**2. Preliminaries**

Before we proceed with the error analysis, we require some auxiliary results that will be used in our analysis.

#### **2.1 Functional spaces**

Let *z t*ð Þ , *x* is a function of time *t* and space *χ*, we introduce the Bochner space *LP*ð Þ 0, *T*, � X where (X is some real Banach space equipped with the norm ∥ � ∥*X*Þ which is the collection of all measurable functions *v*: 0, ð Þ! *T* X, more precisely, for any number *r*≥1

$$L\_P(\mathbf{0}, \ T; \ X) = \left\{ z \; : \; \begin{array}{c} (\mathbf{0}, \ T) \to \mathbf{X} \; : \; \int\_0^T \|\mathbf{z}\|^2 dt \le \infty \right\}, \tag{1}$$

such that

$$\begin{aligned} \|\boldsymbol{z}\|\_{L^{p}(0,T;X)} &:= \left(\int\_{0}^{T} \|\boldsymbol{z}\|^{2} dt\right)^{1/2} < \infty \quad \text{for } 1 \le p < \infty, \\\|\boldsymbol{z}\|\_{L^{p}(0,T;X)} &:= \max\_{t \in [0,T]} \|\boldsymbol{z}(t)\|\_{X} < \infty \quad \text{for } p = \infty. \end{aligned} \tag{2}$$

Lemma 1.1 (Continuous Gronwall inequality). Let *C*0, *C*<sup>1</sup> ∈*L*<sup>1</sup> ð Þ 0, *T* for all *<sup>T</sup>* <sup>&</sup>gt;0 and *<sup>z</sup>*∈*W*1,1, then for almost every *<sup>t</sup>*<sup>∈</sup> ð � 0, *<sup>T</sup>* , reads

$$z'(t) \le \mathcal{C}\_0(t) + \mathcal{C}\_1(t)z(t) \quad , \tag{3}$$

then

$$z(t) \le F(\mathbf{0}, \ T)z(\mathbf{0}) + \int\_0^T F(\mathbf{0}, \ T)z(s)ds,\tag{4}$$

where *<sup>F</sup>*ð Þ¼ 0, *<sup>T</sup>* exp <sup>Ð</sup> *<sup>T</sup>* <sup>0</sup> *C*1ð Þ *ξ*ð Þ*t dξ* � . Furthermore, if *C*<sup>0</sup> and *C*<sup>1</sup> are nonnegatives, gives

$$z(T) \le F(\mathbf{0}, \ T) \left( z(\mathbf{0}) + \int\_0^T \mathbf{C}\_0(s) ds \right). \tag{5}$$

**Proof:** See [15].

*A Posteriori Error Analysis in Finite Element Approximation for Fully Discrete Semilinear… DOI: http://dx.doi.org/10.5772/intechopen.94369*

Theorem 1.2 Given some *p*≥2, we have

$$\begin{aligned} \|\boldsymbol{\nu}\|\_{L^{p}(\Omega)}^{p} &\leq C \|\nabla\boldsymbol{\nu}\|\_{\frac{p^{d}-2d}{2}\|\boldsymbol{\nu}\|^{\frac{2p+2d-pd}{2}}}^{\frac{p^{d}-2d}{2}\|\boldsymbol{\nu}\|^{\frac{2p+2d-pd}{2}}}\\ \|\boldsymbol{\nu}\|\_{L^{p}(\Omega)}^{p} &\leq C \|\nabla\boldsymbol{\nu}\|^{p-2}\|\boldsymbol{\nu}\|^{2}, d = 2 \\ \|\boldsymbol{\nu}\|\_{L^{p}(\Omega)}^{p} &\leq C \|\nabla\boldsymbol{\nu}\|^{\frac{3p-6}{2}}\|\boldsymbol{\nu}\|^{\frac{6-p}{2}}, \; d = 3, p \leq 6. \end{aligned}$$

**Proof:** See [16].

## **3. Model problem**

Consider the semilinear parabolic problem as

$$\begin{aligned} \frac{\partial u}{\partial t} - \Delta u &= f(u), \quad \text{in } \Omega \cup [0, \ T], \\ u &= 0, \quad \text{on } \partial \Omega, \\ u(0, \boldsymbol{x}) &= u\_0(\boldsymbol{x}), \quad \text{on } \{0\} \times \Omega, \end{aligned} \tag{6}$$

where Ω is a plane convex domain subset of *<sup>k</sup>*, Ω ⊂ *<sup>k</sup>* with smooth boundary condition <sup>∂</sup>Ω, where *ut* <sup>¼</sup> *<sup>∂</sup>u=∂t*, *<sup>T</sup>* <sup>&</sup>gt;0 and *<sup>f</sup>* <sup>∈</sup>*C*<sup>1</sup> ð Þ . Let *Lp*ð Þ *ω* , 1≤*p* ≤ ∞ and *Hr* ð Þ *ω* , *r*∈ , denote the standard Lebesgue and Hilbertian Sobolev spaces on a domain *<sup>ω</sup>* <sup>⊂</sup> <sup>Ω</sup>. For brevity, the norm of *<sup>L</sup>*2ð Þ� *<sup>ω</sup> <sup>H</sup>*<sup>0</sup>ð Þ *<sup>ω</sup>* , *<sup>ω</sup>*<sup>⊂</sup> <sup>Ω</sup>, will be denoted by ∥ � ∥*ω*, and is induced by the standard *L*2ð Þ *ω* -inner product, denoted by ð Þ �, � *<sup>ω</sup>*; when *ω* ¼ Ω, we shall use the abbreviations ∥ � ∥ � ∥ � ∥<sup>Ω</sup> and ð Þ� � �, � ð Þ , � <sup>Ω</sup>.

Returning to the (6), multiplying by a test function *v*∈ *H*<sup>1</sup> <sup>0</sup>ð Þ Ω and then integrate by parts, we arrive to (7) in weak form, which reads: find *u*∈ *L*<sup>2</sup> 0, *T*, *H*<sup>1</sup> 0 � �ð Þ <sup>Ω</sup> <sup>∩</sup> *<sup>H</sup>*<sup>1</sup> <sup>0</sup>ð0, *T*, *L*2ð Þ Ω for almost every *t*∈ð � 0, *T* , this becomes

$$\int\_{\Omega} \frac{\partial \mathbf{z}}{\partial t} v d\mathbf{x} + D(t; \mathbf{z}, \boldsymbol{\nu}) = \int\_{\Omega} f(\mathbf{z}) v d\mathbf{x},\tag{7}$$

for all *v*∈ *H*<sup>1</sup> <sup>0</sup>ð Þ Ω . Here,

$$D(t; z, \ \nu) = \int\_{\Omega} \nabla z \cdot \nabla v d\mathbf{x}.\tag{8}$$

By using Cauchy-Schwarz inequality, the convercitivity and continuity of the bilnear form *D*, viz.

$$\begin{aligned} |D(v, \ v) \ge \mathcal{C}\_{\text{coer}} \| \nabla v \| ^2 &\quad \text{for all } v \in H\_0^1(\Omega), \\ |D(v, \ w)| \le \mathcal{C}\_{\text{cont}} \| \nabla v \| \| \nabla w \| &\quad \text{for all } v, \ w \in H\_0^1(\Omega), \end{aligned} \tag{9}$$

with *C*cont,*C*coer positive constants independent of *w*, *v*.

#### **4. Fully discrete backward Euler formulation**

To introduce a backward Euler approximation of the time derivative paired with the standard conforming finite element method of the spatial operator. To this end, we will discretize the time interval 0, ½ � *T* into subintervals ð � *tn*�1, *tn* , *n* ¼ 1, … , *N* with *t* <sup>0</sup> <sup>¼</sup> 0 and *tN* <sup>¼</sup> *<sup>T</sup>*, and we denote by *<sup>κ</sup><sup>n</sup>* <sup>¼</sup> *tn* � *tn*�<sup>1</sup> the local time step. We associate to each time-step *tN* a spatial mesh <sup>T</sup> *<sup>n</sup>* and the respective finite element space *<sup>V</sup>n*; <sup>¼</sup> *<sup>V</sup><sup>p</sup> <sup>h</sup>* <sup>T</sup> *<sup>n</sup>* ð Þ. The fully discrete scheme is defined as follows. Set *<sup>Z</sup>*ð Þ <sup>0</sup> to be a projection of *<sup>z</sup>*<sup>0</sup> onto some space *<sup>V</sup>*<sup>0</sup> subordinate to a mesh <sup>T</sup> <sup>0</sup> employed for the discretization of the initial condition. For *k* ¼ 1, … , *n*, find *Z* ∈*S<sup>n</sup>* such that the fully discrete, then reads as follows

$$\left(\frac{Z^n - Z^{n-1}}{K\_n}, \ \phi^n\right) + D(Z^n, \ \phi^n) = \left(\ f^n(Z^n), \ \phi^n\right), \ \forall \phi^n \in V^n \tag{10}$$

where *<sup>D</sup>n*ð Þ¼ �, � *D t*ð Þ *<sup>n</sup>*, �, � denotes the cG bilinear form defined on the mesh T *n* . Since *<sup>Z</sup><sup>n</sup>* <sup>∈</sup>*Vn*, there exist *<sup>α</sup>i*ð Þ*<sup>t</sup>* <sup>∈</sup> , *<sup>j</sup>* <sup>¼</sup> 0, 1, 2, … , *Nh*, so that

