Alternating Direction Implicit Galerkin Methods for an Evolution Equation with a Positive-Type Memory Term

Abstract

We formulate and analyze new methods for the solution of a partial integrodifferential equation with a positive-type memory term. These methods combine the finite element Galerkin (FEG) method for the spatial discretization with alternating direction implicit (ADI) methods based on the Crank–Nicolson (CN) method and the second order backward differentiation formula for the time stepping. The ADI FEG methods are proved to be of optimal accuracy in time and in the $L^2$ norm in space. Furthermore, the analysis is extended to include an ADI CN FEG method with a graded mesh in time for problems with a nonsmooth kernel. Numerical results confirm the predicted convergence rates and also exhibit optimal spatial accuracy in the $L^{\infty }$ norm.

Appendices

Appendix 1

Lemma 7.1

With $\zeta _{n}=k_{n+1}/{k_n}$,

$$\begin{aligned} \sum _{n=1}^{J-1}\left( \zeta _n^{2(1+\alpha )} - 1 \right) \le C. \end{aligned}$$

Proof

First recall that, for $a, b \in \mathcal{R}$,

$$\begin{aligned} a^n + b^n \le (a+b)^n. \end{aligned}$$

With

$$\begin{aligned} a= x^{1/n} - y^{1/n}, \quad b=y^{1/n}, \qquad x \ge y, \end{aligned}$$

we have

$$\begin{aligned} \left( x^{1/n} - y^{1/n}\right) ^n + y \le x \end{aligned}$$

or

$$\begin{aligned} x^{1/n} - y^{1/n}\le (x-y)^{1/n}. \end{aligned}$$

(7.1)

With $t_n=(nk)^{\gamma }$,

$$\begin{aligned} k_{n+1} = t_{n+1}-t_n = [(n+1)k]^{\gamma }-(nk)^{\gamma } = (nk)^{\gamma }\left[ \left( 1+\frac{1}{n}\right) ^{\gamma } -1 \right] \end{aligned}$$

and

$$\begin{aligned} k_n = t_{n}-t_{n-1}= (nk)^{\gamma }\left[ 1-\left( 1-\frac{1}{n}\right) ^{\gamma } \right] . \end{aligned}$$

Thus

$$\begin{aligned} \zeta _n^{2(1+\alpha )} - 1 =\frac{k_{n+1}^{2(\alpha +1)} -k_n^{2(\alpha +1)} }{k_{n}^{2(\alpha +1)}} \le \frac{( k_{n+1} +k_n )^{2(\alpha +1)} }{k_{n}^{2(\alpha +1)}}, \end{aligned}$$

(7.2)

on using (7.1). Then, from (7.2),

$$\begin{aligned} \zeta _n^{2(1+\alpha )} - 1\le & {} \frac{\left[ \left( 1 + \frac{1}{n}\right) ^{\gamma } -2 +\left( 1 - \frac{1}{n}\right) ^{\gamma } \right] ^{2(\alpha +1)}}{\left[ 1-\left( 1 - \frac{1}{n}\right) ^{\gamma }\right] ^{2(\alpha +1)}}\\= & {} \frac{ \frac{\gamma (\gamma -1)}{n^2}\left[ 1+\sum _{j=3}^{\infty }\frac{1+ (-1)^j}{\gamma (\gamma -1)}\left( \begin{array}{c} \gamma \\ j \end{array} \right) \frac{1}{n^{j-1}}\right] ^{2(\alpha +1)}}{\frac{\gamma }{n}\left[ 1+ \sum _{j=2}^{\infty }\frac{(-1)^j}{\gamma }\left( \begin{array}{c} \gamma \\ j \end{array} \right) \frac{1}{n^{j-1}}\right] ^{2(\alpha +1)}} \\\le & {} C\left( \frac{k}{nk}\right) ^{2(\alpha +1)} \\= & {} Ck^{2(\alpha +1)}t_n^{-\frac{2(\alpha +1)}{\gamma }}. \end{aligned}$$

Thus

$$\begin{aligned} \sum _{n=1}^{J-1}\left( \zeta _n^{2(1+\alpha )} - 1 \right)\le & {} C\sum _{n=1}^{J-1}k^{2(\alpha +1)}t_n^{-\frac{2(\alpha +1)}{\gamma }}\\= & {} C\sum _{n=1}^{J-1}\left( \frac{k}{k_n}\right) ^{2(\alpha +1)}k_n^{2(\alpha +1)}t_n^{-\frac{2(\alpha +1)}{\gamma }}. \end{aligned}$$

