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<h1>Approximate Solution of Nonlinear Hyperbolic Equations with Homogeneous Jump Conditions</h1>
<p class="authors">
<span class="author">Matthew O. Adewole\(^\ast \)</span>
</p>
<p class="date">February 12, 2019; accepted: May 28, 2019; published online: January 21, 2020.</p>
</div>
<div class="abstract"><p> We present the error analysis of a class of second order nonlinear hyperbolic interface problems where the spatial and time discretizations are based on a finite element method and linearized backward difference scheme respectively. Both semi discrete and fully discrete schemes are analyzed with the assumption that the interface is arbitrary but smooth. Almost optimal convergence rate in the \(H^1\)-norm is obtained. Numerical examples are given to support the theoretical result. </p>
<p><b class="bf">MSC.</b> 65M60, 65M12, 65M06 </p>
<p><b class="bf">Keywords.</b> Almost optimal, nonlinear hyperbolic equation, linearized backward difference. </p>
</div>
<p>\(^\ast \)Department of Computer Science and Mathematics, Mountain Top University, Prayer City, Ogun State, Nigeria, e-mail: <span class="tt">olamatthews@ymail.com</span>. </p>
<h1 id="a0000000002">1 Introduction</h1>
<p> In this work, we study finite element solution of the nonlinear hyperbolic equation of the form </p>
<div class="equation" id="eq1">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq1} u_{tt} - \nabla \cdot (a(x,u)\nabla u) + b(x,u)u= f(x,t) \quad \mbox{in} \quad \Omega \times (0,T] \end{equation}
  </div>
  <span class="equation_label">1.1</span>
</p>
</div>
<p> with initial and boundary conditions </p>
<div class="equation" id="a0000000003">
<p>
  <div class="equation_content">
    \begin{equation}  \left\{  \begin{array}{rcll} u(x,0) &  = &  u_0(x) &  \mbox{in }\Omega \\ u_t (x,0) &  = &  u_1(x) &  \mbox{in }\Omega \\ u(x,t) &  = &  0 &  \mbox{on }\partial \Omega \times [0,T] \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">1.2</span>
</p>
</div>
<p> and interface conditions </p>
<div class="equation" id="eq2">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq2} \left\{  \begin{array}{rcl} \displaystyle [u]_\Gamma &  = &  0 \\[1mm] \displaystyle \left[a(x,u)\tfrac {\partial u}{\partial n}\right]_\Gamma &  = &  0 \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">1.3</span>
</p>
</div>
<p> where \(0{\lt}T{\lt}\infty \) and \(\Omega \) is a convex polygonal domain in \(\mathbb {R}^2\) with boundary \(\partial \Omega \). \(\Omega _1\subset \Omega \) is an open domain with smooth boundary \(\Gamma = \partial \Omega _1\), \(\Omega _2=\Omega \setminus \bar{\Omega }_1\) is another open domain contained in \(\Omega \) with boundary \(\Gamma \cup \partial \Omega \), see Figure <a href="#fig1">1.</a>. The symbol \([u]\) is the jump of a quantity \(u\) across the interface \(\Gamma \) and \(n\) is the unit outward normal to the boundary \(\partial \Omega _1\). The interface conditions are defined as the difference of the limiting values from each side of the interface ie </p>
<div class="displaymath" id="a0000000004">
  \[  [u]_{m\in \Gamma } := \lim _{x\rightarrow m^+} u_1(x,t) - \lim _{x\rightarrow m^-} u_2(x,t)  \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000005">
  \[  \left[ a(x,u)\tfrac {\partial u}{\partial n} \right]_{m\in \Gamma } := \left[ \lim _{x\rightarrow m^+} a_1\nabla u_1(x,t) - \lim _{x\rightarrow m^-}a_2\nabla u_2(x,t) \right]\cdot n  \]
</div>
<p> where \(u_i(x,t)\), \(a_i(x,u)\), \(b_i(x,u)\) and \(f_i(x,t)\) are the restrictions of \(u(x,t)\), \(a(x,u)\), \(b(x,u)\) and \(f(x,t)\) to \(\Omega _i\), \(i=1,2\). The input functions \(a_i(x,u)\), \(b_i(x,u)\) and \(f_i(x,t)\) are assumed continuous on \(\Omega _i\), \(i=1,2\) for \(t\in [0,T]\). </p>
<figure id="fig1">
  <div class="centered"> <img src="img-0001.png" alt="\includegraphics[width=4cm]{Dom1.png}" style="width: 906px; height: 827px" />
 </div>
<p> <br /> </p>
<figcaption>
  <span class="caption_title">Figure</span> 
  <span class="caption_ref">1.</span> 
  <span class="caption_text">A polygonal domain \(\Omega = \Omega _1 \cup \Omega _2 \) with interface \(\Gamma \).</span> 
</figcaption>


</figure>
<p> We impose the following <div class="ass_thmwrapper theorem-style-remark" id="res1">
  <div class="ass_thmheading">
    <span class="ass_thmcaption">
    Assumption
    </span>
    <span class="ass_thmlabel">1.1</span>
  </div>
  <div class="ass_thmcontent">
  
<ol class="enumerate">
  <li><p>\(\Omega \) is a bounded convex polygonal domain in \(\mathbb {R}^2\), the interface \(\Gamma \subset \Omega \) and the boundary \(\partial \Omega \) are piecewise smooth, Lipschitz continuous and 1-dimensional. </p>
</li>
  <li><p>\(f(x,t)\in H^1(0,T;L^2(\Omega ))\). Functions \(a\) and \(b\) satisfy </p>
<div class="displaymath" id="a0000000006">
  \[  a_i(x,\xi )\geq \mu _1, \qquad b_i(x,\xi )\geq \mu _1, \qquad \| a_i(x,0)\| _{L^\infty (\Omega )} + \| b_i(x,0)\| _{L^\infty (\Omega )}\leq \mu _2,  \]
</div>
<div class="displaymath" id="a0000000007">
  \[  |a_i(x,\xi )-a_i(x,\psi )|+|b_i(x,\xi )-b_i(x,\psi )|\leq \mu _3\| \xi -\psi \| _{L^2(\Omega _i)} ,  \]
</div>
<p> for \(\xi ,\psi \in \mathbb {R}\), \(x\in \Omega _i\), \(t\in \mathbb {R}^+\) with positive constants \(\mu _1\), \(\mu _2\) and \(\mu _3\) independent of \(t,x,\xi ,\psi \). </p>
</li>
</ol>