$$Z^{n}(\mathbf{x},\ t) = \sum\_{j=0}^{N\_{\text{ln}}N\_{d}} a\_{j}^{n}(t)\Phi\_{j}(\mathbf{x}),\ \ \Phi\_{j},\ \ j=0,1,2\ldots N\_{h} \tag{11}$$

is the basis functions. After plugging (11) into (10), yields a nonlinear system of ordinary differential equations

$$\begin{aligned} (\mathbf{M}\_{\ \perp} + \kappa\_n \mathbf{A}) a\_j^n(t) &= \mathbf{M} a\_j^{n-1}(t) + \kappa\_n \mathbf{F} \\ a(\mathbf{0}) &= \delta, \end{aligned} \tag{12}$$

where *Mi*,*<sup>j</sup>* ¼ Φ *<sup>j</sup>*, Φ*<sup>j</sup>* � � and *Ai*,*<sup>j</sup>* <sup>¼</sup> *<sup>D</sup>* <sup>Φ</sup>*j*, <sup>Φ</sup> *<sup>j</sup>* � � are called the mass and stiffness matrices with element *F <sup>j</sup>*,*<sup>k</sup>* ¼ *f* Φ *<sup>j</sup>* � �, Φ*<sup>k</sup>* � �. We define the piecewise linear interpolant *Z* and time-dependent elliptic reconstruction *w t*ð Þ as by the linear interpolant with respect to *t* of the values *Z<sup>n</sup>*�<sup>1</sup> and *Z<sup>n</sup>*, viz.,

$$Z(\mathbf{t}) \coloneqq \boldsymbol{\varepsilon}\_{n-1}(\mathbf{t}) Z^{n-1} + \boldsymbol{\varepsilon}\_{n}(\mathbf{t}) Z^{n}, \qquad w(\mathbf{t}) \coloneqq \boldsymbol{\varepsilon}\_{n-1} \mathbf{R}\_{be}^{n-1} Z^{n-1} + \boldsymbol{\varepsilon}\_{n} \mathbf{R}\_{be}^{n} Z^{n}, \tag{13}$$

where f g ℓ*<sup>n</sup>*�1, ℓ*<sup>n</sup>* denotes the linear Lagrange interpolation basis on the interval *In* are defined as

$$\ell\_n := \frac{t\_n - t}{K\_n}, \quad \ell\_{n-1} := \frac{t - t\_{n-1}}{K\_n}. \tag{14}$$

We give here some essential definitions in the error analysis of the discrete parabolic equations.

i. *L*<sup>2</sup> projection operator Π*<sup>n</sup>* 0; The operator defined Π*<sup>n</sup>* 0: *<sup>L</sup>*<sup>2</sup> ! *<sup>V</sup><sup>n</sup>*, 1<sup>≤</sup> *<sup>n</sup>*<sup>≤</sup> *<sup>N</sup>* such that

$$\left(\Pi\_0^n \nu, \ \phi^n\right) = \left(\upsilon, \ \phi^n\right) \quad \forall \phi^n \in V^n,\tag{15}$$

for all *v*∈ *L*<sup>2</sup> ð Þ Ω .

ii. Discrete elliptic operator: The elliptic operator defined A*<sup>n</sup> <sup>h</sup>*: *H*<sup>1</sup> <sup>0</sup>ð Þ! <sup>Ω</sup> *<sup>V</sup><sup>n</sup>* such that for *v*∈ *H*<sup>1</sup> <sup>0</sup>ð Þ Ω , reads

$$D\left(\mathbf{A}\_h^n v,\ \phi^n\right) = D(v,\ \phi^n)\ \ \forall \phi^n \in V^n. \tag{16}$$

*A Posteriori Error Analysis in Finite Element Approximation for Fully Discrete Semilinear… DOI: http://dx.doi.org/10.5772/intechopen.94369*

Using the above projections, (10) can be expressed in distributional form as

$$\frac{Z^n - \Pi\_0^n Z^{n-1}}{K\_n} + \mathbf{A}\_h^n Z^n = \Pi\_0^n f^n(Z^n). \tag{17}$$

## **5. Elliptic reconstruction**

The aim of this section will be introduced the elliptic reconstruction operator and then discuss the related aposteriori error analysis for the backward Euler approximation. To do this, we define the elliptic reconstruction R*<sup>n</sup> be* ∈ *H*<sup>1</sup> <sup>0</sup>ð Þ <sup>Ω</sup> of *<sup>Z</sup><sup>n</sup>* as the solution of elliptic problem

$$D\left(\mathbf{R}\_{b\epsilon}^{n}v,\ \phi\right) = \left(\mathbf{g}^{n},\ \phi\right),\tag{18}$$

for a given *<sup>v</sup>*∈*V<sup>n</sup>* and *<sup>g</sup><sup>n</sup>* <sup>¼</sup> <sup>Π</sup>*<sup>n</sup>* 0 *f <sup>n</sup> <sup>Z</sup><sup>n</sup>* ð Þ� *<sup>Z</sup>n*�Π*<sup>n</sup>* 0*Zn*�<sup>1</sup> *kn* . The crucial property, this operator R*<sup>n</sup> be* is orthogonal with respect to *D* such that

$$D\left(\mu - \mathbb{R}\_{b\epsilon}^{n}\mu, \ v\right) = 0 \quad \mu, \ v \in V^{n}. \tag{19}$$

The following lemma is the elliptic reconstruction error bound in the *H*<sup>1</sup> and *L*2-norms To see the proof, we refer the reader to [3] for details.

Lemma 1.3 (Posteriori error estimates). For any *Z<sup>n</sup>* ∈*V<sup>n</sup>*, the following elliptic a posteriori bounds hold:

$$\begin{aligned} \| \left( R\_{b\epsilon}^n Z^n - Z^n \right) \| \le \mathcal{C} \Phi\_{n, L\_2}^2 \\ \| \nabla \left( \left( R\_{b\epsilon}^n Z^n - Z^n \right) \right) \| \le \mathcal{C} \Phi\_{n, \mathcal{H}^1}^2 \end{aligned} \tag{20}$$

where

$$\begin{aligned} \boldsymbol{\Phi}\_{n,L\_2}^2 &:= \|\boldsymbol{h}\_n^2(\mathbf{g}^n + \boldsymbol{\Delta}^n \boldsymbol{Z}^n)\| + \|\boldsymbol{h}\_n^{3/2}[\boldsymbol{Z}^n]\|\_{\Sigma\_n}, \\ \boldsymbol{\Phi}\_{n,\mathcal{H}^1}^2 &:= \|\boldsymbol{h}\_n(\mathbf{g}^n + \boldsymbol{\Delta}^n \boldsymbol{Z}^n)\| + \|\boldsymbol{h}\_n^{1/2}[\boldsymbol{Z}^n]\|\_{\Sigma\_n}, \end{aligned} \tag{21}$$

and *g<sup>n</sup>* defined in (18).

Lemma 1.4 (Main semilinear parabolic error equation). The following error bounds hold

$$\begin{split} \left(\frac{\partial \rho}{\partial t}, \; \psi \right) + D(\rho, \; \phi) &= \left(f(\mathbf{z}) - f^n(\mathbf{Z}^n), \; \phi\right) + \left(\frac{\partial \varepsilon}{\partial t}, \; \phi\right) + D(w - w^n, \; \phi) \\ &+ \left(\Pi\_0^n f^n(\mathbf{Z}^n) - f^n(\mathbf{Z}^n) + \frac{\Pi\_0^n Z^{n-1} - Z^{n-1}}{K\_n}, \; \phi\right). \end{split} \tag{22}$$

**Proof:** To begin with, we first decompose the error as

$$
\varepsilon := \rho - \varepsilon, \quad \rho := \varepsilon - w, \quad \varepsilon := w - Z. \tag{23}
$$