Since

$$\begin{aligned} \left( \frac{k}{k_n}\right) ^{2(\alpha +1)}\le \left( \frac{2^{\gamma -1}}{\gamma } \right) ^{2(\alpha +1)}t_n^{2(-1+\frac{1}{\gamma })(\alpha +1)}, \end{aligned}$$

(7.3)

it follows that

$$\begin{aligned} \sum _{n=1}^{J-1}\left( \zeta _n^{2(1+\alpha )} - 1 \right)\le & {} C\left( \frac{2^{\gamma -1}}{\gamma } \right) ^{2(\alpha +1)} \sum _{n=1}^{J-1}k_n^{2(\alpha +1)}t_n^{2(\alpha +1)(-1+\frac{1}{\gamma })}t_n^{-\frac{2}{\gamma }(\alpha +1)} \end{aligned}$$

Then using (7.3) again,

$$\begin{aligned} \sum _{n=1}^{J-1}\left( \zeta _n^{2(1+\alpha )} -1 \right)\le & {} Ck^{2\alpha +1}\sum _{n=1}^{J-1}k_nt_n^{-(1+\frac{1}{\gamma } + \frac{2\alpha }{\gamma })}\\\le & {} Ck^{2\alpha +1}\int _{t_1}^{t_J}t^{-(1+\frac{1}{\gamma }+ \frac{2\alpha }{\gamma })}\,dt\\\le & {} Ck^{2\alpha +1} \left[ \frac{t_1^{-(\frac{1}{\gamma } + \frac{2\alpha }{\gamma })} -t_J^{-(\frac{1}{\gamma } + \frac{2\alpha }{\gamma })}}{(\frac{1}{\gamma } + \frac{2\alpha }{\gamma })} \right] \\\le & {} Ck^{2\alpha +1}k^{-(2\alpha +1)}\\= & {} C, \end{aligned}$$

which completes the proof.$\square$

Appendix 2

For any continuous function $\psi$,

$$\begin{aligned} \psi (t_{n-1/2}) \approx \frac{1}{k_n}\int _{t_{n-1}}^{t_n} \psi (t)\;dt, \end{aligned}$$

from the midpoint rule. Thus, if

$$\begin{aligned} \mathcal{I}(\phi ) = \int _0^t \beta (t-s)\varphi (s)\;ds, \end{aligned}$$

then

$$\begin{aligned} \mathcal{I}^{n+{\frac{1}{2}}}(\varphi )\approx & {} \frac{1}{k_n}\int _{t_{n-1}}^{t_n} \int _0^t \beta (t-s)\varphi (s)\;ds\;dt\\= & {} \frac{1}{k_n}\int _{t_{n-1}}^{t_n}\left\{ \int _{0}^{t_1} \beta (t-s)\varphi (s)\;ds +\sum _{j=2}^{n-1}\int _{t_{j-1}}^{t_j}\beta (t-s)\varphi (s)\;ds\right. \\&\left. +\int _{t_{n-1}}^{t} \beta (t-s)\varphi (s)\;ds\right\} dt\\\approx & {} \frac{1}{k_n}\int _{t_{n-1}}^{t_n}\left\{ \varphi (t_1)\int _{0}^{t_1} \beta (t-s)\;ds +\sum _{j=2}^{n-1}\varphi ^{j-{\frac{1}{2}}}\int _{t_{j-1}}^{t_j}\beta (t-s)\varphi (s)\;ds\right. \\&\left. +\,\varphi ^{n-{\frac{1}{2}}}\int _{t_{n-1}}^{t}\beta (t-s)\varphi (s)\;ds\right\} dt\\= & {} \omega _{n1} k_1 \varphi ^{1}+ \sum _{j=2}^{n} \omega _{nj} k_j \varphi ^{j-{\frac{1}{2}}}\\\equiv & {} \tilde{Q}^{n-{\frac{1}{2}}}(\varphi ), \end{aligned}$$

where

$$\begin{aligned} \omega _{nj}= & {} \frac{1}{k_n k_j} \int _{t_{n-1}}^{t_n} \int _{t_{j-1}}^{t_j} \beta (t-s) \,ds\,dt, \quad j=1,\ldots ,n-1,\nonumber \\ \quad \omega _{nn}= & {} \frac{1}{k_n^2} \int _{t_{n-1}}^{t_n} \int _{t_{n-1}}^{t} \beta (t-s) \,ds\,dt. \end{aligned}$$