$. We focus on the continuous $k$-th order finite element space defined on $\Omega_{h}$, i.e.

$$S_{h}=\{v_{h}\in C^{0}(\Omega_{h}):v_{h}|_{T}\in\mathbb{P}_{k}(T),\quad\forall T\in\Omega_{h}\}.$$

(1.8)

$\mathbb{P}_{k}(T)$ denotes the set of all $k$-th order polynomials in $T$.

The finite element discretization of the nonlocal diffusion model is to find $u_{h}\in S_{h}$ such that

$\displaystyle\left<L_{\delta}u_{h},v_{h}\right>_{\Omega_{h}}=\left<\bar{f},v_{h}\right>_{\Omega_{h}},\quad\forall v_{h}\in S_{h},$

with $\bar{f}(\bm{x})=\int_{\Omega}\bar{R}_{\delta}(\bm{x},\bm{y})f(\bm{y})\mathrm{d}\bm{y}+2\int_{\partial\Omega}\bar{R}_{\delta}(\bm{x},\bm{y})g(\bm{y})\mathrm{d}\tau_{\bm{y}}$ and

$\displaystyle L_{\delta}u(\bm{x})=\frac{1}{\delta^{2}}\int_{\Omega}R_{\delta}(\bm{x},\bm{y})(u(\bm{x})-u(\bm{y}))\mathrm{d}\bm{y}+\int_{\Omega}\bar{R}_{\delta}(\bm{x},\bm{y})u(\bm{y})\mathrm{d}\bm{y}.$

(1.9)

$<\cdot,\cdot>_{\Omega_{h}}$ denotes the inner product in $\Omega_{h}$,

$\displaystyle\left<u,v\right>_{\Omega_{h}}=\int_{\Omega_{h}}u(\bm{x})v(\bm{x})\mathrm{d}\bm{x}$

(1.10)

For the sake of simplification, we focus on the case $\Omega=\Omega_{h}$ which means that we do not consider the error from domain approximation. In the rest of the paper, we do not distinguish $\Omega$ and $\Omega_{h}$.

Let $e_{h}=u^{*}-u_{h}$, $u^{*}$ is the solution of Poisson equation (1.1). From Theorem 1.1, we have

$\displaystyle\left<L_{\delta}e_{h},v_{h}\right>_{\Omega}=\left<r_{in}+r_{bd},v_{h}\right>_{\Omega},\quad\forall v_{h}\in S_{h}.$

(1.11)

Now, we introduce some notations and technical results which will be used later.

$\displaystyle S_{\delta}u(\bm{x})=\frac{1}{w_{\delta}(\bm{x})}\int_{\Omega}R_{\delta}(\bm{x},\bm{y})u(\bm{y})\mathrm{d}\bm{y},\qquad w_{\delta}(\bm{x})=\int_{\Omega}R_{\delta}(\bm{x},\bm{y})\mathrm{d}\bm{y}$

(1.12)

and

$\displaystyle E_{\delta}(u)^{2}=\frac{1}{2\delta^{2}}\int_{\Omega}\int_{\Omega}R_{\delta}(\bm{x},\bm{y})(u(\bm{x})-u(\bm{y}))^{2}\mathrm{d}\bm{x}\mathrm{d}\bm{y}+\int_{\Omega}\int_{\Omega}\bar{R}_{\delta}(\bm{x},\bm{y})u(\bm{x})u(\bm{y})\mathrm{d}\bm{x}\mathrm{d}\bm{y}=\left<L_{\delta}u,u\right>_{\Omega}$

(1.13)

And we also have

• 1.

  Proof can be found in [SS17]

  $\displaystyle\|u\|_{L^{2}(\Omega)}\leq CE_{\delta}(u),\quad\|\nabla(S_{\delta}u)\|_{L^{2}(\Omega)}\leq CE_{\delta}(u),\quad E_{\delta}(u)\leq\frac{C}{\delta}\|u\|_{L^{2}(\Omega)}$

  (1.14)

• 2.