By recalling (17), this becomes

$$\left(\frac{\partial Z}{\partial t}, \ \phi\right) + D(w^n, \ \phi) = \left(\frac{\Pi\_0^n Z^{n-1} - Z^{n-1}}{k\_n}, \ \phi\right) + \left(\Pi\_0^n f^n(Z^n), \ \phi\right) \ \forall \phi \in H\_0^1(\Omega), \tag{24}$$

$$
\left(\frac{\partial}{\partial t}[Z-z],\ \phi\right) + D(w^n - z,\ \phi) = \left(\Pi\_0^n f''(Z^n) - f(z),\ \phi\right) + \left(\frac{\Pi\_0^n Z^{n-1} - Z^{n-1}}{\kappa\_n},\ \phi\right),\tag{25}
$$

$$\begin{split} \left( \frac{\partial}{\partial t} [-x - w + w + Z^n], \ \phi \right) + D(w^n - w + w - z, \ \phi) &= \left( \Pi\_0^n f^n(Z^n) - f^n(Z^n), \ \phi \right) \\ + \left( f^n(Z^n) - f(z), \ \phi \right) + \left( \frac{\Pi\_0^n Z^{n-1} - Z^{n-1}}{K\_n}, \ \phi \right). \end{split} \tag{26}$$

$$T\_{n,1} \coloneqq \int\_{t\_{n-1}}^{t\_n} \left| D \left( w - w^n, \ \ \ \frac{\partial \rho}{\partial t} \right) \right| dt,\tag{27}$$

$$T\_{n,1} \le \left( \int\_{t\_{n-1}}^{t\_n} \left\| \frac{\partial \rho}{\partial t} \right\|^2 dt \right)^{1/2} (\kappa\_n)^{1/2} \Phi\_{n,2},\tag{28}$$

$$\Phi\_{n,2} \coloneqq \begin{cases} \frac{\sqrt{3}}{3} \vartheta \left( \left\| \prod\_{0}^{n} f^{n}(Z^{n}) - \frac{Z^{n} - \prod\_{0}^{n} Z^{n-1}}{k\_{n}} \right\| \right) \text{ for } n \in [2:N],\\ \frac{\sqrt{3}}{3} \left( \left\| \prod\_{0}^{1} f^{1}(Z^{1}) - \frac{Z^{1} - \prod\_{0}^{1} Z^{0}}{k1} \right\| \right) \text{ for } n = 1. \end{cases} \tag{29}$$

$$T\_{n,2} \coloneqq \int\_{t\_{n-1}}^{t\_n} \left| \begin{array}{cc} \left(\frac{\partial \varepsilon}{\partial t}, \ \frac{\partial \rho}{\partial t}\right) \end{array} \right| dt,\tag{30}$$

$$T\_{n,2} \le \left( \int\_{t\_{n-1}}^{t\_n} \left\| \frac{\partial \rho}{\partial t} \right\|^2 dt \right)^{1/2} (\kappa\_n)^{1/2} \Upsilon\_{n,2},\tag{31}$$

$$\Upsilon\_{n,2} := \mathbf{C}\left(\frac{d}{dt} \left\| h\_n^2 (\mathbf{g}^n + \Delta^n Z^n) \right\| \right) + \mathbf{C} \| \dot{h}\_n^{3/2} [Z^n - Z^{n-1}] \|\_{\Sigma\_n} + \mathbf{C} \| \dot{h}\_n^{3/2} [Z^n - Z^{n-1}] \|\_{\bar{\Sigma}\_n \bar{\Sigma}\_n} \tag{32}$$

$$T\_{n,3} \coloneqq \int\_{t\_{n-1}}^{t\_n} \left| \begin{array}{c} \Pi\_0^n f^n(Z^n) - f^n(Z^n) + \frac{\Pi\_0^n Z^{n-1} - Z^{n-1}}{\kappa\_n}, \ \frac{\partial \rho}{\partial t} \end{array} \right| dt,\tag{33}$$

$$T\_{n,3} \le \kappa\_n \max\_{t \in [0, t\_m]} \|\nabla \rho\| \left(\delta\_{n,\bullet} + \sum\_{n=2}^m \kappa\_n \delta\_{n,1} + \delta\_{\bullet,1}\right) \tag{34}$$

$$\begin{split} \delta\_{n,1} & \coloneqq \| h\_n^{\wedge} \partial \left( \Pi\_0^n - I \right) \left( f^n(Z^n) - \kappa\_n Z^{n-1} \right) \|, \\ \delta\_{n,\infty} & \coloneqq \| h\_n \left( \Pi\_0^n - I \right) \left( f^n(Z^n) - \kappa\_n Z^{n-1} \right) \|. \end{split} \tag{35}$$

$$|f(z\_1) - f(z\_2)| \le \mathcal{C}\_\emptyset |z\_1 - z\_2|,\tag{36}$$

$$\begin{split} T\_{n,4} &= \int\_{t\_{n-1}}^{t\_n} \left| \left( f(\mathbf{z}) - f^n(\mathbf{Z}^n), \frac{\partial \rho}{\partial t} \right) \right| dt \leq \frac{\sqrt{C\_\mathbf{g}}}{2\beta} \kappa\_n \| \nabla \rho \|^2 + \frac{\beta \sqrt{C\_\mathbf{g}}}{2} \int\_{t\_{n-1}}^{t\_n} \left\| \frac{\partial \rho}{\partial t} \right\|^2 dt \\ &+ \kappa\_n \Psi\_{n,1} \left( \int\_{t\_{n-1}}^{t\_n} \left\| \frac{\partial \rho}{\partial t} \right\|^2 dt \right)^{1/2} + \kappa\_n \Psi\_{n,2} \left( \int\_{t\_{n-1}}^{t\_n} \left\| \frac{\partial \rho}{\partial t} \right\|^2 dt \right)^{1/2}, \end{split} \tag{37}$$

$$\begin{cases} \Psi\_{n,1} := \sqrt{\mathbb{C}\_{\mathfrak{F}}} \left\{ \|\varepsilon^{n-1}\|, \quad \|\varepsilon^{n}\| \right\}, \\\Psi\_{n,2} := \frac{1}{\kappa\_{n}} \int\_{t\_{n-1}}^{t\_{n}} \|f(Z) - f^{n}(Z^{n})\|. \end{cases} \tag{38}$$

$$\begin{split} T\_{n,4} &= \int\_{t\_{n-1}}^{t\_n} \left| \left( f(\mathbf{z}) - f^n(\mathbf{Z}^n), \frac{\partial \rho}{\partial t} \right) \right| dt \leq \int\_{t\_{n-1}}^{t\_n} \left| \left( f(\mathbf{z}) - f(\mathbf{w}), \frac{\partial \rho}{\partial t} \right) \right| dt \\ &+ \int\_{t\_{n-1}}^{t\_n} \left| \left( f(\mathbf{w}) - f(\mathbf{Z}), \frac{\partial \rho}{\partial t} \right) \right| dt + \int\_{t\_{n-1}}^{t\_n} \left| \left( f(\mathbf{Z}) - f^n(\mathbf{Z}^n), \frac{\partial \rho}{\partial t} \right) \right| dt \\ &:= L\_{n,1} + L\_{n,2} + L\_{n,3} .\end{split} \tag{39}$$

$$\begin{split} L\_{n,1} &= \int\_{t\_{n-1}}^{t\_n} \left| \left( f(x) - f(w), \frac{\partial \rho}{\partial t} \right) \right| dt \leq \int\_{t\_{n-1}}^{t\_n} \| f(x) - f(w) \| \left| \left| \frac{\partial \rho}{\partial t} \right| \right| dt \\ &\leq \frac{\sqrt{C\_{\mathfrak{g}}}}{2\beta} \kappa\_n \| |\nabla \rho| \|^{2} + \frac{\beta \sqrt{C\_{\mathfrak{g}}}}{2} \int\_{t\_{n-1}}^{t\_n} \left\| \frac{\partial \rho}{\partial t} \right\|^{2} dt. \end{split} \tag{40}$$

$$L\_{n,2} = \int\_{t\_{n-1}}^{t\_n} \left| \left( f(\omega) - f(Z), \frac{\partial \rho}{\partial t} \right) \right| dt \le \int\_{t\_{n-1}}^{t\_n} \|\omega - Z\| \left\| \left| \frac{\partial \rho}{\partial t} \right| \right\| dt$$

$$\le \sqrt{C\_{\mathfrak{g}}} \int\_{t\_{n-1}}^{t\_n} \left( \left| \frac{t\_n - t}{\kappa\_n} \right| \|\epsilon^{n-1}\| + \left| \frac{t - t\_{n-1}}{\kappa\_n} \right| \|\epsilon^n\| \right) \left\| \frac{\partial \rho}{\partial t} \right\| dt \tag{41}$$

$$\le \frac{\sqrt{C\_{\mathfrak{g}}}}{2} \kappa\_n \left( \|\|\epsilon^{n-1}\| + \|\|\epsilon^n\| \right) \left( \int\_{t\_{n-1}}^{t\_n} \left| \left| \frac{\partial \rho}{\partial t} \right| \right|^2 dt \right)^{1/2}.$$

$$L\_{n,3} = \int\_{t\_{n-1}}^{t\_n} \left| \left( f(Z) - f^n(Z^n), \frac{\partial \rho}{\partial t} \right) \right| dt \le \| f(Z) - f^n(Z^n) \| \left( \int\_{t\_{n-1}}^{t\_n} \left\| \frac{\partial \rho}{\partial t} \right\|^2 dt \right)^{1/2}. \tag{42}$$

$$\left(\max\_{t \in [0, t\_m]} \left\| \nabla \rho(t) \right\|^2 + \int\_0^{t\_m} \left\| \frac{\partial \rho}{\partial t} \right\|^2 dt\right)^{1/2} \le \left\{ 2\mathcal{E}\_G(m) \left\| \nabla \rho \right\|^2 \right\}^{1/2} + 2\mathcal{E}\_G(m) \left( \mathcal{F}\_{1,m}^2 + \mathcal{F}\_{2,m}^2 \right) \tag{43}$$

$$\begin{aligned} \mathcal{F}\_{1,m} &:= 2 \max\_{t \in [0, t\_m]} \delta\_{m, \infty} + 2 \sum\_{n=2}^{m} \kappa\_n \delta\_{n, 1}, \\\\ \mathcal{F}\_{2,m}^2 &:= \sum\_{n=1}^{m} \kappa\_n \left( \Phi\_{n,2}^2 + \Upsilon\_{n,2}^2 + \Psi\_{n,1}^2 + \Psi\_{n,2}^2 \right). \end{aligned} \tag{44}$$

$$\begin{split} \frac{1}{2} \frac{d}{dt} \|\nabla \rho(t)\|^2 + \frac{\mathcal{C}\_{\text{corr}}}{2} \left| \frac{\partial \rho}{\partial t} \right|^2 \leq \left| \left( \frac{\partial \varepsilon}{\partial t}, \frac{\partial \rho}{\partial t} \right) \right| + \left| \left( f(\mathbf{z}) - f^n(\mathbf{Z}^n), \frac{\partial \rho}{\partial t} \right) \right| + \left| D \left( w - w^n, \frac{\partial \rho}{\partial t} \right) \right| \\ + \left| \left( \Pi\_0^n f(\mathbf{Z}^n) - f^n(\mathbf{Z}^n) + \frac{P\_0^n \mathbf{Z}^{n-1} - \mathbf{Z}^{n-1}}{k\_n}, \frac{\partial \rho}{\partial t} \right) \right|. \end{split} \tag{45}$$