  </div>
</div> </p>
<p>Hyperbolic partial differential equations arise in many physical problems such as vibrating string, vibrating membrane, shallow water waves, etc <span class="cite">
	[
	<a href="#debnath2012" >13</a>
	, 
	<a href="#leissa_mohamad2011" >23</a>
	, 
	<a href="#rao2007" >24</a>
	]
</span> and become interface problems when medium or materials with different properties are involved <span class="cite">
	[
	<a href="#brekhovskikh1980" >10</a>
	, 
	<a href="#deka_sinha2012" >16</a>
	, 
	<a href="#deka_ahmed2013" >15</a>
	]
</span>. The solutions of interface problems have low regularity globally but may have higher regularities in each individual material region because of the discontinuities across the interface <span class="cite">
	[
	<a href="#kellogg1971" >21</a>
	]
</span>. Thus, obtaining exact solutions or approximate solutions with higher order accuracy may be difficult. </p>
<p>Finite element solutions of non-interface hyperbolic problems have been extensively discussed in <span class="cite">
	[
	<a href="#baker1976" >7</a>
	, 
	<a href="#baker_bramble1979" >8</a>
	, 
	<a href="#baker_dougalis1980" >9</a>
	, 
	<a href="#dupont1973" >17</a>
	, 
	<a href="#georgoulis_etal2013" >19</a>
	, 
	<a href="#larsson_thomee2003" >22</a>
	, 
	<a href="#rauch1985" >25</a>
	]
</span>. The convergence of finite element solutions of linear hyperbolic interface problems has been considered in <span class="cite">
	[
	<a href="#1adewole2018" >3</a>
	, 
	<a href="#adewole2019_1" >4</a>
	, 
	<a href="#deka2017" >14</a>
	, 
	<a href="#deka_ahmed2013" >15</a>
	, 
	<a href="#deka_sinha2012" >16</a>
	]
</span>. In <span class="cite">
	[
	<a href="#deka_sinha2012" >16</a>
	]
</span>, the authors assumed that the interface can be fitted exactly using interface elements with curved edges and established convergence rates of optimal order for both semi and full discretizations. Time discretization was based on symmetric difference approximation around the nodal points. In <span class="cite">
	[
	<a href="#deka_ahmed2013" >15</a>
	]
</span>, approximation properties of interpolation and projection operators were used to establish convergence rates of optimal order for finite element solution of an homogenous hyperbolic interface problem. Their time discretization was also based on symmetric difference approximation around the nodal points. Linear finite element with time discretization based on implicit scheme was presented for wave equation with discontinuous coefficient in <span class="cite">
	[
	<a href="#deka2017" >14</a>
	]
</span>. In <span class="cite">
	[
	<a href="#1adewole2018" >3</a>
	]
</span>, we investigated the error contributed by semi discretization to the finite element solution of linear hyperbolic interface problems. With low regularity assumptions on the solution across the interface and with the assumption that the interface could not be fitted exactly, almost optimal convergence rates in \(L^2(\Omega )\) and \(H^1(\Omega ) \) norms were established. In <span class="cite">
	[
	<a href="#adewole2019_1" >4</a>
	]
</span>, we proposed finite element solution of a linear hyperbolic interface problem where the interface was approximated by straight lines. Quasi-uniform triangular elements were used for the spatial discretization and time discretization was based on a three-step implicit scheme. The proposed scheme was proved to be stable and preserves the discrete maximum principle under certain conditions on the input data. Almost optimal convergence rates in \(L^2(\Omega )\) and \(H^1(\Omega )\) norms were obtained. Inspite of the wide applicability of nonlinear hyperbolic equations, the discussion on finite element solutions of nonlinear hyperbolic interface problems of the form <a href="#eq1" class="eqref">1.1</a>–<a href="#eq2" class="eqref">1.3</a> is scarce in literature. </p>
<p>The objective of this paper is to establish convergence in the \(H^1\)-norm for the approximate solution of nonlinear hyperbolic interface problems of the form <a href="#eq1" class="eqref">1.1</a>–<a href="#eq2" class="eqref">1.3</a> on finite elements. Both semi discrete and fully discrete schemes are analyzed. Full discretization of <a href="#eq1" class="eqref">1.1</a>–<a href="#eq2" class="eqref">1.3</a> results to a system of nonlinear equations due to the presence of \(a(x,u)\) and \(b(x,u)\). We propose a linearized scheme in order to avoid this difficulty, and for practicability of the scheme, we do not assume that the interface could be perfectly fitted. The interface is first approximated by piecewise continuous straight lines and the mesh is fitted to this approximation. In this study, we use the standard notations and properties of Sobolev spaces as contained in <span class="cite">
	[
	<a href="#adams1975" >1</a>
	]
</span>. Other tools used in this paper are the linear theories of interface and non-interface problems, as well as approximation properties of the elliptic projection operator. </p>
<p>Let \(v_i\) be the restriction of \(v\) to \(\Omega _i\), \(i=1,2\), we shall need the following spaces for the convergence analysis </p>
<div class="equation" id="a0000000008">
<p>
  <div class="equation_content">
    \begin{equation} \notag X=\left\{  v:v\in H^1(\Omega ), v_i\in H^2(\Omega _i) \right\}  \;  ,\quad Y=\left\{  v:v\in L^2(\Omega ), v_i\in H^1(\Omega _i) \right\}  \end{equation}
  </div>
  <span class="equation_label">1.4</span>
</p>
</div>
<p> equipped with the norms </p>
<div class="equation" id="a0000000009">
<p>
  <div class="equation_content">
    \begin{equation} \notag \| v\| _X = \| v\| _{H^1(\Omega )} + \| v_1\| _{H^2(\Omega _1)} + \| v_2\| _{H^2(\Omega _2)} \quad \forall \;  v\in X, \end{equation}
  </div>
  <span class="equation_label">1.4</span>
</p>
</div>
<div class="displaymath" id="a0000000010">
  \begin{equation*}  \| v\| _Y = \| v\| _{L^2(\Omega )} + \| v_1\| _{H^1(\Omega _1)} + \| v_2\| _{H^1(\Omega _2)} \quad \forall \;  v\in Y. \end{equation*}
</div>
<p>The weak form of <a href="#eq1" class="eqref">1.1</a>\(-\)<a href="#eq2" class="eqref">1.3</a> is to find \(u(t)\in H^1_0(\Omega )\), \(t\in (0,T]\) such that </p>
<div class="equation" id="eq3">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq3} (u_{tt},v) + A(u:u,v) = (f,v) \quad \forall \;  v(t)\in H^1_0(\Omega ), \;  t\in (0,T] \end{equation}
  </div>
  <span class="equation_label">1.4</span>
</p>
</div>
<p> where </p>
<div class="displaymath" id="a0000000011">
  \[  (\phi ,\psi ) = \int _\Omega \phi \psi \;  dx \quad A(\xi :\phi ,\psi ) = \int _\Omega \left[ a(x,\xi ) \nabla \phi \cdot \nabla \psi +b(x,\xi )\phi \psi \right]\;  dx . \]
</div>
<p> For <a href="#eq3" class="eqref">1.4</a>, we have the following energy estimate <div class="thm_thmwrapper theorem-style-plain" id="a0000000012">
  <div class="thm_thmheading">
    <span class="thm_thmcaption">
    Theorem
    </span>
    <span class="thm_thmlabel">1.2</span>
  </div>
  <div class="thm_thmcontent">
  <p>Suppose that the conditions of Assumption <span class="rm"><a href="#res1">1.1</a></span> are satisfied for \(a:\Omega \times \mathbb {R}\rightarrow \mathbb {R}\), \(b:\Omega \times \mathbb {R}\rightarrow \mathbb {R}\), and \(f:\Omega \times \mathbb {R}\rightarrow \mathbb {R}^+\). Then there exists a \(C{\gt}0\) such that </p>
<div class="displaymath" id="eq10">
  \begin{multline} \label{eq10} \| u\| _{L^2(0,T;X)}+\| u_t\| _{L^2(0,T;Y)}+\| u_{tt}\| _{L^2(0,T;L^2(\Omega _1)\cap L^2(\Omega _2))}\leq \\ \leq C\left(\| f\| _{H^1(0,T;L^2(\Omega ))} + \| f(x,0)\| _{L^2(\Omega )} + \| u_0\| _X + \| u_1\| _Y \right) \end{multline}
</div>

