Nonsmooth data optimal error estimates by energy arguments for subdiffusion equations with memory

Abstract

This paper considers the semidiscrete Galerkin finite element approximation for time fractional diffusion equations with memory in a bounded convex polygonal domain. We use novel energy arguments in conjunction with repeated applications of time integral operators to study the error analysis. Our error estimates cover both smooth and nonsmooth initial data cases. Since the continuous solution u of such models has singularity at t = 0 even for smooth initial data, we use t^j, j = 1,2, type of weights to overcome the singular behavior at t = 0. Optimal order error estimates in both L²(Ω)- and H¹(Ω)-norms are proved with respect to both convergence order of the approximate solution and regularity of the initial data. Moreover, a quasi-optimal pointwise in time error bound in the maximum norm is shown to hold for smooth initial data. In the end, we provide numerical results to support our theoretical findings.

Appendix

In this part we calculate a bound of $\|{u^{\prime }(t)}\|_{p}$, p = 1, 2 for both smooth and nonsmooth initial data. For f = 0 with nonsmooth initial data u₀, a bound of $\|{u^{\prime }(t)}\|_{2}$ is given in (2.3). To obtain a bound for $\|{u^{\prime }(t)}\|_{1}$, we apply interpolation of estimates $\|{u^{\prime }(t)}\|\le Ct^{-1}\|{u_{0}}\|$ and $\|{u^{\prime }(t)}\|_{2}\le Ct^{-1-\alpha }\|{u_{0}}\|$ to get

$$\begin{array}{@{}rcl@{}} \|{u^{\prime}(t)}\|_{1}\le Ct^{-1-\alpha/2}\|{u_{0}}\|, t>0. \end{array}$$

(A.1)

We next turn our attention to nonhomogeneous problem (1.1). Decompose (1.1) in two parts as

$$\begin{array}{@{}rcl@{}} \partial_{t}^{\alpha}v(t)+Av(t) &=&{{\int}_{0}^{t}}B(t,s)v(s) ds \quad \text{in}\quad {{\varOmega}}, \quad 0<t \leq T,\quad 0<\alpha<1, \\ v(t) &=& 0 \quad \text{on}\quad \partial{{\varOmega}}, \quad 0<t\leq T,\\ v(0)&=& u_{0}(x) \quad \text{in} \quad {{\varOmega}}, \end{array}$$

(A.2)

and

$$\begin{array}{@{}rcl@{}} \partial_{t}^{\alpha}w(t)+Aw(t) &=& f(t) \quad \text{in}\quad {{\varOmega}}, \quad 0<t \leq T,\quad 0<\alpha<1, \\ w(t) &=& 0 \quad \text{on}\quad \partial{{\varOmega}}, \quad 0<t\leq T,\\ w(0)&=&0 \quad \text{in} \quad {{\varOmega}}. \end{array}$$

(A.3)

Following [18, Theorem 3.4], for smooth initial data we can obtain $\|{v^{\prime }(t)}\|\le C t^{-1+\alpha }\|{u_{0}}\|_{2}$ and $\|{v^{\prime }(t)}\|_{2}\le C t^{-1}\|{u_{0}}\|_{2}$ for t > 0. By interpolation we get

$$\begin{array}{@{}rcl@{}} \|{v^{\prime}(t)}\|_{1}\le Ct^{-1+\alpha/2}\|{u_{0}}\|_{2}, t>0. \end{array}$$

(A.4)

Now, the solution for problem (A.3) (cf. [12]) is given by

$$\begin{array}{@{}rcl@{}} w(t)={{\int}_{0}^{t}}\widehat{E}(t-s)f(s) ds={{\int}_{0}^{t}}\widehat{E}(s)f(t-s) ds, \end{array}$$

(A.5)

where the operator $\widehat {E}(t)$ is defined by

$$\begin{array}{@{}rcl@{}} \widehat{E}(t)\chi=\sum\limits_{j=1}^{\infty} t^{\alpha -1}E_{{\alpha},\alpha}(-\lambda_{j}t^{\alpha})(\chi,\phi_{j})\phi_{j}(x), t>0, \chi\in L^{2}({{\varOmega}}), \end{array}$$

with $E_{{\alpha },\alpha }(-\lambda _{j}t^{\alpha })={\sum }_{j=0}^{\infty }\frac {(-\lambda _{j}t^{\alpha })^{j}}{\Gamma (j\alpha +\alpha )}$. Differentiating (A.5) with respect to time to have

$$\begin{array}{@{}rcl@{}} w^{\prime}(t)={{\int}_{0}^{t}}\widehat{E}(s)f^{\prime}(t-s) ds + \widehat{E}(t)f(0)={{\int}_{0}^{t}}\widehat{E}(t-s)f^{\prime}(s) ds + \widehat{E}(t)f(0). \end{array}$$

Invoking [12, Lemma 2.2] with p = 1,q = 0, we obtain for t > 0,

$$\begin{array}{@{}rcl@{}} \|{w^{\prime}(t)}\|_{1}&\le& {{\int}_{0}^{t}}\|{\widehat{E}(t-s)f^{\prime}(s)}\|_{1} ds + \|{\widehat{E}(t)f(0)}\|_{1}\\ &\le& C{{\int}_{0}^{t}}(t-s)^{-1+\alpha/2}\|{f^{\prime}(s)}\| ds + Ct^{-1+\alpha/2}\|{f(0)}\|\\ &\le& Ct^{\alpha/2}\|{f}\|_{C^{1}(L^{2})}+ Ct^{-1+\alpha/2}\|{f(0)}\|. \end{array}$$

(A.6)

In view of (A.4) and (A.6), we get for t > 0,

$$\begin{array}{@{}rcl@{}} \|{u^{\prime}(t)}\|_{1}\le \|{v^{\prime}(t)}\|_{1}+\|{w^{\prime}(t)}\|_{1}\le Ct^{-1+\alpha/2}\big(\|{u_{0}}\|_{2}+\|{f(0)}\|\big) + Ct^{\alpha/2}\|{f}\|_{C^{1}(L^{2})}. \end{array}$$

(A.7)

Similarly, putting p = 2,q = 1, in Lemma 2.2 [12, p. 565], we obtain

$$\begin{array}{@{}rcl@{}} \|{w^{\prime}(t)}\|_{2}&\le Ct^{\alpha/2}\|{f}\|_{C^{1}(\dot{H}^{1})}+ Ct^{-1+\alpha/2}\|{f(0)}\|_{1}, t>0. \end{array}$$

Thus, for positive time t,

$$\begin{array}{@{}rcl@{}} \|{u^{\prime}(t)}\|_{2}\le \|{v^{\prime}(t)}\|_{2}+\|{w^{\prime}(t)}\|_{2}\le Ct^{-1}\big(\|{u_{0}}\|_{2}+t^{\alpha/2}\|{f(0)}\|_{1}\big) + Ct^{\alpha/2}\|{f}\|_{C^{1}(\dot{H}^{1})}. \end{array}$$

(A.8)