*A Posteriori Error Analysis in Finite Element Approximation for Fully Discrete Semilinear… DOI: http://dx.doi.org/10.5772/intechopen.94369*

Integrate the above from *tn*�<sup>1</sup> to *tn* then, we have

$$\frac{1}{2} \|\nabla \rho(t\_n)\|^2 - \frac{1}{2} \|\nabla \rho(t\_{n-1})\|^2 + \frac{C\_{\text{cer}}}{2} \int\_{t\_{n-1}}^{t\_n} \left\|\frac{\partial \rho}{\partial t}\right\|^2 dt \le T\_{n,1} + T\_{n,2} + T\_{n,3} + T\_{n,4}, \tag{46}$$

where *Tn*,*i*, *i* ¼ 1, 2, 3, 4 defined in Lemmas 1.5, 1.6, 1.7 and 1.8, respectively. Summing up over *n* ¼ 1: *m* so that

$$\|\nabla\rho(t\_m)\|^2 + \mathcal{C}\_{cor}\int\_0^{t\_m} \left\|\frac{\partial\rho}{\partial t}\right\|^2 dt \le \|\nabla\rho(\mathbf{0})\|^2 + 2\sum\_{n=1}^m (T\_{n,1} + T\_{n,2} + T\_{n,3} + T\_{n,4}).\tag{47}$$

By introducing

$$\|\|\nabla(\rho\_m^\*)\|\|;=\|\nabla\rho(t\_m^\*)\|\|=\max\_{t\in[0,t\_m]}\|\nabla\rho(t)\|,\tag{48}$$

therefore

$$\max\_{t \in [0, t\_m]} \|\nabla \rho(t)\| + C\_{cor} \int\_0^{t\_m} \left\|\frac{\partial \rho}{\partial t}\right\|^2 dt \le 2 \|\nabla \rho(\mathbf{0})\|^2 + 4 \sum\_{n=1}^m (T\_{n,1} + T\_{n,2} + T\_{n,3} + T\_{n,4}).\tag{49}$$

Now, using Lemmas 1.5, 1.6, 1.7 and 1.8, reads

$$\begin{split} \max\_{t \in [0, t\_n]} \|\nabla \rho(t)\|^2 &\le 2\|\nabla \rho(0)\|^2 + \left(2\beta\sqrt{\mathsf{C}\_{\mathsf{g}}} - \mathsf{C}\_{\mathsf{cor}}\right) \int\_0^{t\_n} \left\|\frac{\partial \rho}{\partial t}\right\|^2 dt + 2\max\_{t \in [0, t\_n]} \|\nabla \rho(t)\| \mathcal{F}\_{1, \mathsf{m}} \\ &+ \frac{2\sqrt{\mathsf{C}\_{\mathsf{g}}}}{\beta} \sum\_{n=1}^m K\_n \max\_{t \in [0, t\_n]} \|\nabla \rho(t)\|^2 + 4 \left(\int\_{t\_{n-1}}^{t\_n} \left\|\frac{\partial \rho}{\partial t}\right\|^2 dt\right)^{1/2} (\kappa\_n)^{1/2} (\varPhi\_{n, 2} + \mathsf{Y}\_{n, 2} + \varPsi\_{n, 1} + \varPsi\_{n, 2}). \end{split} \tag{50}$$

Selecting now *β* > 0 be such that 2*β* ffiffiffiffiffi *Cg* <sup>p</sup> � *Ccoer* � �<sup>&</sup>gt; 0 and using Gronwall'<sup>s</sup> inequality, imply

$$\begin{split} \max\_{t \in [0, t\_n]} & \left\| \nabla \rho(t) \right\|^2 + \mathcal{E}\_G(m) \int\_0^{t\_n} \left\| \frac{\partial \rho}{\partial t} \right\|^2 dt \leq 2 \mathcal{E}\_G(m) \left\| \nabla \rho(\mathbf{0}) \right\|^2 + 2 \mathcal{E}\_G(m) \max\_{t \in [0, t\_n]} \left\| \nabla \xi(t) \right\| \mathcal{F}\_{1, m} \\ & + 4 \mathcal{E}\_G(m) \sum\_{n=1}^m \left( \int\_{t\_{n-1}}^{t\_n} \left\| \frac{\partial \rho}{\partial t} \right\|^2 dt \right)^{1/2} (\kappa\_n)^{1/2} (\Phi\_{n, 2} + \Upsilon\_{n, 2} + \Psi\_{n, 1} + \Psi\_{n, 2}), \end{split} \tag{51}$$

with <sup>E</sup>*G*ð Þ *<sup>m</sup>* <sup>≔</sup> 1, <sup>P</sup>*<sup>m</sup> n*¼1 <sup>2</sup> ffiffiffiffi *Cg* p *<sup>β</sup> κ<sup>n</sup>* exp <sup>2</sup> ffiffiffiffi *Cg* p *<sup>β</sup>* Σ*n*<*j*<sup>&</sup>lt; *mk <sup>j</sup>* � � � � � � . To finish the proof of lemma, we use a standard inequlty. For ð Þ *<sup>a</sup>*0, *<sup>a</sup>*1, … , *an* , ð Þ *<sup>b</sup>*0, *<sup>b</sup>*1, … , *bn* <sup>∈</sup> *<sup>m</sup>*þ<sup>1</sup> .

$$\left|\mathfrak{a}\right|^{2} \leq c^{2} + ab,\tag{52}$$

then

$$|a| \le |c| + |b|, \tag{53}$$

and by taking

$$a\_0 \coloneqq \max\_{t \in [0, t\_m]} \|\nabla \rho(t)\|, \quad a\_n \coloneqq \left\{ \mathcal{E}\_G(m) \int\_0^{t\_m} \left\| \frac{\partial \rho}{\partial t} \right\|^2 dt \right\}^{1/2}, \ c \coloneqq \left\{ 2\mathcal{E}\_G(m) \|\nabla \rho(\mathcal{O})\|^2 \right\}^{1/2} \tag{54}$$

$$b\_0 \coloneqq \sqrt{2} \mathcal{E}\_G(m) \mathcal{F}\_{1, m}, \quad b\_n \coloneqq 4 \mathcal{E}\_G(m) \sum\_{n=1}^m (\kappa\_n)^{1/2} (\Phi\_{n, 2} + \Upsilon\_{n, 2} + \Psi\_{n, 1} + \Psi\_{n, 2}).$$

The proof already will be finished.

Theorem 1.10 Let *z* be the exact solution of (7) and let *Z<sup>n</sup>* be its finite element approximation obtained by the backward Euler approximation (10). Then, for 1≤*n* ≤ *N*, the following a posteriori error bounds hold:

$$\begin{split} & \max\_{t \in [0, t\_m]} \|\nabla(\boldsymbol{z}(t) - \boldsymbol{Z}(t))\|^2 \leq 2\mathcal{E}\_G(\boldsymbol{m}) \Big( \Phi^2\_{n, \mathcal{H}^1}(\mathbf{0}) + \|\nabla(\boldsymbol{z}(\mathbf{0}) - \boldsymbol{Z}(\mathbf{0}))\|^2 \Big) \\ & + 2\mathcal{E}\_G(\boldsymbol{m}) \Big( \mathcal{F}^2\_{1, \boldsymbol{m}} + \mathcal{F}^2\_{2, \boldsymbol{m}} \Big) \Big) + 2 \max\_{t \in [0, t\_m]} \Phi^2\_{n, \mathcal{H}^1}, \end{split} \tag{55}$$

where Φ<sup>2</sup> *<sup>n</sup>*,H<sup>1</sup> defined in (20).