  </div>
</div> </p>
<div class="proof_wrapper" id="a0000000013">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  <p>Let \(v=u_t\) in <a href="#eq3" class="eqref">1.4</a>. For \(t\in [0,T]\), a simple calculation shows that </p>
<div class="equation" id="eq14">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq14} \| u_t\| ^2_{L^2(\Omega )} + \| u\| ^2_{H^1(\Omega )} \leq C \left[ \| u_0\| ^2_{H^1(\Omega )} + \| u_1\| ^2_{L^2(\Omega )} + \| f\| ^2_{L^2(0,t;L^2(\Omega ))} \right] \;  . \end{equation}
  </div>
  <span class="equation_label">1.7</span>
</p>
</div>
<p> Let </p>
<div class="equation" id="a0000000014">
<p>
  <div class="equation_content">
    \begin{equation}  \notag \left. a_1(x,u)\tfrac {\partial u}{\partial n}\right|_\Gamma = g_1 \qquad \mbox{and}\qquad \left. a_2(x,u)\tfrac {\partial u}{\partial n}\right|_\Gamma = -g_2. \end{equation}
  </div>
  <span class="equation_label">1.8</span>
</p>
</div>
<p> It is clear from <a href="#eq2" class="eqref">1.3</a> that \(g_1 +g_2=0\). From <a href="#eq1" class="eqref">1.1</a>\(-\)<a href="#eq2" class="eqref">1.3</a>, we have </p>
<div class="equation" id="a0000000015">
<p>
  <div class="equation_content">
    \begin{equation} \notag \int _{\Omega _i} u_{ttt} v + \int _{\Omega _i} (a\nabla u_t \cdot \nabla v + bu_tv ) = \int _{\Omega _i} f_tv + \int _{\Gamma }g_{it}v \;  , \qquad i=1,2. \end{equation}
  </div>
  <span class="equation_label">1.8</span>
</p>
</div>
<p> We take \(v=u_{tt}\) and obtain </p>
<div class="equation" id="a0000000016">
<p>
  <div class="equation_content">
    \begin{equation} \notag \tfrac {d}{dt}\| u_{tt}\| ^2_{L^2(\Omega _i)} + \mu _1 \tfrac {d}{dt}\| u_t\| ^2_{H^1(\Omega _i)} - \| u_{tt}\| ^2_{L^2(\Omega _i)} \;  \leq \;  \tfrac {1}{4}\| f_t\| ^2_{L^2(\Omega _i)} + \int _{\Gamma }g_{it}u_{tt} \end{equation}
  </div>
  <span class="equation_label">1.8</span>
</p>
</div>
<p> which implies </p>
<div class="displaymath" id="eq17">
  \begin{align}  \label{eq17} & \| u_{tt}\| ^2_{L^2(\Omega _i)} + \| u_t\| ^2_{H^1(\Omega _i)} \leq \\ \nonumber & \leq C \left( \| u_0\| ^2_{H^2(\Omega _i)} + \| u_1\| ^2_{H^1(\Omega _i)} + \int _{0}^{t}\| f_t\| ^2_{L^2(\Omega _i)} dt + \| f(x,0)\| ^2_{L^2(\Omega _i)} \right)\\ \nonumber & \quad + \int _{0}^{t}\exp (-s)\int _{\Gamma }g_{it}u_{tt}\; dt.\nonumber \end{align}
</div>
<p> It follows directly that </p>
<div class="displaymath" id="eq19">
  \begin{align} \label{eq19} & \int _{0}^{T}\left(\| u_{tt}\| ^2_{L^2(\Omega _1)} + \| u_{tt}\| ^2_{L^2(\Omega _2)}+ \| u_t\| ^2_{H^1(\Omega _1)} + \| u_t\| ^2_{H^1(\Omega _2)} \right) \;  dt \leq \\ & \leq C \left[\| u_0\| ^2_X + \| u_1\| ^2_Y + \| f(x,0)\| ^2_{L^2(\Omega )}+ \int _{0}^{T} \| f_t\| ^2_{L^2(\Omega )} \;  dt \right]\nonumber \end{align}
</div>
<p> Now, we multiply <a href="#eq1" class="eqref">1.1</a> by \(-u_{tt}\), integrate over \(\Omega _i\) then simplify the resulting equation and obtain </p>
<div class="equation" id="eq21">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq21} \mu _1^2 \| u\| ^2_{H^2(\Omega _i)} \leq 2\mu _3^2 \| u\| ^2_{H^1(\Omega _i)} + 2\| u_{tt}\| ^2_{L^2(\Omega _i)} + 2\| f\| ^2_{L^2(\Omega _i)} + 2\int _{\Gamma }bg_{i}u. \end{equation}
  </div>
  <span class="equation_label">1.9</span>
</p>
</div>
<p> It follows from <a href="#eq14" class="eqref">1.7</a>, <a href="#eq17" class="eqref">1.8</a> and <a href="#eq21" class="eqref">1.9</a> that </p>
<div class="displaymath" id="eq20">
  \begin{multline} \label{eq20} \int _{0}^{T}\left(\| u\| ^2_{H^2(\Omega _1)} + \| u\| ^2_{H^2(\Omega _2)} \right) \;  dt \leq \\ \leq C \left[\| u_0\| ^2_X + \| u_1\| ^2_Y + \| f(x,0)\| ^2_{L^2(\Omega )}+ \int _{0}^{T} \left( \| f\| ^2_{L^2(\Omega )} + \| f_t\| ^2_{L^2(\Omega )} \right) dt \right]. \end{multline}
</div>
<p> <a href="#eq10" class="eqref">1.5</a> follows from <a href="#eq14" class="eqref">1.7</a>, <a href="#eq19" class="eqref">1.8</a> and <a href="#eq20" class="eqref">1.10</a>. </p>

  </div>
</div>
<p><div class="rem_thmwrapper theorem-style-remark" id="a0000000017">
  <div class="rem_thmheading">
    <span class="rem_thmcaption">
    Remark
    </span>
    <span class="rem_thmlabel">1.3</span>
  </div>
  <div class="rem_thmcontent">
  <p>Estimate <a href="#eq10" class="eqref">1.5</a> establishes that a weak solution exists. For \(u\in L^\infty (0,T;H^{m+1}(\Omega ))\), \(m\in \mathbb {N}\), standard energy argument for hyperbolic equations requires that \(u_0\in H^{m+1}(\Omega )\) and \(u_1\in H^m(\Omega )\) <span class="cite">
	[
	<a href="#Evans1998" >18</a>
	, 
	Theorems 5 and 6, pages 389-391
	]
</span>. However, this level of global regularity is not guaranteed for interface problems as such problems are more regular on the individual domain than the entire domain <span class="cite">
	[
	<a href="#babuska1970" >6</a>
	, 
	<a href="#kellogg1971" >21</a>
	]
</span>. </p>