  	$\displaystyle\left|\left<L_{\delta}u,v\right>_{\Omega}\right|=$	$\displaystyle\frac{1}{2\delta^{2}}\left|\int_{\Omega}\int_{\Omega}R_{\delta}(\bm{x},\bm{y})(u(\bm{x})-u(\bm{y}))(v(\bm{x})-v(\bm{y}))\mathrm{d}\bm{y}\mathrm{d}\bm{x}\right|$
  	$\displaystyle\leq$	$\displaystyle\frac{1}{2\delta^{2}}\left(\int_{\Omega}\int_{\Omega}R_{\delta}(\bm{x},\bm{y})(u(\bm{x})-u(\bm{y}))^{2}\mathrm{d}\bm{y}\mathrm{d}\bm{x}\right)^{1/2}\left(\int_{\Omega}\int_{\Omega}R_{\delta}(\bm{x},\bm{y})(v(\bm{x})-v(\bm{y}))^{2}\mathrm{d}\bm{y}\mathrm{d}\bm{x}\right)^{1/2}$
  	$\displaystyle\leq$	$\displaystyle CE_{\delta}(u)\|v\|_{H^{1}(\Omega)}$		(1.15)

• 3.

  	$\displaystyle\|L_{\delta}u\|_{L_{2}(\Omega)}^{2}=$	$\displaystyle\int_{\Omega}\frac{1}{\delta^{4}}\left|\int_{\Omega}R_{\delta}(\bm{x},\bm{y})(u(\bm{x})-v(\bm{y}))\mathrm{d}\bm{y}\right|^{2}\mathrm{d}\bm{x}$
  	$\displaystyle\leq$	$\displaystyle\frac{C}{\delta^{4}}\int_{\Omega}\int_{\Omega}R_{\delta}(\bm{x},\bm{y})(u(\bm{x})-u(\bm{y}))^{2}\mathrm{d}\bm{y}\mathrm{d}\bm{x}$
  	$\displaystyle\leq$	$\displaystyle\frac{C}{\delta^{2}}E_{\delta}(u)^{2}$		(1.16)

For the boundary error $r_{bd}$, we have another estimate.

$I_{h}$ denotes the projection operator onto $S_{h}$. Then, we have

	$\displaystyle\left<L_{\delta}e_{h},e_{h}\right>_{\Omega}=$	$\displaystyle\left<L_{\delta}e_{h},u^{*}-I_{h}u^{*}\right>_{\Omega}+\left<r_{in}+r_{bd},I_{h}u^{*}-u_{h}\right>_{\Omega}$
	$\displaystyle\leq$	$\displaystyle E_{\delta}(e_{h})\|u^{*}-I_{h}u^{*}\|_{H^{1}(\Omega)}+\|r_{in}\|_{L^{2}(\Omega)}\|I_{h}u^{*}-u_{h}\|_{L^{2}(\Omega)}+C\delta\|u^{*}\|_{H^{3}(\Omega)}E_{\delta}(I_{h}u^{*}-u_{h})$
	$\displaystyle\leq$	$\displaystyle Ch^{k}E_{\delta}(e_{h})\|u^{*}\|_{H^{k+1}(\Omega)}+C\delta\|u^{*}\|_{H^{3}(\Omega)}E_{\delta}(I_{h}u^{*}-u_{h})$
	$\displaystyle\leq$	$\displaystyle Ch^{k}E_{\delta}(e_{h})\|u^{*}\|_{H^{k+1}(\Omega)}+C\delta\|u^{*}\|_{H^{3}(\Omega)}E_{\delta}(u^{*}-I_{h}u^{*})+C\delta\|u^{*}\|_{H^{3}(\Omega)}E_{\delta}(e_{h})$
	$\displaystyle\leq$	$\displaystyle Ch^{k}E_{\delta}(e_{h})\|u^{*}\|_{H^{k+1}(\Omega)}+C\delta\|u^{*}\|_{H^{3}(\Omega)}\|u^{*}-I_{h}u^{*}\|_{H^{1}({\Omega})}+C\delta\|u^{*}\|_{H^{3}(\Omega)}E_{\delta}(e_{h})$
	$\displaystyle\leq$	$\displaystyle Ch^{k}E_{\delta}(e_{h})\|u^{*}\|_{H^{k+1}(\Omega)}+C\delta h^{k}\|u^{*}\|_{H^{3}(\Omega)}\|u^{*}\|_{H^{k+1}(\Omega)}+C\delta\|u^{*}\|_{H^{3}(\Omega)}E_{\delta}(e_{h})$

