Optimal A Priori Error Estimates for the Finite Element Approximation of Dual-Phase-Lag Bio Heat Model in Heterogeneous Medium

Abstract

Galerkin finite element method is applied to dual-phase-lag bio heat model in heterogeneous medium. Well-posedness of the model interface problem and a priori estimates of its solutions are established. Optimal a priori error estimates for both semidiscrete and fully discrete schemes are proved in $L^\infty (L^2)$ norm. The fully discrete space-time finite element discretizations is based on second order in time Newmark scheme. Finally, numerical results for two dimensional test problems are presented in support of our theoretical findings. Finite element algorithm presented here can contribute to a variety of engineering and medical applications.

Appendix

Proof of Lemma 4.1:

Taking $t\rightarrow 0^+$ in (4.1) and then using definition of $\mathcal{Q}_h$ operator, we obtain

$$\begin{aligned} (u_h^{\prime \prime }(0),v_h)= & {} -\mathcal{B}_{\sigma h}(u_h^\prime (0),v_h)-\mathcal{B}_{\delta h}(u_h(0),v_h)\nonumber \\&-\mathcal{A}_{\epsilon h}(u_h^\prime (0),v_h)-\mathcal{A}_{\beta h}(u_h(0),v_h)+(f(0),v_h)\nonumber \\= & {} -\mathcal{B}_{\sigma h}(\mathcal{Q}_{\epsilon h}v_0,v_h)-\mathcal{B}_{\beta h}(\mathcal{Q}_{\beta h}u_0,v_h)\nonumber \\&-\mathcal{A}_{\epsilon h}(\mathcal{Q}_{\epsilon h}v_0,v_h)-\mathcal{A}_{\beta h}(\mathcal{Q}_{\beta h}u_0,v_h)+(f(0),v_h)\nonumber \\= & {} -\mathcal{B}_{\sigma h}(\mathcal{Q}_{\epsilon h}v_0,v_h)-\mathcal{B}_{\beta h}(\mathcal{Q}_{\beta h}u_0,v_h)\nonumber \\&-\mathcal{A}_{\epsilon }(v_0,v_h)-\mathcal{A}_{\beta }(u_0,v_h)+(f(0),v_h). \end{aligned}$$

(7.3)

Here, we have used the fact that $u_h\in C^2(J; V_h)$. For the third term and fourth term in (7.3), we use Green’s formula and boundary condition to derive

$$\begin{aligned}&\mathcal{A}_{\epsilon }(v_0,v_h)=-(\nabla \cdot (\epsilon \nabla v_0),v_h)\le C\Vert v_0\Vert _2\Vert v_h\Vert ,\\&\mathcal{A}_{\beta }(u_0,v_h)=-(\nabla \cdot (\beta \nabla u_0),v_h)\le C\Vert u_0\Vert _2\Vert v_h\Vert . \end{aligned}$$

Hence, (7.3) yields

$$\begin{aligned} \Vert u_h^{\prime \prime }(0)\Vert \le C\big (\Vert u_0\Vert _2+\Vert v_0\Vert _2+\Vert f\Vert _{H^1(L^2)} \big ). \end{aligned}$$

(7.4)

In the previous estimate, we have used the fact that

$$\begin{aligned} \sup _{0\le t \le T}\Vert f(t)\Vert \le C(T)\Vert f\Vert _{H^1(J;W)}. \end{aligned}$$

In fact, for any Banach space $\mathcal{B}$, we know that (cf. [30], Proposition 7.1)

$$\begin{aligned} \sup _{0\le t \le T}\Vert v(t)\Vert _\mathcal{B}\le C(T)\Vert v\Vert _{H^1(J;\mathcal{B})}\;\forall v\in H^1(J;\mathcal{B}). \end{aligned}$$

(7.5)

Also, from the definition of $\mathcal{Q}_h$ operator, we can easily derive

$$\begin{aligned} \Vert u_h^\prime (0)\Vert _1=\Vert \mathcal{Q}_{\epsilon h}v_0\Vert _1\le C\Vert v_0\Vert _1. \end{aligned}$$