  </div>
</div> </p>
<p>This paper is organized as follows. In Section <a href="#FED">2</a>, we describe a finite element discretization of the problem and state a result on the elliptic projection operator used for the error analysis. In Section <a href="#EE">3</a>, we give the discrete versions of <a href="#eq3" class="eqref">1.4</a> then establish the convergence rates of almost optimal order for both semi discrete and fully discrete schemes. We confirm our theoretical analysis with numerical examples in Section <a href="#Ex">4</a>. Throughout this paper, \(C\) is a generic positive constant (which is independent of the mesh parameter \(h\) and the time step size \(k\)) and may take on different values at different occurrences. </p>
<h1 id="FED">2 Finite Element Discretization</h1>
<p> \(\mathcal{T}_h\) denotes a conforming triangulation of \(\Omega \). Let \(h_K\) be the diameter of an element \(K\in \mathcal{T}_h\) and \(h=\max _{K\in \mathcal{T}_h} h_K\). Let \(\mathcal{T}^\star _h\) denote the set of all elements that are intersected by the interface \(\Gamma \) (see Fig <a href="#fig12">2.</a>); </p>
<div class="displaymath" id="a0000000018">
  \[  \mathcal{T}^\star _h = \{  K\in \mathcal{T}_h:K\cap \Gamma \neq \emptyset \}   \]
</div>
<p> \(K\in \mathcal{T}^\star _h\) is called an interface element and we write \(\Omega ^\star _h = \bigcup _{K\in \mathcal{T}^\star _h} K\). </p>
<p>The domain \(\Omega _1\) is approximated by a domain \(\Omega ^h_1\) with a polygonal boundary \(\Gamma _h\) whose vertices all lie on the interface \(\Gamma \). \(\Omega ^h_2\) represents the domain with \(\partial \Omega \) and \(\Gamma _h\) as its exterior and interior boundaries respectively. The triangulation \(\mathcal{T}_h\) of the domain \(\Omega \) is fitted to \(\Omega ^h_1\) and satisfies the following conditions </p>
<ol class="enumerate">
  <li><p>\(\displaystyle \bar{\Omega }=\bigcup _{K\in \mathcal{T}_h}\bar{K}\) </p>
</li>
  <li><p>If \(\bar{K}_1, \bar{K}_2 \in \mathcal{T}_h\) and \(\bar{K}_1\neq \bar{K}_2\), then either \(\bar{K}_1\cap \bar{K}_2 = \emptyset \) or \(\bar{K}_1\cap \bar{K}_2\) is a common vertex or a common edge. </p>
</li>
  <li><p>Each \(K\in \mathcal{T}_h\) is either in \(\Omega ^h_1\) or \(\Omega ^h_2\), and has at most two vertices lying on \(\Gamma _h\). </p>
</li>
  <li><p>For each element \(K\in \mathcal{T}_h\), let \(r_K\) and \(\bar{r}_K\) be the diameters of its inscribed and circumscribed circles respectively. It is assumed that, for some fixed \(h_0{\gt}0\), there exists two positive constants \(C_0\) and \(C_1\), independent of \(h\), such that </p>
<div class="displaymath" id="a0000000019">
  \[  C_0r_K \leq h \leq C_1 \bar{r}_K \quad \forall \;  h\in (0,h_0)  \]
</div>
</li>
</ol>
<figure id="fig12">
  <div class="centered"> <img src="img-0002.png" alt="\includegraphics[width=5cm]{work121.png}" style="width: 600px; height: 400px" />
 </div>
<p> <br /> </p>
<figcaption>
  <span class="caption_title">Figure</span> 
  <span class="caption_ref">2.</span> 
  <span class="caption_text">A typical interface element.</span> 
</figcaption>


</figure>
<p>Let \(S_h \subset H^1_0(\Omega )\) denote the space of continuous piecewise linear functions on \(\mathcal{T}_h\) vanishing on \(\partial \Omega \). <br />The FE solution \(u_h(x,t)\in S_h\) is represented as </p>
<div class="displaymath" id="a0000000020">
  \[  u_h(x,t)=\sum ^{N_h}_{j=1}\alpha _j(t)\phi _j(x) \; ,  \]
</div>
<p> where each basis function \(\phi _j\), \((j=1,2,\ldots ,N_h)\) is a pyramid function with unit height. For the approximation \(\hat{g}(t)\), let \(\{ z_j\} ^{n_h}_{j=1}\) be the set of all nodes of the triangulation \(\mathcal{T}_h\) that lie on the interface \(\Gamma \) and \(\{ \psi _j\} ^{n_h}_{j=1}\) be the hat functions corresponding to \(\{ z_j\} ^{n_h}_{j=1}\) in the space \(S_h\). </p>
<p>Let \(P_h:X\cap H^1_0(\Omega )\rightarrow S_h\) be the elliptic projection of the exact solution \(\nu \) in \(S_h\) defined by </p>
<div class="equation" id="eq22">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq22} A(u:\nu -P_h\nu , \phi ) = 0 \quad \forall \;  \phi \in S_h, \;  t\in [0,T]. \end{equation}
  </div>
  <span class="equation_label">2.12</span>
</p>
</div>
<p> For this projection, we have <div class="lem_thmwrapper theorem-style-plain" id="res3">
  <div class="lem_thmheading">
    <span class="lem_thmcaption">
    Lemma
    </span>
    <span class="lem_thmlabel">2.1</span>
  </div>
  <div class="lem_thmcontent">
  <p> Let \(a=a(x,u)\), \(b=b(x,u)\) satisfy Assumption <span class="rm"><a href="#res1">1.1</a></span> and let \(a_{tt}\), \(b_{tt}\) be continuous on \(\Omega _i\times (0,T]\), \(i=1,2\). Assume that \(u\in X\cap H^1_0\) and let \(P_hu\) be defined as in <a href="#eq22" class="eqref">2.12</a>, then </p>
<div class="displaymath" id="a0000000021">
  \begin{eqnarray*}  \| \tfrac {\partial ^n}{\partial t^n}(P_hu-u)\| _{H^1(\Omega )}& \leq &  Ch\left(1+\tfrac {1}{|\ln h|}\right)^{1/2} \sum ^n_{i=1}\| \tfrac {\partial ^i u}{\partial t^i}\| _X \\ \| \tfrac {\partial ^n}{\partial t^n}(P_hu-u)\| _{L^2(\Omega )}& \leq &  Ch^2\left(1+\tfrac {1}{|\ln h|}\right) \sum ^n_{i=1}\| \tfrac {\partial ^i u}{\partial t^i}\| _X \end{eqnarray*}
</div>
<p> for \(n=0,1,2\). </p>

  </div>
</div></p>
<div class="proof_wrapper" id="a0000000022">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  <p>It can be proved using the interpolation error estimate <span class="cite">
	[
	<a href="#adewole2017" >2</a>
	, 
	Lemma 2.1
	]
</span> and a similar argument to the proof of <span class="cite">
	[
	<a href="#adewole_payne2018" >5</a>
	, 
	Lemmas 2.4 and 2.5
	]
</span> but with little modification due to different assumptions on \(a(x,u)\) and \(b(x,u)\). </p>

  </div>
</div>
<p> <div class="rem_thmwrapper theorem-style-remark" id="a0000000023">
  <div class="rem_thmheading">
    <span class="rem_thmcaption">
    Remark
    </span>
    <span class="rem_thmlabel">2.2</span>
  </div>
  <div class="rem_thmcontent">
  <p>The term \(|\ln h|\) in Lemma <a href="#res3">2.1</a> is due to the fact that the mesh in Section <a href="#FED">2</a> cannot perfectly fit the interface. However, with the use of interface elements with curved edges along the interface, convergence rate of optimal is obtainable (see <span class="cite">
	[
	<a href="#deka_sinha2012" >16</a>
	]
</span> for example). In practice, the use of curved interface elements that perfectly fits the interface may be computationally difficult or impossible particularly when the interface is irregular in shape&#160;<span class="cite">
	[
	<a href="#ciarlet1978" >12</a>
	]
</span>.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h1 id="EE">3 Error Estimates</h1>