For second and third line, we use Theorem 1.1, Theorem 1.2 and the classical result that

$$\|u^{*}-I_{h}u^{*}\|_{H^{1}(\Omega)}\leq Ch^{k}\|u^{*}\|_{H^{k+1}(\Omega)}.$$

Then, we can get

$\displaystyle E_{\delta}(e_{h})^{2}\leq$

$\displaystyle Ch^{k}E_{\delta}(e_{h})\|u^{*}\|_{H^{k+1}(\Omega)}+C\delta h^{k}\|u^{*}\|_{H^{3}(\Omega)}\|u^{*}\|_{H^{k+1}(\Omega)}+C\delta\|u^{*}\|_{H^{3}(\Omega)}E_{\delta}(e_{h})$

which implies that

$\displaystyle E_{\delta}(e_{h})\leq C\left(h^{k}+\delta\right)\|u^{*}\|_{H^{\max\{k+1,3\}}(\Omega)}.$

This gives the $L^{2}$ norm of the error,

$\displaystyle\|e_{h}\|_{L^{2}({\Omega})}\leq CE_{\delta}(e_{h})\leq C\left(h^{k}+\delta\right)\|u^{*}\|_{H^{\max\{k+1,3\}}(\Omega)}.$

Moreover, we can get $H^{1}$ estimate of $u^{*}-S_{\delta}u_{h}$.

$\displaystyle\|u^{*}-S_{\delta}u_{h}\|_{H^{1}(\Omega)}=\|u^{*}-S_{\delta}u^{*}\|_{H^{1}(\Omega)}+\|S_{\delta}e_{h}\|_{H^{1}(\Omega)}$

(2.1)

The second term is easy to bound, since

$\displaystyle\|S_{\delta}e_{h}\|_{H^{1}(\Omega)}\leq CE_{\delta}(e_{h})$

(2.2)

To bound first term, we need more calculation.

		$\displaystyle\|u^{*}-S_{\delta}u^{*}\|_{L^{2}(\Omega)}^{2}$
	$\displaystyle=$	$\displaystyle\int_{\Omega}\frac{1}{w_{\delta}^{2}(\bm{x})}\left(\int_{\Omega}R_{\delta}(\bm{x},\bm{y})(u^{*}(\bm{x})-u^{*}(\bm{y}))\mathrm{d}\bm{y}\right)^{2}\mathrm{d}\bm{x}$
	$\displaystyle\leq$	$\displaystyle C\int_{\Omega}\int_{\Omega}R_{\delta}(\bm{x},\bm{y})(u^{*}(\bm{x})-u^{*}(\bm{y}))^{2}\mathrm{d}\bm{y}\mathrm{d}\bm{x}\leq C\delta^{2}\|u^{*}\|_{H^{1}(\Omega)}^{2}$		(2.3)

and

	$\displaystyle\nabla(u^{*}-S_{\delta}u^{*})(\bm{x})=$	$\displaystyle\nabla u^{*}(\bm{x})-\nabla\left(\frac{1}{w_{\delta}(\bm{x})}\int_{\Omega}R_{\delta}(\bm{x},\bm{y})u^{*}(\bm{y})\mathrm{d}\bm{y}\right)$
	$\displaystyle=$	$\displaystyle\nabla u^{*}(\bm{x})-\frac{1}{w_{\delta}(\bm{x})}\int_{\Omega}R_{\delta}(\bm{x},\bm{y})\nabla u^{*}(\bm{y})\mathrm{d}\bm{y}$
		$\displaystyle+\frac{1}{w^{2}_{\delta}(\bm{x})}\int_{\partial\Omega}\int_{\Omega}R_{\delta}(\bm{x},\mathbf{z})R_{\delta}(\bm{x},\bm{y})(u^{*}(\mathbf{z})-u^{*}(\bm{y}))\mathrm{d}\bm{y}\mathrm{d}S_{\mathbf{z}}$
	$\displaystyle=$	$\displaystyle\frac{1}{w_{\delta}(\bm{x})}\int_{\Omega}R_{\delta}(\bm{x},\bm{y})\left(\nabla u^{*}(\bm{x})-\nabla u^{*}(\bm{y})\right)\mathrm{d}\bm{y}$
		$\displaystyle+\frac{1}{w^{2}_{\delta}(\bm{x})}\int_{\partial\Omega}\int_{\Omega}R_{\delta}(\bm{x},\mathbf{z})R_{\delta}(\bm{x},\bm{y})(u^{*}(\mathbf{z})-u^{*}(\bm{y}))\mathrm{d}\bm{y}\mathrm{d}S_{\mathbf{z}}$		(2.4)

