Finite volume methods for a Keller–Segel system: discrete energy, error estimates and numerical blow-up analysis

Abstract

We consider the finite volume approximation for a non-linear parabolic-elliptic system, which describes the aggregation of slime molds resulting from their chemotactic features, called a simplified Keller–Segel system. First, we present a linear finite volume scheme that satisfies both positivity and mass conservations, which are important features of the original system. We derive some inequalities on the discrete free energy. Then, under some assumptions on the regularity of solution, admissible mesh and a priori estimates of the discrete solution, we establish error estimates in $L^p$ norm with a suitable $p>2$ for the two dimensional case. In the last part of this paper, we restrict our attention to the radially symmetric solution of chemotaxis system, and we derive some inequalities concerned with the blow-up phenomenon of numerical solution. Several numerical experiments are presented to verify the theoretical results.

Appendix 1: The \(W^{1,p}\) error estimate of the finite volume method with Voronoi mesh for elliptic problem

In this appendix section, we consider the $W^{1,p}$ error estimate of finite volume method for the elliptic problem, which is applied to prove Lemma 4.1. Given any $u_h \in X_h$, let $\hat{u}_h = M_h^{-1}u_h \in \hat{X}_h$. We consider the solutions $v_h \in X_h$ and $\hat{w}_h$ of discrete problems (3.17) and (4.12), respectively, i.e. $v_h = G_h u_h$ and $\hat{w}_h = L_h^{-1}\hat{u}_h$. Let $\hat{v}_h = M_h^{-1} v_h$. Lemma 4.1 claims the error estimate

$$\begin{aligned} \Vert \hat{v}_h -\hat{w}_h\Vert _{1,p} \le Ch\Vert \hat{u}_h\Vert _{1,p}. \end{aligned}$$

Subtracting (3.17) from (4.12) yields: for any $\hat{\chi }_h \in \hat{X}_h$,

$$\begin{aligned} (\nabla (\hat{v}_h - \hat{w}_h), \nabla \hat{\chi }_h) + (\hat{v}_h - \hat{w}_h,\hat{\chi }_h) = (\hat{u}_h + \hat{w}_h, \hat{\chi }_h) - (\hat{u}_h + \hat{w}_h, \hat{\chi }_h)_h. \end{aligned}$$

(7.1)

Substituting $\hat{\chi }_h = \hat{v}_h - \hat{w}_h$ into (7.1), together with (3.14), we obtain

$$\begin{aligned} \Vert \nabla (\hat{v}_h - \hat{w}_h\Vert _2^2 + \Vert \hat{v}_h - \hat{w}_h\Vert ^2 \le Ch \Vert \hat{u}_h + \hat{w}_h\Vert _{1,2}, \end{aligned}$$

(7.2)

which implies (in view of $\Vert \hat{w}_h\Vert _{1,2} \le C\Vert \hat{u}_h\Vert _2$)

$$\begin{aligned} \Vert \nabla (\hat{v}_h - \hat{w}_h\Vert _{1,2} \le Ch^{1/2} \Vert \hat{u}_h\Vert _{1,2}. \end{aligned}$$

(7.3)

Since $M_h$ does not satisfies (3.3), we only obtain the $W^{1,2}$ norm error of order $O(h^{1/2})$. However, Lemma 4.1 claims the $W^{1,p}$ norm error is of order O(h) for $p \in [2,\mu )$.

To tackle this problem, we introduce the solution of continuous elliptic problem (2.1) with u replaced by $u_h$ (resp. $\hat{u}_h$), where the solution is $G u_h \in \mathscr {W}^p$ (resp. $G \hat{u}_h \in \mathscr {W}^p$) with $2 \le p < \mu$. Here, G is defined by (2.3). By the regularity (2.2) and the error estimate of the finite element method (cf. [32]), we have the error estimate of $G(u_h - \hat{u}_h)$ and $\hat{w}_h - G \hat{u}_h$. Then by the triangle inequality, we are left to estimate $\Vert M_h^{-1}(v_h - G_h u_h)\Vert _{1,p}$ (see the proof of Lemma 4.1), which is the error estimate of the finite volume method with Voronoi mesh for elliptic problem.