**Proof:** By decomposing *Z t*ðÞ� *z t*ð Þ into *ρ* and *ε*, so that

$$\left\|\nabla(Z(t) - z(t))\right\|^2 \le 2\left\|\nabla\varepsilon\right\|^2 + 2\left\|\nabla\rho\right\|^2. \tag{56}$$

To be able to bound the first term on the right hand side of (56), using (13), this becomes

$$\begin{split} \left\| \nabla \epsilon(t) \right\|^{2} &= \left\| \nabla (w(t) - Z(t)) \right\|^{2} = \left\| \nabla \left( \ell\_{n} \mathbf{R}\_{\mathbf{k}}^{n} Z^{n} + \ell\_{n-1} \mathbf{R}\_{\mathbf{k}}^{n-1} Z^{n-1} - \ell\_{n-1} (t) Z^{n-1} - \ell\_{n} (t) Z^{n} \right) \right\|^{2} \\ &\leq \ell\_{n} \left\| \nabla \left( \mathbf{R}\_{\mathbf{k}}^{n} Z^{n} - Z^{n} \right) \right\|^{2} + \ell\_{n-1} \left\| \nabla \left( \mathbf{R}\_{\mathbf{k}}^{n-1} Z^{n} - Z^{n-1} \right) \right\|^{2} \\ &\leq \max\_{t \in \left[ 0, t\_{n} \right]} \left\{ \left\| \nabla \left( \mathbf{R}\_{\mathbf{k}}^{n-1} Z^{n-1} - Z^{n-1} \right) \right\|^{2} , \left\| \nabla \left( \mathbf{R}\_{\mathbf{k}}^{n} Z^{n} - Z^{n} \right) \right\|^{2} \right\} \\ &\leq \max\_{t \in \left[ 0, t\_{n} \right]} \left\{ \left\| \nabla \left( \mathbf{R}\_{\mathbf{k}}^{n} Z^{n} - Z^{n} \right) \right\|^{2} \right\} \\ &\leq \max\_{t \in \left[ 0, t\_{n} \right]} \Phi\_{u, \mathbf{k}^{1}}^{2} . \end{split} \tag{57}$$

and ∥∇*ρ*ð Þ <sup>0</sup> <sup>∥</sup><sup>2</sup> <sup>¼</sup> ∥∇ð Þ *<sup>w</sup>*ð Þ� <sup>0</sup> *<sup>z</sup>*ð Þ <sup>0</sup> <sup>∥</sup><sup>2</sup> <sup>≤</sup>2∥∇*ε*ð Þ <sup>0</sup> <sup>∥</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>∥∇ð Þ *<sup>z</sup>*ð Þ� <sup>0</sup> *<sup>Z</sup>*ð Þ <sup>0</sup> <sup>∥</sup><sup>2</sup> . Finally, the second term on the right hand side of (56) will be estimated via Lemma 1.9.

#### **6.2 A posteriori error analysis for the locally Lipschitz continuity case**

Let *f*: *R* ! *R* is locally Lipschitz continuous for a.e. ð Þ *x*, *t* ∈ Ω ∪ ½ � 0, *T* , in the sense that there exist real numbers *CL* >0 and *γ* ≥0 such that

$$|f(u) - f(v)| = \mathcal{C}\_L(t) \left( \mathbb{1} + |u|^\gamma + |v|^\gamma \right) |u - v|. \tag{58}$$

Lemma 1.11 (Estimation of the nonlinear term). If the nonlinear reaction *f* is satisfying the growth condition (58) with 0 ≤*r*<2 for *d* ¼ 2, and with 0 ≤*r*≤4*=*3 for *d* ¼ 3, we have the bound

$$\begin{split} \|\|f(\mathbf{z}) - f^n(Z^n)\|\| &\lesssim \mathcal{N}\_1(t) \Big\{ \mathcal{N}\_2(Z)(\|\rho\| + \|\epsilon\|) + \sqrt{3} \|\|\rho\|\| \nabla \rho\|\|^r + \sqrt{5} \|\epsilon\| \|\| \nabla \epsilon\|\|^r \Big\} \\ &+ \Theta\_{n,3} \Big( \int\_{t\_{n-1}}^{t\_n} \left\|\left|\frac{\partial \rho}{\partial t}\right|\right\|^2 dt \Big)^{1/2}, \end{split} \tag{59}$$

*A Posteriori Error Analysis in Finite Element Approximation for Fully Discrete Semilinear… DOI: http://dx.doi.org/10.5772/intechopen.94369*

where \$\mathcal{N}\_1(t) \coloneqq \frac{1}{\sqrt{2}} \mathcal{C}\_L(t) \max \{1, 4^{\nu}\}\$, \$N(Z) \coloneqq \frac{1}{\sqrt{2}} \sqrt{1 + 4\gamma |Z|\_{\infty}^{2\gamma}}\$ and \$\Theta\_{n,3} \coloneqq \frac{1}{\kappa\_n} \int\_{t\_{n-1}}^{t\_n} ||f(Z) - f''(Z^n)||.

**Proof:** Applying triangle inequality, reads

$$\begin{split} T\_{L,4} &= \int\_{t\_{n-1}}^{t\_n} \left| \left( f(\mathbf{z}) - f^n(\mathbf{Z}^n), \frac{\partial \rho}{\partial t} \right) \right| dt \leq \int\_{t\_{n-1}}^{t\_n} \left| \left( f(\mathbf{z}) - f(\mathbf{Z}), \frac{\partial \rho}{\partial t} \right) \right| dt \\ &+ \int\_{t\_{n-1}}^{t\_n} \left| \left( f(\mathbf{Z}) - f^n(\mathbf{Z}^n), \frac{\partial \rho}{\partial t} \right) \right| dt \coloneqq \mathcal{J}\_{n,1} + \mathcal{J}\_{n,2} . \end{split} \tag{60}$$

J *<sup>n</sup>*,1 can be bounded as follows

$$\begin{split} \mathcal{I}\_{\mathfrak{n},1} &= \int\_{t\_{\mathfrak{n}-1}}^{t\_{\mathfrak{n}}} \left( \left. f(\mathbf{z}) - f(\mathbf{Z}), \frac{\partial \rho}{\partial t} \right| \leq \int\_{t\_{\mathfrak{n}-1}}^{t\_{\mathfrak{n}}} \left|| \left( f(\mathbf{z}) - f(\mathbf{Z}) \right| \right| \left| \frac{\partial \rho}{\partial t} \right| dt \\ &\leq \frac{1}{2} \left\| \left( f(\mathbf{z}) - f(\mathbf{Z}) \right) \right\|^{2} + \frac{1}{2} \int\_{t\_{\mathfrak{n}-1}}^{t\_{\mathfrak{n}}} \left\| \frac{\partial \rho}{\partial t} \right\|^{2} dt. \end{split} \tag{61}$$

Now, we have

$$\|f(\mathbf{z}) - f(Z)\|^2 = \int\_{t\_{n-1}}^{t\_n} \|f(\mathbf{z}) - f(Z)\|^2 dt \le \int\_{t\_{n-1}}^{t\_n} \|f(\mathbf{z}) - f(w)\|^2 dt + \int\_{t\_{n-1}}^{t\_n} \|f(w) - f(Z)\|^2 dt. \tag{62}$$
 
$$:= Z\_{1,n} + Z\_{2,n}. \tag{63}$$

To estimate Z1,*<sup>n</sup>* on the first term in the right hand side of (62), we use the Cauchy–Schwarz inequality and (58) to obtain

$$\begin{split} \mathbf{Z}\_{1,n} &= \int\_{t\_{n-1}}^{t\_n} \|f(\boldsymbol{z}) - f(\boldsymbol{w})\|^2 dt = C\_L^2(t) \int\_{t\_{n-1}}^{t\_n} \left( \mathbf{1} + |\boldsymbol{z}|^{2\gamma} + |\boldsymbol{w}|^{2\gamma} \right) |\boldsymbol{z} - \boldsymbol{w}|^2 \\ &\le \int\_{t\_{n-1}}^{t\_n} \left( \mathbf{1} + |\boldsymbol{z}|^{2\gamma} \right) |\boldsymbol{z} - \boldsymbol{w}|^2 dt + \int\_{t\_{n-1}}^{t\_n} |\boldsymbol{w}|^{2\gamma} |\boldsymbol{z} - \boldsymbol{w}|^2 dt. \end{split} \tag{63}$$

Applying the elementary inequality j j *Ca* þ *Cb* <sup>2</sup>*<sup>α</sup>* <sup>≤</sup>*C C*j j *<sup>a</sup>* <sup>2</sup>*<sup>α</sup>* <sup>þ</sup> j j *Cb* <sup>2</sup>*<sup>α</sup>* � � with *Ca* <sup>¼</sup> *z* � *w* and *Cb* ¼ *w*, so that j j *z* <sup>2</sup>*<sup>α</sup>* <sup>≤</sup>*C z*j j � *<sup>w</sup>* <sup>2</sup>*<sup>α</sup>* <sup>þ</sup> *C w*j j<sup>2</sup>*<sup>α</sup>* , this becomes

$$\begin{split} \mathbb{E}\left[\mathbf{Z}\_{1,\mathsf{w}}\leq & C\_{L}^{2}(t)\mathcal{C}\Big|\_{t\_{k-1}}^{t\_{k}}\left(1+|\boldsymbol{z}-\boldsymbol{w}|^{2^{2}}\right)|\boldsymbol{z}-\boldsymbol{w}|^{2}\mathrm{d}t+C\_{L}^{2}(t)\mathcal{C}\Big|\_{t\_{k-1}}^{t\_{k}}\left(2|\boldsymbol{w}-\boldsymbol{Z}|^{2^{2}}+2|\boldsymbol{Z}|^{2^{2}}\right)|\boldsymbol{z}-\boldsymbol{w}|^{2}\mathrm{d}t \\ \leq & C\_{L}^{2}(t)\mathcal{C}\max\left\{1,\mathbf{1}\left|\mathbf{f}'\right|\right\}\Big(\left(1+\boldsymbol{4}^{\prime}|\boldsymbol{Z}|^{2^{2}}\right)\left\|\boldsymbol{\rho}\right\|^{2}+\left\|\boldsymbol{\rho}\right\|\_{2+2\gamma}^{2+2\gamma}+2\int\_{t\_{k-1}}^{t\_{k}}\left\|\boldsymbol{\varepsilon}\right\|^{2^{\gamma}}\|\boldsymbol{\rho}\|^{2}\Big). \end{split} \tag{64}$$