<h2 id="a0000000024">3.1 Continuous-in-Time Approximation</h2>
<p> We may pose the semi discrete problem as: find \(u_h:[0,T]\rightarrow S_h\) such that \(u_h(0)=u_{h,0}\) and </p>
<div class="equation" id="eq5">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq5} (u_{h,tt},v_h) + A(u_h:u_h,v_h)=(f,v_h) \quad \forall \;  v_h\in S_h, \;  \mbox{a.e } t\in [0,T] \end{equation}
  </div>
  <span class="equation_label">3.13</span>
</p>
</div>
<p>Below is the main results concerning the convergence of <a href="#eq5" class="eqref">3.13</a> to the exact solution in the \(L^\infty (0,T;H^1(\Omega ))\)-norm. </p>
<p><div class="thm_thmwrapper theorem-style-plain" id="a0000000025">
  <div class="thm_thmheading">
    <span class="thm_thmcaption">
    Theorem
    </span>
    <span class="thm_thmlabel">3.1</span>
  </div>
  <div class="thm_thmcontent">
  <p>Suppose that the conditions of Assumption <a href="#res1">1.1</a> are satisfied for \(a:\Omega \times \mathbb {R}\rightarrow \mathbb {R}\), \(b:\Omega \times \mathbb {R}\rightarrow \mathbb {R}\) and \(f:\Omega \times \mathbb {R}^+\rightarrow \mathbb {R}\) and let \(u\) and \(u_h\) be the solutions of <a href="#eq3" class="eqref">1.4</a> and <a href="#eq5" class="eqref">3.13</a> respectively, then for \(u_0\in H^1_0(\Omega )\cap X\), \(u_1\in H^1_0(\Omega )\) and \(0{\lt}h{\lt}h_0\), there exists a positive constant \(C\), independent of \(h\), such that </p>
<div class="displaymath" id="a0000000026">
  \[  \max _{0\leq t\leq T}\| u-u_h\| _{H^1(\Omega )}\;  \leq \;  Ch\left( 1+\tfrac {1}{|\ln h|} \right)^{1/2}  \]
</div>

  </div>
</div> </p>
<div class="proof_wrapper" id="a0000000027">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  <p>Subtract <a href="#eq5" class="eqref">3.13</a> from <a href="#eq3" class="eqref">1.4</a> </p>
<div class="displaymath" id="a0000000028">
  \[  (u_t-u_{h,tt},v_h)+A(u:u,v_h)-A(u_h:u_h,v_h)=0\qquad \forall \;  v_h\in S_h  \]
</div>
<p> Let \(e(t)=u-u_h\), \(v_h=(P_hu-u_h)_t\)<br /> \(\displaystyle \tfrac {1}{2}\tfrac {d}{dt} \| e'(t)\| ^2_{L^2(\Omega )} + \tfrac {\mu _1}{2}\tfrac {d}{dt} \| e(t)\| ^2_{H^1(\Omega )}=\) </p>
<div class="displaymath" id="eq8">
  \begin{eqnarray} \label{eq8} \notag & =& (u_{h,tt}-u_{tt},(P_hu-u)_t) + A(u_h;e(t),(u-P_hu)_t)\\ \notag & & +\;  A(u_h:u,(P_hu-u)_t)-A(u:u,(P_hu-u)_t) \\ & \leq &  I_1+I_2 +I_3 \end{eqnarray}
</div>
<p> where </p>
<div class="displaymath" id="a0000000029">
  \begin{eqnarray*}  I_1 & =&  |(u_{tt}-u_{h,tt},(P_hu-u)_t)|, \qquad I_2 \; =\;  |A(u_h:e(t),(u-P_hu)_t)|, \\ I_3& =& |A(u_h:u,(P_hu-u)_t)-A(u:u,(P_hu-u)_t)| \end{eqnarray*}
</div>
<p> For \(I_1\), we have </p>
<div class="displaymath" id="eq6">
  \begin{eqnarray} \label{eq6} \notag I_1 & =&  |\tfrac {d}{dt}(e’(t),(P_hu-u)_t) - (e’(t),(P_hu-u)_{tt})| \\ \notag & \leq &  \tfrac {1}{4}\tfrac {d}{dt}\| e’(t)\| ^2_{L^2(\Omega )} + \tfrac {d}{dt}\| (P_hu-u)_t\| ^2_{L^2(\Omega )} + \tfrac {1}{4}\| e’(t)\| ^2_{L^2(\Omega )} \\ \notag & & +\;  \| (P_hu-u)_{tt}\| ^2_{L^2(\Omega )}\\ \notag \label{eq18}& \leq &  \tfrac {1}{4}\tfrac {d}{dt}\| e’(t)\| ^2_{L^2(\Omega )} + \tfrac {1}{4}\| e’(t)\| ^2_{L^2(\Omega )} + \| (P_hu-u)_t\| ^2_{L^2(\Omega )}\\ & &  +2\| (P_hu-u)_{tt}\| ^2_{L^2(\Omega )} \end{eqnarray}
</div>
<div class="displaymath" id="a0000000030">
  \begin{eqnarray}  \notag I_2 &  \leq &  (\mu _3\| u_h\| _{L^2(\Omega )}+\mu _2)\| e(t)\| _{H^1(\Omega )} \| (u-P_hu)_t\| _{H^1(\Omega )} \\ & \leq &  \tfrac {\mu _1}{4} \| e(t)\| ^2_{H^1(\Omega )} + \tfrac {1}{\mu _1}(\mu _3\| u_h\| _{L^2(\Omega )}+\mu _2)^2\| (P_hu-u)_t\| ^2_{H^1(\Omega )} \end{eqnarray}
</div>
<p> For \(I_3\), we use Young’s inequality and obtain </p>
<div class="displaymath" id="eq7">
  \begin{eqnarray} \label{eq7} \notag I_3 & \leq &  \mu _3\| e(t)\| _{L^2(\Omega )}\| u\| _{H^1(\Omega )}\| (u-P_hu)_t\| _{H^1(\Omega )} \\ & \leq &  \tfrac {\mu _1}{4}\| e(t)\| ^2_{H^1(\Omega )} + \tfrac {\mu ^2_3}{\mu _1} \| u\| _{H^1(\Omega )}^2\| (u-P_hu)_t\| ^2_{H^1(\Omega )} \end{eqnarray}
</div>
<p> We substitute <a href="#eq6" class="eqref">3.15</a>\(-\)<a href="#eq7" class="eqref">3.17</a> into <a href="#eq8" class="eqref">3.14</a> and obtain </p>
<div class="displaymath" id="a0000000031">
  \begin{align*} & \tfrac {1}{4} \tfrac {d}{dt} \| e’(t)\| ^2_{L^2(\Omega )} + \tfrac {\mu _1}{2} \tfrac {d}{dt}\| e(t)\| ^2_{H^1(\Omega )}\leq \\ \leq &  \tfrac {1}{4}\| e’(t)\| ^2_{L^2(\Omega )}+ \tfrac {\mu _1}{2}\| e(t)\| ^2_{H^1(\Omega )} + 2\| (P_hu-u)_{tt}\| ^2_{L^2(\Omega )}\\ &  + \;  \| (P_hu-u)_t\| ^2_{L^2(\Omega )}+\tfrac {1}{\mu _1}(\mu _3\| u_h\| _{L^2(\Omega )}+\mu _2)^2\| (P_hu-u)_t\| ^2_{H^1(\Omega )} \\ &  +\;  \tfrac {\mu ^2_3}{\mu _1} \| u\| _{H^1(\Omega )}^2\| (u-P_hu)_t\| ^2_{H^1(\Omega )}. \end{align*}
</div>
<p> It is obvious that </p>
<div class="displaymath" id="a0000000032">
  \[  h^2\left( 1+\tfrac {1}{|\ln h|} \right) \leq h\left( 1+\tfrac {1}{|\ln h|} \right)^{1/2} \quad \Leftrightarrow \quad 0{\lt}h{\lt}0.58857838891.  \]
</div>
<p>Therefore using Lemma <a href="#res3">2.1</a> for \(0{\lt}h{\lt}0.58857838891\), it follows that </p>
<div class="displaymath" id="a0000000033">
  \begin{align*} & \tfrac {1}{4} \tfrac {d}{dt} \| e’(t)\| ^2_{L^2(\Omega )} + \tfrac {\mu _1}{2} \tfrac {d}{dt}\| e(t)\| ^2_{H^1(\Omega )} \leq \\ \leq &  \tfrac {1}{4}\| e’(t)\| ^2_{L^2(\Omega )}+ \tfrac {\mu _1}{2}\| e(t)\| ^2_{H^1(\Omega )}\\ &  + \;  Ch^2\left( 1+\tfrac {1}{|\ln h|} \right)\left[\left(1+\| u\| _{H^1(\Omega )}^2+(\mu _3\| u_h\| _{L^2(\Omega )}+\mu _2)^2\right) \right.\\ &  \times \left. \left(\| u\| ^2_X +\| u_t\| ^2_X\right) + \| u_{tt}\| ^2_X \right]. \end{align*}
</div>
<p> After a simple calculation, we have </p>
<div class="displaymath" id="a0000000034">
  \begin{eqnarray*}  \| e(t)\| ^2_{H^1(\Omega )} & \leq &  \exp (T)\| e’(0)\| ^2_{L^2(\Omega )} + \exp (T)\| e(0)\| ^2_{H^1(\Omega )} \\ & &  + \;  Ch^2\int _{0}^{t} \exp (t-s)\left( 1+\tfrac {1}{|\ln h|} \right)\\ & &  \times \bigg[\left(1+\| u\| _{H^1(\Omega )}^2+(\mu _3\| u_h\| _{L^2(\Omega )}+\mu _2)^2\right) \\ & &  \times \left(\| u\| ^2_X +\| u_t\| ^2_X\right) + \| u_{tt}\| ^2_X \frac{}{} \bigg] \;  ds. \end{eqnarray*}
</div>
<p> The result follows by taking \(u_{0,h}=P_hu_0\) and \(u_{1,h}=P_hu_1\). </p>