In [13], the $W^{1,2}$ norm error of order O(h) is obtained. In this appendix, we derive the error estimate O(h) of $W^{1,p}$ norm.

In the following, we extend the method of Chou et al. [8, Theorem 2.1] to obtain the error estimate of $W^{1,p}$ norm of the finite volume method for elliptic problem.

Lemma 7.1

Let $u_h \in X_h$, set $v_h = G_h u_h$, $\hat{v}_h = M_h^{-1}v_h$. Let $V = Gu_h$, be the solution of

$$\begin{aligned} -\Delta V + V = u_h \text { in } \Omega , \quad \partial _\nu V = 0 \text { on } \Gamma . \end{aligned}$$

(7.4)

Then, we have

$$\begin{aligned} \Vert \hat{v}_h - V\Vert _{1,p} \le Ch\Vert V\Vert _{2,p}. \end{aligned}$$

(7.5)

Proof

We define a bilinear form $a^*(\cdot ,\cdot )$: for all $w \in H^1(\Omega )$, $\chi _h \in X_h$,

$$\begin{aligned} a^*(w,\chi _h) := -\sum _{K \in \mathscr {T}} \chi _K \left( -\int _{\partial K} \nabla w \cdot \nu ~ds\right) + (w,\chi _h), \end{aligned}$$

(7.6)

where $\nu$ is the unit outer normal vector to $\partial K$. For any $\chi _h \in X_h$, multiplying (7.4) with $\chi _h$ yields

$$\begin{aligned}&\sum _{K \in \mathscr {T}} \chi _K \int _K -\Delta V~dx + (V,\chi _h) = (u_h, \chi _h)\nonumber \\&\quad = -\sum _{K \in \mathscr {T}} \chi _K (\int _{\partial K} \nabla V \cdot \nu ~ds) + (V,\chi _h)\nonumber \\&\quad = a^*(V,\chi _h). \end{aligned}$$

(7.7)

Let $\hat{V}_h$ be the solution of the equation: find $\hat{V}_h \in \hat{X}_h$ such that

$$\begin{aligned} a^*(\hat{V}_h,\chi _h) = (u_h,\chi _h), \quad \forall \chi _h \in X_h. \end{aligned}$$

(7.8)

From (7.7) and (7.8), we have

$$\begin{aligned} a^*(V-\hat{V}_h,\chi _h) = 0, \quad \forall \chi _h \in X_h. \end{aligned}$$

(7.9)

On the other hand, V also satisfies,

$$\begin{aligned} a(V,\chi ) = (u_h,\chi ), \quad \forall \chi \in H^1(\Omega ), \end{aligned}$$

(7.10)

where

$$\begin{aligned} a(\phi ,\varphi ) := (\nabla \phi , \nabla \varphi ) + (\phi , \varphi ), \quad \forall \phi ,\varphi \in H^1(\Omega ). \end{aligned}$$

(7.11)

We shall prove the following inequality: for any $\hat{\chi }_h \in \hat{X}_h$,

$$\begin{aligned}&|a(V-\hat{V}_h, \hat{\chi }_h ) - a^*(V-\hat{V}_h, M_h \hat{\chi }_h )| \nonumber \\&\quad \le Ch(\Vert V - \hat{V}_h\Vert _p + \Vert V\Vert _{2,p})\Vert \hat{\chi }_h \Vert _{1,q}. \end{aligned}$$

(7.12)

For any $\hat{w}_h, \hat{\chi }_h \in \hat{X}_h$, with $w_h = M_h \hat{w}_h$, $\chi _h = M_h \hat{\chi }_h \in X_h$, we have:

$$\begin{aligned}&(\nabla \hat{w}_h, \nabla \hat{\chi }_h )= A_h(w_h,\chi _h) \quad (\text {by (3.22)}) \nonumber \\&\quad = \sum _{K \in \mathscr {T}} \chi _K \sum _{L \in \mathscr {N}_K} \tau _{K,L} (w_K - w_L) = -\sum _{K \in \mathscr {T}} \chi _K \int _{\partial K} \nabla \hat{w}_h \cdot \nu ~ds, \end{aligned}$$