Similarly, Z2,*<sup>n</sup>* follows as

$$\begin{split} \mathbf{Z}\_{2,n} &= \int\_{t\_{n-1}}^{t\_n} \|f(w) - f(Z)\|^2 dt = \mathbf{C}\_L^2(t) \mathbf{C} \Big|\_{t\_{n-1}}^{t\_n} \left( \mathbf{1} + |w|^{2\gamma} + |Z|^{2\gamma} \right) |w - Z|^2 \\ &\le \mathbf{C}\_L^2(t) \mathbf{C} \max\left\{ \mathbf{1}, \mathbf{1} \mathbf{6}^{\gamma} \right\} \Big( \left( \mathbf{1} + \mathbf{4}^{\nu} |Z|\_{\infty}^{2\gamma} \right) \|\varepsilon\|^2 + \|\varepsilon\|\_{2+2\gamma}^{2+2\gamma} \right). \end{split} \tag{65}$$

#### Collecting all these terms, we obtain

$$\begin{split} \|f(\boldsymbol{x}) - f(\boldsymbol{Z})\|^2 &\leq \mathcal{C}\_{\boldsymbol{L}}^2(\boldsymbol{t}) \mathsf{C} \max\left\{ \mathbf{1}, \mathsf{1} \mathsf{G}' \right\} \left( \mathbf{1} + \mathsf{4}^{\boldsymbol{\epsilon}} |\boldsymbol{Z}|\_{\boldsymbol{\epsilon}\boldsymbol{\epsilon}}^{2\boldsymbol{\gamma}} \right) \left( \|\boldsymbol{\rho}\|^2 + \|\boldsymbol{\epsilon}\|^2 \right) \\ &+ \mathsf{C}\_{\boldsymbol{L}}^2(\boldsymbol{t}) \mathsf{C} \max\left\{ \mathbf{1}, \mathsf{1} \mathsf{G}' \right\} \left( \|\boldsymbol{\rho}\|\_{2+2\boldsymbol{\gamma}}^{2+2\boldsymbol{\gamma}} + \mathsf{J} \|\boldsymbol{\epsilon}\|\_{2+2\boldsymbol{\gamma}}^{2+2\boldsymbol{\gamma}} + 2 \int\_{t\_{n-1}}^{t\_n} \|\boldsymbol{\epsilon}\|^2 \|\boldsymbol{\rho}\|^{2\boldsymbol{\gamma}} d\boldsymbol{t} \right). \end{split} \tag{66}$$

Using Holder's inequality and Young's inequality, we deduce that

$$\int\_{t\_{n-1}}^{t\_n} \|\alpha\|^{2r} \|\beta\|^2 d\mathbf{x} \le \frac{\|\alpha\|\_{2+2r}^{2+2r}}{r+1} + \frac{r\|\beta\|\_{2+2r}^{2+2r}}{r+1}.\tag{67}$$

Therefore,

$$\int\_{t\_{n-1}}^{t\_n} \left\| \varepsilon \right\|^{2r} \left\| \rho \right\|^2 \le \frac{\left\| \varepsilon \right\|\_{2+2\gamma}^{2+2\gamma}}{\gamma+1} + \frac{\gamma \left\| \rho \right\|\_{2+2\gamma}^{2+2\gamma}}{\gamma+1} \tag{68}$$

$$\le \left\| \varepsilon \right\|\_{2+2\gamma}^{2+2\gamma} + \left\| \rho \right\|\_{2+2\gamma}^{2+2\gamma}.$$

Substituting this into our grand inequality yields

$$\|f(\mathbf{z}) - f(\mathbf{Z})\|^2 \le \mathcal{N}\_1^2(\mathbf{t}) \left( \mathcal{N}\_2^2(\mathbf{Z}) \left( \|\rho\|^2 + \|\boldsymbol{\varepsilon}\|^2 \right) + \mathbf{3} \|\rho\|\_{2+2\gamma}^{2+2\gamma} + \mathbf{5} \|\boldsymbol{\varepsilon}\|\_{2+2\gamma}^{2+2\gamma} \right),\tag{69}$$

where <sup>N</sup> <sup>2</sup> <sup>1</sup>ðÞ¼ *<sup>t</sup>* <sup>1</sup> 2*C*<sup>2</sup> *<sup>L</sup>*ð Þ*<sup>t</sup> <sup>C</sup>* max 1, 16*<sup>γ</sup>* f g and <sup>N</sup> <sup>2</sup> <sup>2</sup>ð Þ¼ *<sup>Z</sup>* <sup>1</sup> <sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>4</sup>*<sup>r</sup>* j j *<sup>Z</sup>* <sup>2</sup>*<sup>r</sup>* ∞ � �. From Gagliardo-Nirenberg inequality in Theorem 1.2, implies that

$$\|\rho\|\_{2+2\gamma} \le C \|\nabla \rho\|^{\frac{(2+2\gamma)d-2d}{2}} \|\rho\|^{\frac{4+4q+2d-2d-2d\gamma}{2}},\tag{70}$$

valid for all *γ* ≥0 for *d* ¼ 2 and 0 ≤*γ* ≤2 for *d* ¼ 3. Combining this with the Poincar'e-Friedrichs inequality ∥*ρ*∥ ≤*C*∥∇*ρ*∥, yields

$$\|\|\rho\|\|\_{2+2\gamma} \le C \|\nabla \rho\|\,. \tag{71}$$

Finally,

$$\mathcal{I}\_{n,2} = \int\_{t\_{n-1}}^{t\_n} \left| \left( f(Z) - f^n(Z^n), \frac{\partial \rho}{\partial t} \right) \right| dt \le \left\| \left( f(Z) - f^n(Z^n) \right) \left( \int\_{t\_{n-1}}^{t\_n} \left\| \frac{\partial \rho}{\partial t} \right\|^2 dt \right)^{1/2} . \tag{72}$$

Putting all of the results together the proof will be finished.

Theorem 1.12 Let *z* be the exact solution of (7) and let *Z<sup>n</sup>* be its finite element approximation obtained by the backward Euler approximation (10). Then, for 1≤*n* ≤ *N*, the following a posteriori error bounds hold

$$\begin{split} & \max\_{t \in [0, t\_m]} \| \nabla (\boldsymbol{z}(t) - \boldsymbol{Z}(t)) \|^2 \le 4 \mathcal{E}(t\_n, \ \boldsymbol{Z}) \Big( \| \nabla (\boldsymbol{z}(0) - \boldsymbol{Z}(0)) \|^2 + \Phi^2\_{n, \mathcal{H}^1}(\mathbf{0}) \Big) \\ & + 4 \mathcal{E}(t\_n, \boldsymbol{Z}) \sum\_{n=1}^m \mathcal{F}^2\_{1, m} + 4 \mathcal{E}(t\_n, \ \boldsymbol{Z}) \sum\_{n=1}^m \kappa\_n^2 \Big\{ \Phi^2\_{n, 2} + \Upsilon^2\_{n, 2} + \Psi^2\_{n, 1} + \Psi^2\_{n, 2} \Big\} \\ & + 4 \mathcal{N}^2\_1(t) \mathcal{E}(t\_n, \ \boldsymbol{Z}) \sum\_{n=1}^m \Big( \mathcal{N}^2\_2(\boldsymbol{Z}) \Phi^2\_{n, L\_2} + \Phi^2\_{n, L\_2} \Phi^{2\gamma}\_{n, \mathcal{H}^1} \Big) + 2 \max\_{t \in [0, t\_m]} \Phi^2\_{n, \mathcal{H}^1}, \end{split} \tag{73}$$

where Φ<sup>2</sup> *<sup>n</sup>*,*L*<sup>2</sup> and Φ<sup>2</sup> *<sup>n</sup>*,H1 are given in (20). *A Posteriori Error Analysis in Finite Element Approximation for Fully Discrete Semilinear… DOI: http://dx.doi.org/10.5772/intechopen.94369*