Notice that

$\displaystyle\left\|\frac{1}{w_{\delta}(\bm{x})}\int_{\Omega}R_{\delta}(\bm{x},\bm{y})\left(\nabla u^{*}(\bm{x})-\nabla u^{*}(\bm{y})\right)\mathrm{d}\bm{y}\right\|_{L^{2}(\Omega)}\leq C\delta\|u^{*}\|_{H^{2}(\Omega)}$

(2.5)

and

		$\displaystyle\left\|\frac{1}{w^{2}_{\delta}(\bm{x})}\int_{\partial\Omega}\int_{\Omega}R_{\delta}(\bm{x},\mathbf{z})R_{\delta}(\bm{x},\bm{y})(u^{*}(\mathbf{z})-u^{*}(\bm{y}))\mathrm{d}\bm{y}\mathrm{d}S_{\mathbf{z}}\right\|_{L^{2}(\Omega)}^{2}$
	$\displaystyle\leq$	$\displaystyle C\int_{\Omega}\int_{\partial\Omega}\int_{\Omega}R_{\delta}(\bm{x},\mathbf{z})R_{\delta}(\bm{x},\bm{y})(u^{*}(\mathbf{z})-u^{*}(\bm{y}))^{2}\mathrm{d}\bm{y}\mathrm{d}S_{\mathbf{z}}\mathrm{d}\bm{x}$
	$\displaystyle\leq$	$\displaystyle C\int_{\Omega}\int_{\partial\Omega}\int_{\Omega}R_{\delta}\left(\bm{x},\frac{\bm{y}+\mathbf{z}}{2}\right)R_{\delta}(\bm{y},\mathbf{z})(u^{*}(\mathbf{z})-u^{*}(\bm{y}))^{2}\mathrm{d}\bm{y}\mathrm{d}S_{\mathbf{z}}\mathrm{d}\bm{x}$
	$\displaystyle\leq$	$\displaystyle C\int_{\partial\Omega}\int_{\Omega}R_{\delta}(\bm{y},\mathbf{z})(u^{*}(\mathbf{z})-u^{*}(\bm{y}))^{2}\mathrm{d}\bm{y}\mathrm{d}S_{\mathbf{z}}$
	$\displaystyle\leq$	$\displaystyle C\delta^{2}\|u^{*}\|_{H^{2}(\Omega)}^{2}$		(2.6)

Combining all above calculation together, we have

$\displaystyle\|u^{*}-S_{\delta}u_{h}\|_{H^{1}(\Omega)}\leq C\left(h^{k}+\delta\right)\|u^{*}\|_{H^{\max\{k+1,3\}}(\Omega)}$

In the analysis above, we need to require that $h/\rho$ is bounded. For the irregular mesh without boundness of $h/\rho$, we can also get error estimate. First,