(7.13)

Applying (7.13), we obtain:

$$\begin{aligned}&a(V-\hat{V}_h, \hat{\chi }_h ) - a^*(V-\hat{V}_h, M_h \hat{\chi }_h ) \nonumber \\&\quad = (V-\hat{V}_h,\hat{\chi }_h - M_h \hat{\chi }_h ) + (\nabla V,\nabla \hat{\chi }_h ) + \sum _{K \in \mathscr {T}} \chi _K \int _{\partial K} \nabla V \cdot \nu ~ds. \nonumber \\&\quad = (V-\hat{V}_h, \hat{\chi }_h - M_h \hat{\chi }_h ) + (-\Delta V, \hat{\chi }_h ) + (\Delta V, M_h \hat{\chi }_h). \end{aligned}$$

(7.14)

Since $\Vert \hat{\chi }_h - M_h \hat{\chi }_h \Vert _q \le Ch \Vert \hat{\chi }_h \Vert _{1,q}$, we conclude (7.12). Furthermore, in view of (7.9), we have

$$\begin{aligned} |a(V-\hat{V}_h, \hat{\chi }_h )|= & {} | a(V-\hat{V}_h, \hat{\chi }_h )- a^*(V-\hat{V}_h, M_h \hat{\chi }_h )| \nonumber \\\le & {} Ch(\Vert V - \hat{V}_h\Vert _p + \Vert V\Vert _{2,p})\Vert \hat{\chi }_h \Vert _{1,q}, \end{aligned}$$

(7.15)

which implies (cf. [8, The proof of Theorem 2.1]):

$$\begin{aligned} \Vert V - \hat{V}_h\Vert _{1,p} \le C h \Vert V\Vert _{2,p}. \end{aligned}$$

(7.16)

Our goal is to obtain the error estimate of $\Vert V - \hat{v}_h\Vert _{1,p}\Vert _{1,p}$. By triangle inequality and (7.16), we are left to estimate $\Vert \hat{V}?h - \hat{v}_h\Vert$. We see that $\hat{v}_h$ and $\hat{V}_h$ satisfy the following tow equations respectively:

$$\begin{aligned}&(\nabla \hat{v}_h, \nabla \hat{\chi }_h) + ( M_h \hat{v}_h, \chi _h) = (u_h,\chi _h), \quad \forall \chi _h \in X_h, \end{aligned}$$

(7.17a)

$$\begin{aligned}&(\nabla \hat{V}_h, \nabla \hat{\chi }_h) + (\hat{V}_h, \chi _h) = (u_h,\chi _h), \quad \forall \chi _h \in X_h, \end{aligned}$$

(7.17b)

where $\hat{\chi }_h = M_h^{-1} \chi _h$. Therefore, we have

(7.18)

which implies (applying Lemma 4.3 with )

$$\begin{aligned} \Vert \hat{V}_h - \hat{v}_h\Vert _{1,p} \le C \Vert \hat{V}_h - M_h \hat{V}_h\Vert _p \le C h \Vert \hat{V}_h\Vert _{1,p} \le Ch \Vert u_h\Vert _p. \end{aligned}$$

(7.19)

By the triangle inequality, (7.16) and (7.19), we conclude the error estimate (7.5). $\square$

Remark 7.1

In [8], the authors consider lumping operator $\bar{M}_h$ (defined by (3.3)) instead of $M_h$. However, (7.12) and (7.5) hold for both $M_h$ and $\bar{M}_h$, since we have only used the properties (3.8) and (3.14), which are satisfied for both $M_h$ and $\bar{M}_h$. With the help of (3.12), one can obtain a higher-order error estimate of $L^p$ norm (cf. [8]), which is not necessary for our case since we need the error estimate in $W^{1,p}$ norm (see Lemma 4.1).