**Proof:** Now, setting *<sup>v</sup>* <sup>¼</sup> *<sup>∂</sup><sup>ρ</sup> <sup>∂</sup><sup>t</sup>* in 22, and integrate from *tn*�<sup>1</sup> to *tn* along with summing up over *n* ¼ 1: *m* we have

$$\max\_{t \in [0, t\_n]} \|\nabla \rho(t)\|^2 + \mathcal{C}\_{\alpha er} \int\_0^{t\_n} \|\frac{\partial \rho}{\partial t}\|^2 dt \le \|\nabla \rho(\mathbf{0})\|^2 + 2\sum\_{n=1}^m \int\_{t\_{n-1}}^{t\_n} \|f(\mathbf{z}) - f^n(\mathbf{Z}^n)\|^2 \tag{74}$$

$$+ 2\sum\_{n=1}^m (T\_{n,1} + T\_{n,2} + T\_{n,3}).$$

Using Lemma 1.11, along with lemmas 1.3, 1.5, 1.6 and 1.7, imply

$$\begin{split} \max\_{t \in [0, t\_n]} & \|\nabla \rho(t)\|^2 + \int\_0^{t\_n} \left\|\frac{\partial \rho}{\partial t}\right\|^2 dt \le \|\nabla \rho(0)\|^2 + \sum\_{n=1}^m \mathcal{F}\_{1, m}^2 \\ & + \sum\_{n=1}^m \kappa\_n^2 (\Phi\_{n, 2}^2 + \Upsilon\_{n, 2}^2 + \Psi\_{n, 1}^2 + \Psi\_{n, 2}^2) + \mathcal{N}\_1^2(t) \sum\_{n=1}^m \left(\mathcal{N}\_2^2(\mathbf{Z}) \Phi\_{n, L\_2}^2 + \mathbf{5} \Phi\_{n, L\_2}^2 \Phi\_{n, \mathbf{H}^1}^{2r}\right) \\ & + \sum\_{n=1}^m \int\_{t\_{n-1}}^{t\_n} \left(\mathcal{N}\_1^2(t) \mathcal{N}\_2^2(\mathbf{Z}) \|\nabla \rho\|^2 + 3 \mathcal{N}\_1^2(t) \|\rho\|^2 \|\nabla \rho\|^{2r}\right). \end{split} \tag{75}$$

Setting

$$\begin{split} \mathcal{F}(t\_n, \ Z, \ \varepsilon)^2 &= \|\nabla\rho(\mathbf{0})\|^2 + \sum\_{n=1}^m \mathcal{F}\_{1,m}^2 + \sum\_{n=1}^m \kappa\_n^2 \{\Phi\_{n,2}^2 + \Upsilon\_{n,2}^2 + \Psi\_{n,1}^2 + \Psi\_{n,2}^2\} \\ &\quad + \mathcal{N}\_1^2(t) \sum\_{n=1}^m \Big(\mathcal{N}\_2^2(Z)\Phi\_{n,L\_2}^2 + \mathsf{5}\Phi\_{n,L\_2}^2\Phi\_{n,\mathbf{i}^1}^{\mathcal{H}}\Big). \end{split} \tag{76}$$

Upon observing that

$$\begin{split} \left| \int\_{t\_{n-1}}^{t\_n} \left\| \nabla \rho \right\| \right| ^{2r} \left\| \rho \right\| ^2 &\leq \max\_{t \in \left[0, t\_n\right]} \left\| \nabla \rho \right\| ^{2r} \int\_{t\_{n-1}}^{t\_n} \left\| \rho \right\| ^2 \right) \mathrm{d}s \\ &\leq \left( \max\_{t \in \left[0, t\_n\right]} \left\| \nabla \rho \right\| ^2 + \int\_{t\_{n-1}}^{t\_n} \left\| \rho \right\| ^2 \right) d t \right)^{r+1} . \end{split} \tag{77}$$

Now combining two equations, we obtain

$$\begin{split} & \max\_{t \in [0, t\_m]} \|\nabla \rho(t)\|^2 + \int\_0^{t\_m} \left\|\frac{\partial \rho}{\partial t}\right\|^2 dt \leq \mathcal{F}(t\_m, \ Z, \ e)^2 + \sum\_{n=1}^m \int\_{t\_{n-1}}^{t\_n} \mathcal{N}\_1^2(t) \mathcal{N}\_2^2(Z) \|\nabla \rho\|^2 \\ & + 3\mathcal{N}\_1^2(t) \sum\_{n=1}^m \left( \max\_{t \in [0, t\_m]} \|\nabla \rho(t)\|^2 + \int\_{t\_{n-1}}^{t\_n} \|\rho\|^2 dt \right)^{r+1} . \end{split} \tag{78}$$

To bound of the nonlinear term of above equation, we shall employ a continuation argument in the spirit of [17, 18]. To do that, we consider the set

$$\mathcal{M}\_n = \left\{ \lim\_{t \in [0, t\_m]} \left\| \nabla \rho(t) \right\|^2 + \mathcal{C}\_{cor} \int\_0^{t\_m} \left\| \frac{\partial \rho}{\partial t} \right\|^2 dt \le 4 \mathcal{F}(t\_m, Z, \ \mathrm{e})^2 \mathcal{E}(t\_m, Z) \right\},\tag{79}$$

where Eð Þ¼ *tm*, *<sup>Z</sup> exp* <sup>Ð</sup>*tm* <sup>0</sup> <sup>N</sup> <sup>2</sup> <sup>1</sup>ð ÞN*<sup>t</sup>* <sup>2</sup> <sup>2</sup>ð Þ *<sup>Z</sup> dt* � �. Since the left hand side of (78) depends continuously on *t*, and our aim is to show that M*<sup>n</sup>* ¼ ½ � 0, *T* . To do this, assuming *t* <sup>∗</sup> *<sup>m</sup>* <sup>¼</sup> maxM*<sup>n</sup>* <sup>&</sup>gt;0 and *<sup>t</sup>* <sup>∗</sup> *<sup>m</sup>* <*T*, imply

$$\max\_{t \in \left[0, t\_m^\*\right]} \|\nabla \rho(t)\|^2 + \int\_0^{t\_m^\*} \left\|\frac{\partial \rho}{\partial t}\right\|^2 dt \le \mathcal{F}(t\_n, \ Z, \ \varepsilon)^2 + \{4\mathcal{F}(t\_m, \ Z, \ \varepsilon)\mathcal{E}(t\_m, \ Z)\}^{r+1}$$

$$+ \mathcal{N}\_1^2(t)\mathcal{N}\_2^2(Z) \int\_0^{t\_m^\*} \|\nabla \rho\|^2 dt,\tag{80}$$

and Grönwall inequality, thus, implies

$$\begin{split} & \max\_{t \in \left[0, t\_m^\*\right]} \|\nabla \rho(t)\|^2 + \int\_0^{t\_m^\*} \left\|\frac{\partial \rho}{\partial t}\right\|^2 dt \leq \\ & \mathcal{E}\left(t\_m, \mathcal{Z}\right) \Big\{ \Big( 4\mathcal{N}\_1^2(t)\mathcal{F}\Big(t\_m, \mathcal{Z}, \ \operatorname{\boldsymbol{\varepsilon}}\big) \mathcal{E}(t\_m, \mathcal{Z}) \big)^{\operatorname{\boldsymbol{\gamma}}+1} + \mathcal{F}^2(t\_m, \mathcal{Z}, \ \operatorname{\boldsymbol{\varepsilon}})^2 \Big\}. \end{split} \tag{81}$$

Since <sup>E</sup> *<sup>t</sup>* <sup>∗</sup> *<sup>m</sup>*, *<sup>Z</sup>* � �≤Eð Þ *tm*, *<sup>Z</sup>* and, suppose that the maximum size *hmax* of the mesh is small enough that, for *h*<*h*max, satisfy

$$\mathcal{F}(t\_m, Z, \ \varepsilon) \le \left(\frac{1}{\mathcal{N}\_1^2(t)}\right)^r \left(\frac{1}{4\mathcal{F}(t\_m, Z, \varepsilon)^2 \mathcal{E}(t\_m, Z)}\right)^{r+1}.\tag{82}$$

This leads to

$$\mathcal{N}\_1^2(t) \quad \left(4\mathcal{F}(t\_m, Z, \ \varepsilon)^2 \mathcal{E}(t\_m, Z)\right)^{r+1} \le \mathcal{F}(t\_m, \ Z, \ \varepsilon)^2. \tag{83}$$

Then, (81), becomes

$$\max\_{\mu \in \left[0, t\_m^\*\right]} \left\lVert \nabla \rho(t) \right\rVert^2 + \int\_0^{t\_m^\*} \left\lVert \frac{\partial \rho}{\partial t} \right\rVert^2 dt \le 2\mathcal{E}(t\_m, \ Z) \mathcal{F}(t\_m, \ Z, \ \varepsilon)^2. \tag{84}$$

This leads to contradictions, because of *t* <sup>∗</sup> *<sup>m</sup>* suppose to be *t* <sup>∗</sup> *<sup>m</sup>* ¼ maxM*n*. The triangle inequality along with Lemma 1.3, imply that

$$\begin{split} \max\_{t \in [0, t\_m]} \left\lVert \nabla \boldsymbol{\varepsilon} \right\rVert^2 &\leq 2 \max\_{t \in [0, t\_m]} \left\lVert \nabla \rho \right\rVert^2 + 2 \max\_{t \in [0, t\_m]} \left\lVert \nabla \boldsymbol{\varepsilon} \right\rVert^2 \\ &\leq 4 \mathcal{F}(t\_m, \,\,\, Z, \,\,\, \varepsilon)^2 \mathcal{E}(t\_m, Z) + 2 \max\_{t \in [0, t\_m]} \boldsymbol{\Phi}^2\_{n, H^1}. \end{split} \tag{85}$$

By recalling (76), the proof already finished.