(7.6)

For $i=3$, taking $t\rightarrow 0^+$ in (2.3) and using (7.3), we have

$$\begin{aligned} (u_h^{\prime \prime }(0)-u^{\prime \prime }(0),v_h)= & {} \mathcal{B}_{\sigma }(v_0,v_h)-\mathcal{B}_{\sigma h}(\mathcal{Q}_{\epsilon h}v_0,v_h)+\mathcal{B}_{\delta }(u_0,v_h)\nonumber \\&-\mathcal{B}_{\delta h}(\mathcal{Q}_{\beta h}u_0,v_h)\nonumber \\= & {} \mathcal{B}_{\sigma }^\Delta (v_0,v_h)+\mathcal{B}_{\sigma h}(v_0-\mathcal{Q}_{\epsilon h}v_0,v_h)+\mathcal{B}_{\delta }^\Delta (u_0,v_h)\nonumber \\&+\mathcal{B}_{\delta h}(u_0-\mathcal{Q}_{\beta h}u_0,v_h)\nonumber \\\le & {} C (h^2+\lambda )(\Vert u_0\Vert _2+\Vert v_0\Vert _2)\Vert v_h\Vert . \end{aligned}$$

(7.7)

In the last inequality, we have used Lemmas 3.2 and 3.4, and the fact that $u\in C^2(J; W)$. Then use definition of $L^2$ projection and (7.7) to obtain

$$\begin{aligned} (u_h^{\prime \prime }(0)-\mathcal{L}_hu^{\prime \prime }(0),v_h)= & {} (u_h^{\prime \prime }(0)-u^{\prime \prime }(0),v_h)\nonumber \\\le & {} C (h^2+\lambda )(\Vert u_0\Vert _2+\Vert v_0\Vert _2)\Vert v_h\Vert , \end{aligned}$$

(7.8)

which imply

$$\begin{aligned} \Vert u_h^{\prime \prime }(0)-\mathcal{L}_hu^{\prime \prime }(0)\Vert \le C h(\Vert u_0\Vert _2+\Vert v_0\Vert _2). \end{aligned}$$

(7.9)

Estimate (7.9) together with inverse inequality and (3.19) yields

$$\begin{aligned} \Vert u_h^{\prime \prime }(0)\Vert _1\le & {} Ch^{-1}\Vert u_h^{\prime \prime }(0)-\mathcal{L}_hu^{\prime \prime }(0)\Vert +\Vert \mathcal{L}_hu^{\prime \prime }(0)\Vert _1\nonumber \\\le & {} C(\Vert u_0\Vert _3+\Vert v_0\Vert _3+\Vert f\Vert _{H^1(H^1)}). \end{aligned}$$

(7.10)

Next, for $u_h\in C^3(J; V_h)$, we differentiate (4.1) with respect to t and then take $t\rightarrow 0^+$ to have

$$\begin{aligned} (u_h^{\prime \prime \prime }(0),v_h)= & {} -\mathcal{B}_{\sigma h}(u_h^{\prime \prime }(0),v_h)-\mathcal{B}_{\delta h}(u_h^\prime (0),v_h)\nonumber \\&-\mathcal{A}_{\epsilon h}(u_h^{\prime \prime }(0),v_h)-\mathcal{A}_{\beta h}(u_h^\prime (0),v_h)+(f^{\prime }(0),v_h)\nonumber \\= & {} -\mathcal{B}_{\sigma h}(u_h^{\prime \prime }(0),v_h)-\mathcal{B}_{\delta h}(\mathcal{Q}_{\epsilon h}v_0,v_h)\nonumber \\&-\mathcal{A}_{\epsilon h}(u_h^{\prime \prime }(0)-\mathcal{Q}_{\epsilon h}u^{\prime \prime }(0),v_h)-\mathcal{A}_{\beta h}(\mathcal{Q}_{\epsilon h}v_0-\mathcal{Q}_{\beta h}u^\prime (0),v_h)\nonumber \\&-\sum _{l=1}^{2}\Big \{\mathcal{A}_{\epsilon }^l(u^{\prime \prime }(0),v_h)+\mathcal{A}_{\beta }^l(u^{\prime }(0),v_h)\Big \}+(f^{\prime }(0),v_h). \end{aligned}$$