  </div>
</div>
<h2 id="a0000000035">3.2 Discrete-in-Time Approximation</h2>
<p> In this section, we propose a linearized scheme for the solution of <a href="#eq3" class="eqref">1.4</a> due to the presence of the nonlinear terms. An almost optimal order error estimate in the \(H^1(\Omega )\)-norm is analyzed. </p>
<p>The interval \([0,T]\) is divided into \(M\) equally spaced (for simplicity) subintervals: </p>
<div class="displaymath" id="a0000000036">
  \[  0=t_0{\lt}t_1{\lt}\ldots {\lt}t_M=T  \]
</div>
<p> with \(t_n=nk\), \(k=T/M\) being the time step. Let </p>
<div class="displaymath" id="a0000000037">
  \[  u^n=u(x,t_n)\quad \mbox{and}\quad f^n=f(x,t_n) \;  .  \]
</div>
<p> For a given sequence \(\{ w_n\} ^M_{n=0}\subset L^2(\Omega )\), we have the backward difference quotient defined by </p>
<div class="displaymath" id="a0000000038">
  \[  \partial ^2 w^{n} \; =\;  \tfrac {w^{n}-2w^{n-1}+w^{n-2}}{k^2}\;  ,\quad n=2,3,\ldots , M.  \]
</div>
<p> The fully discrete finite element approximation to <a href="#eq3" class="eqref">1.4</a> is to find \(U^n_h\in S_h\), such that </p>
<div class="equation" id="eq9">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq9} (\partial ^2 U^{n}_h , v_h) + A(U^{n}_h:U^{n}_h , v_h) = (f^{n}, v_h) \quad \forall \;  v_h\in S_h\quad n=2,3,\ldots ,M. \end{equation}
  </div>
  <span class="equation_label">3.18</span>
</p>
</div>
<p> Scheme <a href="#eq9" class="eqref">3.18</a> has the disadvantage that a nonlinear system of algebraic equations has to be solved at each time step due to the presence of \(a(x,U^n_h)\) and \(b(x,U^n_h)\). We therefore propose a linearized modification of the scheme in which this difficulty is avoided by replacing \(U^n_h\) by \(U^{n-1}_h\) in these two places. Thus the linearized fully discrete finite element approximation to <a href="#eq3" class="eqref">1.4</a> is to find \(U^n_h\in S_h\), such that </p>
<div class="equation" id="eq4">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq4} (\partial ^2 U^{n}_h , v_h) + A(U^{n-1}_h:U^{n}_h , v_h) = (f^{n}, v_h) \quad \forall \;  v_h\in S_h\quad n=2,3,\ldots ,M. \end{equation}
  </div>
  <span class="equation_label">3.19</span>
</p>
</div>
<p> For the analysis of linearized schemes for nonlinear parabolic interface problems, see <span class="cite">
	[
	<a href="#adewole_payne2018" >5</a>
	, 
	<a href="#yang2015" >26</a>
	]
</span>. </p>
<p>The result below establishes the convergence of the scheme <a href="#eq4" class="eqref">3.19</a> to the exact solution in \(H^1(\Omega )\)-norm. </p>
<p><div class="thm_thmwrapper theorem-style-plain" id="res4">
  <div class="thm_thmheading">
    <span class="thm_thmcaption">
    Theorem
    </span>
    <span class="thm_thmlabel">3.2</span>
  </div>
  <div class="thm_thmcontent">
  <p> Let \(u^n\) and \(U^n_h\) be the solutions of <a href="#eq3" class="eqref">1.4</a> and <a href="#eq4" class="eqref">3.19</a> respectively at \(t_n\) with \(U^0_h=P_h u_0\) and \(U^1_h=U^0_h+ k P_hu_1\). Suppose that the conditions of Assumption <span class="rm"><a href="#res1">1.1</a></span> are satisfied for \(a:\Omega \times \mathbb {R}\rightarrow \mathbb {R}\), \(b:\Omega \times \mathbb {R}\rightarrow \mathbb {R}\) and \(f:\Omega \times \mathbb {R}^+\rightarrow \mathbb {R}\). There exists a positive constant \(C\) independent of \(h\in (0,h_0)\) and \(k\in [0,k_0)\) such that </p>
<div class="equation" id="a0000000039">
<p>
  <div class="equation_content">
    \begin{equation}  \notag \| u^n-U^n_h\| _{H^1(\Omega )} \;  \leq \;  C\left[k+h\left(1+\tfrac {1}{|\ln h|}\right)^{1/2}\right] \; ,\quad n=2,3,\ldots ,M\; . \end{equation}
  </div>
  <span class="equation_label">3.20</span>
</p>
</div>