	$\displaystyle\left<L_{\delta}e_{h},e_{h}\right>_{\Omega}=$	$\displaystyle\left<L_{\delta}e_{h},u^{*}-I_{h}u^{*}\right>_{\Omega}+\left<L_{\delta}e_{h},I_{h}u^{*}-u_{h}\right>_{\Omega}$
	$\displaystyle=$	$\displaystyle\left<L_{\delta}e_{h},u^{*}-I_{h}u^{*}\right>_{\Omega}+\left<r_{in}+r_{bd},I_{h}u^{*}-u_{h}\right>_{\Omega}$
	$\displaystyle\leq$	$\displaystyle\|L_{\delta}e_{h}\|_{L^{2}(\Omega)}\|u^{*}-I_{h}u^{*}\|_{L^{2}(\Omega)}+\|r_{in}\|_{L^{2}(\Omega)}\|I_{h}u^{*}-u_{h}\|_{L^{2}(\Omega)}+C\delta\|u^{*}\|_{H^{3}(\Omega)}E_{\delta}(I_{h}u^{*}-u_{h})$
	$\displaystyle\leq$	$\displaystyle\frac{Ch^{k+1}}{\delta}E_{\delta}(e_{h})\|u^{*}\|_{H^{k+1}(\Omega)}+C\delta\|u^{*}\|_{H^{3}(\Omega)}E_{\delta}(I_{h}u^{*}-u_{h})$
	$\displaystyle\leq$	$\displaystyle\frac{Ch^{k+1}}{\delta}E_{\delta}(e_{h})\|u^{*}\|_{H^{k+1}(\Omega)}+C\delta\|u^{*}\|_{H^{3}(\Omega)}E_{\delta}(u^{*}-I_{h}u^{*})+C\delta\|u^{*}\|_{H^{3}(\Omega)}E_{\delta}(e_{h})$
	$\displaystyle\leq$	$\displaystyle\frac{Ch^{k+1}}{\delta}E_{\delta}(e_{h})\|u^{*}\|_{H^{k+1}(\Omega)}+C\|u^{*}\|_{H^{3}(\Omega)}\|u^{*}-I_{h}u^{*}\|_{L^{2}(\Omega)}+C\delta\|u^{*}\|_{H^{3}(\Omega)}E_{\delta}(e_{h})$
	$\displaystyle\leq$	$\displaystyle\frac{Ch^{k+1}}{\delta}E_{\delta}(e_{h})\|u^{*}\|_{H^{k+1}(\Omega)}+Ch^{k+1}\|u^{*}\|_{H^{3}(\Omega)}\|u^{*}\|_{H^{k+1}(\Omega)}+C\delta\|u^{*}\|_{H^{3}(\Omega)}E_{\delta}(e_{h})$

The third and fourth line are from Theorem 1.1, Theorem 1.2 and (3.) and the fact that

$$\|u^{*}-I_{h}u^{*}\|_{L^{2}(\Omega)}\leq Ch^{k+1}\|u^{*}\|_{H^{k+1}(\Omega)}.$$

Here, $C$ is independent on $h/\rho$. The sixth line is from (1.14), i.e.

$$E_{\delta}(u^{*}-I_{h}u^{*})\leq\frac{C}{\delta}\|u^{*}-I_{h}u^{*}\|_{L^{2}(\Omega)}.$$

Then, using the definition of $E_{\delta}(u)$,(1.13), it is easy to get

$\displaystyle E_{\delta}(e_{h})\leq C\left(\frac{h^{k+1}}{\delta}+\delta\right)\|u^{*}\|_{H^{\max\{k+1,3\}}(\Omega)}$

which gives

$\displaystyle\|e_{h}\|_{L^{2}({\Omega})}\leq CE_{\delta}(e_{h})\leq C\left(\frac{h^{k+1}}{\delta}+\delta\right)\|u^{*}\|_{H^{\max\{k+1,3\}}(\Omega)}$

Under the same argument as that in previous section, we can get

$\displaystyle\|u^{*}-S_{\delta}u_{h}\|_{H^{1}(\Omega)}\leq C\left(\frac{h^{k+1}}{\delta}+\delta\right)\|u^{*}\|_{H^{\max\{k+1,3\}}(\Omega)}$

For nonlocal diffusion model, DG discretization is actually same with conformal finite element discretization since the nonlocal term provides the interaction between elements automatically. The only difference is that now we consider nonconformal $k$-th order finite element space defined on $\Omega_{h}$, i.e.

$$D_{h}=\{v_{h}:v_{h}|_{T}\in\mathbb{P}_{k}(T),\quad\forall T\in\Omega_{h}\}.$$

(4.1)

$\mathbb{P}_{k}(T)$ denotes the set of all $k$-th order polynomials in $T$.

The DG discretization of the nonlocal diffusion model is to find $u_{h}\in D_{h}$ such that

$\displaystyle\left<L_{\delta}u_{h},v_{h}\right>_{\Omega_{h}}=\left<\bar{f},v_{h}\right>_{\Omega_{h}},\quad\forall v_{h}\in D_{h},$

Error analysis in previous section for conformal finite element method also holds for DG method.

We analysis the error of finite element method for nonlocal diffusion model. Our results show that finite element method for nonlocal diffusion model is asymptotic preserving with shape regular mesh. For irregular mesh, the error is bounded by $O(\frac{h^{k+1}}{\delta}+\delta)$.