Now, for $u\in H^3(J; \mathcal{Y})$ or equivalently $u\in C^2(J;\mathcal{Y})$, use the fact that

$$\begin{aligned} \Biggl [\epsilon (x){\partial u^{\prime \prime }(t) \over {\partial \mathbf{n}}}+\beta (x){\partial u^{\prime }(t) \over {\partial \mathbf{n}}}\Biggr ] =0\;\;\;\text{ along }\; \Gamma \times [0,T] \end{aligned}$$

in the Eq. (7.11) to obtain

$$\begin{aligned} (u_h^{\prime \prime \prime }(0),v_h)= & {} -\mathcal{B}_{\sigma h}(u_h^{\prime \prime }(0),v_h)-\mathcal{B}_{\delta h}(\mathcal{Q}_{\epsilon h}v_0,v_h)\nonumber \\&-\mathcal{A}_{\epsilon h}(u_h^{\prime \prime }(0)-\mathcal{Q}_{\epsilon h}u^{\prime \prime }(0),v_h)-\mathcal{A}_{\beta h}(\mathcal{Q}_{\epsilon h}v_0-\mathcal{Q}_{\beta h}v_0,v_h)\nonumber \\&+\sum _{l=1}^{2}\Big \{(\nabla \cdot \epsilon _l \nabla u^{\prime \prime }(0),v_h)_{\Omega _l}+(\nabla \cdot \beta \nabla u^{\prime }(0),v_h)_{\Omega _l}\Big \}+(f^{\prime }(0),v_h)\nonumber \\\le & {} C\Big \{\Vert u^{\prime \prime }_h(0)\Vert +h^{-1}(\Vert u_h^{\prime \prime }(0)-\mathcal{Q}_{\epsilon h}u^{\prime \prime }(0)\Vert _1+\Vert \mathcal{Q}_{\epsilon h}v_0-\mathcal{Q}_{\beta h}v_0\Vert _1)\nonumber \\&\;\;\;\;\;+\sum _{l=1}^{2}\Vert u^{\prime \prime }(0)\Vert _{2, \Omega _l}+\Vert v_0\Vert _2+\Vert f\Vert _{H^2(L^2)}\Big \}\Vert v_h\Vert . \end{aligned}$$

(7.11)

From (7.8) and Remark 3.1, we have

$$\begin{aligned}&\Vert u_h^{\prime \prime }(0)-\mathcal{Q}_{\epsilon h}u^{\prime \prime }(0)\Vert _1\nonumber \\&\quad \le Ch^{-1}\Vert u_h^{\prime \prime }(0)-\mathcal{L}_hu^{\prime \prime }(0)\Vert +\Vert \mathcal{L}_hu^{\prime \prime }(0)-\mathcal{Q}_{\epsilon h}u^{\prime \prime }(0)\Vert _1\nonumber \\&\quad \le C(\Vert u_0\Vert _2+\Vert v_0\Vert _2)+Ch\sum _{l=1}^{2}\Vert u^{\prime \prime }(0)\Vert _{2, \Omega _l}\nonumber \\&\quad \le Ch(\Vert u_0\Vert _4+\Vert v_0\Vert _4+\Vert f\Vert _{H^1(H^2)}). \end{aligned}$$

(7.12)

In the last inequality, we have used the fact that $\Vert u_0\Vert _K\le Ch\Vert u_0\Vert _{2, K}$ for all $K\in \mathcal{T}_h.$ Using (7.12) in (7.11), we obtain

$$\begin{aligned} \Vert u_h^{\prime \prime \prime }(0)\Vert \le C(\Vert u_0\Vert _4+\Vert v_0\Vert _4+\Vert f\Vert _{H^2(H^2)}). \end{aligned}$$

(7.13)

The case $i=4$ can be done in a similar way and hence details are omitted. This completes the rest of the proof. $\square$