  </div>
</div> </p>
<div class="proof_wrapper" id="a0000000040">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  <p>  Let \(z^n=P_hu^n - U^n_h\). From <a href="#eq3" class="eqref">1.4</a> and <a href="#eq4" class="eqref">3.19</a> using <a href="#eq22" class="eqref">2.12</a>, we have </p>
<div class="displaymath" id="a0000000041">
  \begin{eqnarray*}  (\partial ^2 z^{n},v_h) + A(U^{n-1}_h;z^n,v_h)& =&  (\partial ^2(P_hu^n-u^n),v_h) + (\partial ^2 u^n-u^n_{tt},v_h)\\ & &  +\;  A(U^{n-1}_h:P_hu^n,v_h) - A(u^n:P_hu^n,v_h) \end{eqnarray*}
</div>
<p> After a simple calculation using Young’s inequality with \(v_h=\partial ^2 z^n\), we have </p>
<div class="displaymath" id="eq11">
  \begin{eqnarray} \label{eq11} \notag \| \partial ^2 z^n\| ^2_{L^2(\Omega )} + \tfrac {\mu _1}{4k} \| z^n\| ^2_{H^1(\Omega )} &  \leq &  \tfrac {\mu _1}{4k} \|  z^{n-1}\| ^2_{H^1(\Omega )} +\tfrac {\mu _1}{4k} \|  z^{n-2}\| ^2_{H^1(\Omega )} \\ & &  +\;  \tfrac {9\mu _1}{4k} \|  z^{n-1} - z^{n-2} \| ^2_{H^1(\Omega )} + B_1 + B_2 \end{eqnarray}
</div>
<p> where </p>
<div class="displaymath" id="a0000000042">
  \begin{eqnarray*}  B_{1} & =&  (\partial ^2(P_hu^n-u^n),\partial ^2 z^n) + (\partial ^2 u^n-u^n_{tt},\partial ^2 z^n) \\ B_{2} & =&  A(U^{n-1}_h:P_hu^n,\partial ^2 z^n) - A(u^n:P_hu^n,\partial ^2 z^n) \end{eqnarray*}
</div>
<div class="displaymath" id="eq12">
  \begin{eqnarray} \label{eq12} \  \quad B_{1}& \leq &  2\| \partial ^2(P_hu^n-u^n)\| ^2_{L^2(\Omega )} + \tfrac {1}{4}\| \partial ^2 z^n\| ^2_{L^2(\Omega )} + 2\| \partial ^2 u^n - u^n_{tt}\| ^2_{L^2(\Omega )} \end{eqnarray}
</div>
<p> By Taylor’s expansion, there exists \(\lambda {\gt}0\), such that </p>
<div class="displaymath" id="eq13">
  \begin{eqnarray} \label{eq13} \notag B_{2}& \leq &  \mu _3\| U^{n-1}_h-u^n\| _{L^2(\Omega )}\| P_hu^n\| _{H^1(\Omega )}\| \partial ^2 z^n\| _{H^1(\Omega )}\\ \notag & \leq &  \mu _3\lambda k\| u^n_t\| _{L^2(\Omega )}\| P_hu^n\| _{H^1(\Omega )}\| \partial ^2 z^n\| _{H^1(\Omega )} \\ \notag & &  +\;  \mu _3\| z^{n-1}\| _{L^2(\Omega )}\| P_hu^n\| _{H^1(\Omega )}\| \partial ^2 z^n\| _{H^1(\Omega )}\\ \notag & &  +\;  \mu _3 \| P_hu^{n-1}-u^{n-1}\| _{L^2(\Omega )}\| P_hu^n\| _{H^1(\Omega )}\| \partial ^2 z^n\| _{H^1(\Omega )}\\ \notag & \leq &  Ck^2\| u^n_t\| ^2_{L^2(\Omega )}\| u^n\| ^2_{H^1(\Omega )}+ \mu _3^2\| z^{n-1}\| ^2_{L^2(\Omega )}\| u^n\| ^2_{H^1(\Omega )} \\ & & +\;  \mu ^2_3 \| P_hu^{n-1}-u^{n-1}\| ^2_{L^2(\Omega )}\| u^n\| ^2_{H^1(\Omega )}+ \tfrac {3}{4}\| \partial ^2 z^n\| ^2_{H^1(\Omega )} \end{eqnarray}
</div>
<p> Substitute <a href="#eq12" class="eqref">3.21</a> and <a href="#eq13" class="eqref">3.22</a> into <a href="#eq11" class="eqref">3.20</a>, and use inverse estimate <span class="cite">
	[
	<a href="#brenner_scott2008" >11</a>
	, 
	Theorem 4.5.11
	]
</span>, </p>
<div class="displaymath" id="a0000000043">
  \begin{eqnarray*}  \tfrac {\mu _1}{4k} \| z^n\| ^2_{H^1(\Omega )}&  \leq &  \tfrac {\mu _1}{4k} \|  z^{n-1}\| ^2_{H^1(\Omega )} +\tfrac {\mu _1}{4k} \|  z^{n-2}\| ^2_{H^1(\Omega )} + \tfrac {9\mu _1}{4k} \|  z^{n-1} - z^{n-2} \| ^2_{H^1(\Omega )}\\ & &  +\;  2\| \partial ^2(P_hu^n-u^n)\| ^2_{L^2(\Omega )} + 2\| \partial ^2 u^n - u^n_{tt}\| ^2_{L^2(\Omega )} \\ & &  +\;  Ck^2\| u^n_t\| ^2_{L^2(\Omega )}\| u^n\| ^2_{H^1(\Omega )}+ \mu _3^2\| z^{n-1}\| ^2_{L^2(\Omega )}\| u^n\| ^2_{H^1(\Omega )} \\ & &  +\;  Ch^4\left(1+\tfrac {1}{|\ln h|}\right)^2\| u^n\| ^2_{H^1(\Omega )}\| u^{n-1}\| ^2_X . \end{eqnarray*}
</div>
<p> We used Lemma <a href="#res3">2.1</a> to obtain the last inequality. Therefore, </p>
<div class="displaymath" id="a0000000044">
  \begin{eqnarray*}  \left(1-ck\right)\| z^n\| ^2_{H^1(\Omega )}& \leq &  \|  z^{n-1}\| ^2_{H^1(\Omega )} + \|  z^{n-2}\| ^2_{H^1(\Omega )} + 9\|  z^{n-1} - z^{n-2} \| ^2_{H^1(\Omega )}\\ & & +\;  C \left[ k\| \partial ^2(P_hu^n-u^n)\| ^2_{L^2(\Omega )} + k\| \partial ^2 u^n - u^n_t\| ^2_{L^2(\Omega )} \right] \\ & &  +\;  Ck^2\| u^n_t\| ^2_{L^2(\Omega )}\| u^n\| ^2_{H^1(\Omega )}\\ & & +\;  Ch^4k\left( 1+\tfrac {1}{|\ln h|} \right)^2\| u^n\| ^2_{H^1(\Omega )}\| u^{n-1}\| ^2_X \end{eqnarray*}
</div>
<p> where \(c= \frac{4\mu _3^2}{\mu _1}\| u^n\| ^2_{H^1(\Omega )}\). For \(0{\lt}k{\lt}\min \left\{ \frac{1}{2},\frac{1}{2c} \right\} \), there is a \(C{\gt}0\) such that \(\left( 1-ck \right)^{-1}\leq C\), and therefore </p>
<div class="displaymath" id="a0000000045">
  \begin{eqnarray*}  \| z^n\| ^2_{H^1(\Omega )} & \leq &  C\|  z^{n-1}\| ^2_{H^1(\Omega )} + C\|  z^{n-2}\| ^2_{H^1(\Omega )} + C\|  z^{n-1} - z^{n-2} \| ^2_{H^1(\Omega )}\\ & & +\;  C \left[ k\| \partial ^2(P_hu^n-u^n)\| ^2_{L^2(\Omega )} + k\| \partial ^2 u^n - u^n_{tt}\| ^2_{L^2(\Omega )} \right] \\ & &  +\;  Ck^2\| u^n_t\| ^2_{L^2(\Omega )}\| u^n\| ^2_{H^1(\Omega )}\\ & & +\;  Ch^4k\left( 1+\tfrac {1}{|\ln h|} \right)^2\| u^n\| ^2_{H^1(\Omega )}\| u^n\| ^2_X, \quad \mbox{for}\;  n=2,3,\ldots , M. \end{eqnarray*}
</div>
<p> By iteration on \(n\), we have </p>
<div class="displaymath" id="a0000000046">
  \begin{eqnarray*}  \| z^n\| ^2_{H^1(\Omega )} &  \leq &  C\sum _{i=0}^1\| z^i\| ^2_{H^1(\Omega )} + Ch^4k\left( 1+\tfrac {1}{|\ln h|} \right)^2\sum ^n_{j=2}\| u^j\| ^2_X \| u^j\| ^2_{H^1(\Omega )} \\ & &  +\;  C\sum _{j=2}^n \| z^{j-1} - z^{j-2} \| ^2_{H^1(\Omega )} + Ck\sum ^n_{j=2}\| \partial ^2 u^j-u^j_{tt}\| ^2_{L^2(\Omega )}\\ & &  + \;  Ck\sum ^n_{j=2}\| \partial ^2(u^j-P_hu^j)\| ^2_{L^2(\Omega )} + Ck^3\sum _{j=2}^n \| u^j_t\| ^2_{L^2(\Omega )}\| u^j\| ^2_{H^1(\Omega )} \end{eqnarray*}
</div>
<p> Using the discrete version of Gronwall’s inequality and simplifying the resulting expression, we obtain </p>
<div class="displaymath" id="a0000000047">
  \begin{eqnarray*}  \| z^n\| ^2_{H^1(\Omega )} &  \leq &  \sum _{i=0}^1\| z^i\| ^2_{H^1(\Omega )} + C\int ^{t_n}_0\| (u-P_hu)_{tt}\| ^2_{L^2(\Omega )}\; dt \\ & &  +\;  Ck^2\int ^{t_n}_0\| \tfrac {\partial ^3u}{\partial t^3}\| ^2_{L^2(\Omega )}\; dt + Ck^2\int _{0}^{t_n}\| u_t\| ^2_{L^2(\Omega )} \| u\| ^2_{H^1(\Omega )}\; dt\\ & & +\;  Ch^4\left( 1+\tfrac {1}{|\ln h|} \right)^2\int ^{t_n}_0 \| u\| ^2_X\| u\| ^2_{H^1(\Omega )}\;  dt \\ & \leq &  Ck^2\int ^{t_n}_0 \left( \| u_t\| ^2_{L^2(\Omega )}\| u\| ^2_{H^1(\Omega )}+ \| \tfrac {\partial ^3u}{\partial t^3}\| ^2_{L^2(\Omega )} \right)dt\\ & &  +\; C\sum _{i=0}^1\| z^i\| ^2_{H^1(\Omega )}+ Ch^4\left(1+\tfrac {1}{|\ln h|}\right)^2 \\ & &  \times \int ^{t_n}_0\left[ \| u\| ^2_{H^1(\Omega )}\| u\| ^2_X + \| u\| ^2_X+ \| u_t\| ^2_X+ \| u_{tt}\| ^2_X \right] \;  dt. \end{eqnarray*}
</div>
<p> By triangle inequality and Lemma <a href="#res3">2.1</a>, </p>
<div class="displaymath" id="a0000000048">
  \begin{eqnarray*}  \| u^n-U^n_h\| ^2_{H^1(\Omega )} & \leq &  2\| u^n-P_hu^n\| ^2_{H^1(\Omega )}+2\| z^n\| ^2_{H^1(\Omega )} \\ & \leq &  Ch^2\left(1+\tfrac {1}{|\ln h|}\right)\| u^n\| _X \\ & &  +\;  C\sum _{i=0}^1\left(\| u^i-U^i_h\| ^2_{H^1(\Omega )}+ \| u^i-P_hu^i\| ^2_{H^1(\Omega )} \right) \\ & & +\;  Ck^2\int ^{t_n}_0 \left( \| u_t\| ^2_{L^2(\Omega )}\| u\| ^2_{H^1(\Omega )}+ \| \tfrac {\partial ^3u}{\partial t^3}\| ^2_{L^2(\Omega )} \right)dt\\ & &  +\;  Ch^4\left(1+\tfrac {1}{|\ln h|}\right)^2\\ & &  \times \int ^{t_n}_{0} \left[ \| u\| ^2_X\| u\| ^2_{H^1(\Omega )} +\| u\| ^2_X+ \| u_t\| ^2_X+ \| u_{tt}\| ^2_X \right] \;  dt. \end{eqnarray*}
</div>
<p> It is obvious that </p>
<div class="displaymath" id="a0000000049">
  \begin{equation*}  h^4\left( 1+\tfrac {1}{|\ln h|} \right)^2 \leq h^2\left( 1+\tfrac {1}{|\ln h|} \right) \quad \Leftrightarrow \quad 0{\lt}h{\lt}0.58857838891. \end{equation*}
</div>
<p> The result follows taking \(U^0_h=P_h u_0\) and \(U^1_h=U^0_h+ k P_hu_1 \). </p>