#### **7. Adaptive algorithms**

This section aims to explain an adaptive algorithm aiming to investigate the performance of the presented a posteriori bound from Theorems 1.10 and 1.12 for the backward-Euler cG method for the semilinear parabolic problem (6). To this

*A Posteriori Error Analysis in Finite Element Approximation for Fully Discrete Semilinear… DOI: http://dx.doi.org/10.5772/intechopen.94369*

end, the implementation of the adaptive algorithm will be based on the deal. II finite element library [19] to the present setting of semilinear problems. We shall write algorithm for Theorem 1.10. For the Theorem 1.12 will follow the same with some modifcations. To begin with, we have

$$\begin{split} \Psi^{j}\_{mi} &:= \|\nabla(\boldsymbol{z}(\mathbf{0}) - \boldsymbol{Z}(\mathbf{0}))\| + \|\nabla\boldsymbol{\epsilon}(\mathbf{0})\| \\ \Psi^{j}\_{time} &:= \sum\_{j=1}^{m} \left( \kappa\_{j} \frac{\sqrt{3}}{3} \boldsymbol{\partial} \left| \prod\_{j}^{j} f^{j} \left( \mathbf{Z}^{j} \right) - \frac{\mathbf{Z}^{j} - \Pi^{j}\_{0} \mathbf{Z}^{j-1}}{\kappa\_{j}} \right| \right| + \int\_{t\_{j-1}}^{t\_{j}} \|\boldsymbol{f}(\mathbf{Z}) - \boldsymbol{f}^{j}(\mathbf{Z}^{j})\| \\ \Psi^{j}\_{space} &:= \|\boldsymbol{h}\_{j} (\mathbf{g}^{j} + \boldsymbol{\Delta}^{j} \mathbf{Z}^{j})\| + \|\boldsymbol{h}\_{j}^{1/2} [\![\mathbf{Z}^{j}]\!] \|\_{\Sigma^{j}}. \end{split} \tag{86}$$

The adaptive algorithm from [15], starts with an initial uniform mesh in space and with a given initial time step. Starting from a uniform square mesh of 16 � 16 elements, the algorithm adapts the mesh to improve approximation to the initial condition using the initial condition estimator Ψ*ini* until some tolerance is satisfied. To adapt the timestep *κ <sup>j</sup>*, the algorithm bisects a time interval not satisfying a userdefined temporal tolerance Ψ *<sup>j</sup> time* ≤ **ttol**, and leaves a time-interval unchanged if ϒ *j time* ≤ **ttol**.

Once the time-step is adapted, the algorithm performs spatial mesh refinement and coarsening, determined by the space indicator Ψ *<sup>j</sup>* <sup>s</sup>*pace* using the user-defined tolerances **stol**<sup>þ</sup> and **stol**�, corresponding to refinement and coarsening, respectively. More specifically, we select the elements with the largest local contributions which result to Ψ *<sup>j</sup>* <sup>s</sup>*pace* >**stol**<sup>þ</sup> for refinement. The spatial coarsening threshold is set to **stol**� ¼ 0*:*001 ∗ **stol**<sup>þ</sup>; we select the elements with the smallest local contributions which result to Ψ *<sup>j</sup> space* <**stol**� for coarsening. The algorithm iterates for each time-step. We refer to [15] for the algorithm's workflow and all implementation details. The following two algorithms give the backward Euler method to the ODE system (12) and space-time adaptivity for Theorem 1.10.

**Algorithm 1.** The backward Euler method for solving the semilinear parabolic equation


$$M\_{i,j} = \int\_{I\_n} \phi\_j \phi\_i d\mathbf{x}, \quad A\_{i,j} = \int\_{I\_n} \phi'\_j \phi'\_i d\mathbf{x}, \quad F\_{i,j} = \int\_{I\_n} f\left(\phi\_j\right) \phi\_i d\mathbf{x}.\tag{87}$$

6: Solve

$$(M + \kappa\_n A)\alpha\_i^n(t) = M\alpha\_i^{n-1}(t) + \kappa\_n F. \tag{88}$$

7: **end for**

**Algorithm 2.** Space-time adaptivity.

1: Input *<sup>a</sup>*, *<sup>b</sup>*, *<sup>f</sup>*, *<sup>z</sup>*0, *<sup>T</sup>*, <sup>Ω</sup>, *<sup>n</sup>*, <sup>T</sup> , *ttol*, *stol*þ, *stol*� 2: Pick *<sup>κ</sup>*1, … , *<sup>κ</sup><sup>n</sup>* <sup>¼</sup> *<sup>T</sup> n*. 3: Compute *Z*<sup>0</sup> . 4: Compute *Z*<sup>1</sup> from *Z*<sup>0</sup> . 5: **while** Ψ<sup>1</sup> *time* <sup>2</sup> >*ttol*<sup>þ</sup> **or** max Ψ<sup>1</sup> <sup>s</sup>*pace* <sup>2</sup> <sup>&</sup>gt; *stol*<sup>þ</sup> do bisction <sup>T</sup> <sup>0</sup> by refining all elements such that Ψ<sup>1</sup> <sup>s</sup>*pace* <sup>2</sup> >*stol*<sup>þ</sup> and coarsening all elements such that Ψ1 <sup>s</sup>*pace* <sup>2</sup> < *stol*� 6: **if** Ψ<sup>1</sup> *time* <sup>2</sup> > *ttol*, **then**. 7: *n* � 1 *n*. 8: *Kn*<sup>¼</sup> *Kn*�1, … , *κ*<sup>2</sup> ¼ *κ*1. 9: *<sup>κ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>κ</sup>*<sup>1</sup> 2 . 10: *<sup>κ</sup>*<sup>1</sup> *<sup>κ</sup>*<sup>1</sup> 2 . 11: **end if.** 12: Compute *Z*<sup>0</sup> . 13: Compute *Z*<sup>1</sup> from *Z*<sup>0</sup> . 14: **end while** 15: put *<sup>j</sup>* <sup>¼</sup> 1, <sup>T</sup> <sup>1</sup> ¼ T 0, *time* <sup>¼</sup> *<sup>κ</sup>*1. 16: **while** *time*<*T* do 17: Calculute *Z <sup>j</sup>* from *Z <sup>j</sup>*�<sup>1</sup> . 18: **while** Ψ*<sup>i</sup> time* <sup>2</sup> > *ttol* **do** 19: **if** Ψ<sup>1</sup> *time* <sup>2</sup> >*ttol* **then** 20: *n* � 1 *n*. 21; *κ<sup>n</sup>* ¼ *κ<sup>n</sup>*�1, … , *κ <sup>j</sup>*þ<sup>2</sup> ¼ *κ <sup>j</sup>*þ1. 22: *<sup>κ</sup> <sup>j</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>κ</sup> <sup>j</sup>* 2 . 23; *<sup>κ</sup> <sup>j</sup> <sup>κ</sup> <sup>j</sup>* 2 . 24: **end if** 25: Compute *Z <sup>j</sup>* from *Z <sup>j</sup>*�<sup>1</sup> . 26: **end while** 27: Create <sup>T</sup> *<sup>j</sup>* from <sup>T</sup> *<sup>j</sup>*�<sup>1</sup> by refining all elements such that <sup>Ψ</sup>*<sup>i</sup>* <sup>s</sup>*pace* <sup>2</sup> >*stol*<sup>þ</sup> and coarsening all elements such that Ψ*<sup>i</sup>* <sup>s</sup>*pace* <sup>2</sup> <*stol*�. 28: Compute *Z <sup>j</sup>* from *Z <sup>j</sup>*�<sup>1</sup> . 29: *time time* þ *κ <sup>j</sup>*. 30: *j* � 1 *j*. 31: **end while**

#### **8. Conclusion**

The aim of this Chapter is to derive an optimal order a posteriori error estimates in term of the *L*<sup>∞</sup> *H*<sup>1</sup> for the fully semilinear parabolic problems in two cases when *f u*ð Þ Lipschitz and non Lipschitz are proved. The crucial tools in proving this error is the elliptic reconstruction techniques introduced by Makridakis and Nochetto 2003. This is consequently enabling us to use a posteriori error estimators derived for

*A Posteriori Error Analysis in Finite Element Approximation for Fully Discrete Semilinear… DOI: http://dx.doi.org/10.5772/intechopen.94369*

elliptic equation to obtain optimal order in terms of *L*<sup>∞</sup> *H*<sup>1</sup> norm for Lipschitz and non-Lipschitz nonlinearities. Some challenges have to be overcome due to nonlinearity on the forcing term depending on Gronwall's Lemma and Sobolev embedding through continuation argument. Furthermore, this will give insight about designing adaptive algorithm, which allow use to control the cost of computations. In the future, this Chapter can be extended to the fully discrete case for semilinear parabolic interface problems in *<sup>L</sup>*∞ð Þþ *<sup>L</sup>*<sup>2</sup> *<sup>L</sup>*<sup>2</sup> *<sup>H</sup>*<sup>1</sup> and *<sup>L</sup>*∞ð Þ *<sup>L</sup>*<sup>2</sup> norms [18, 20–22].