  </div>
</div>
<p><div class="rem_thmwrapper theorem-style-remark" id="a0000000050">
  <div class="rem_thmheading">
    <span class="rem_thmcaption">
    Remark
    </span>
    <span class="rem_thmlabel">3.3</span>
  </div>
  <div class="rem_thmcontent">
  <p>In the proof of Theorem <a href="#res4">3.2</a>, we used \(v_h=\partial ^2 z^n\). If we choose \(v_h= z^n\), by a similar argument, one can obtain </p>
<div class="equation" id="a0000000051">
<p>
  <div class="equation_content">
    \begin{equation}  \notag \| u^n-U^n_h\| _{L^2(\Omega )} \;  \leq \;  C\left[k+h^2\left(1+\tfrac {1}{|\ln h|}\right)\right] \; ,\quad n=2,3,\ldots ,M\; . \end{equation}
  </div>
  <span class="equation_label">3.23</span>
</p>
</div>

  </div>
</div> </p>
<h1 id="Ex">4 Examples</h1>
<p> Here, we present examples to verify Theorem <a href="#res4">3.2</a>. Globally continuous piecewise linear finite element functions based on triangulation described in Section <a href="#FED">2</a> are used. The mesh generation and computation are done with FreeFEM++ <span class="cite">
	[
	<a href="#hecht2012" >20</a>
	]
</span>. </p>
<p><div class="exa_thmwrapper theorem-style-definition" id="res5">
  <div class="exa_thmheading">
    <span class="exa_thmcaption">
    Example
    </span>
    <span class="exa_thmlabel">4.1</span>
  </div>
  <div class="exa_thmcontent">
  <p> The problem is defined on the domain \(\Omega = (-1,1)\times (-1,1)\) where the interface \(\Gamma \) is a circle centered at \((0,0)\) with radius \(0.5\). \(\Omega _1=\{ (x,y):x^2+y^2{\lt}0.25\} \), \(\Omega _2=\Omega \setminus \overline{\Omega }_1\).<br />On \(\Omega \times (0,50]\), we consider the nonlinear problem <a href="#eq1" class="eqref">1.1</a>\(-\)<a href="#eq2" class="eqref">1.3</a> whose exact solution is </p>
<div class="displaymath" id="a0000000052">
  \begin{equation*}  u=\left\{  \begin{array}{ll} \tfrac {1}{8}(1-4r^2)t\sin (t) &  \mbox{in} \quad \Omega _1 \times (0,50]\\[2mm] \tfrac {1}{4}(1-x^2)(1-y^2)(1-4r^2)t\sin (2t) &  \mbox{in} \quad \Omega _2 \times (0,50] \end{array} \right. \; , \end{equation*}
</div>
<p> where \(r^2=x^2+y^2\). The source function \(f\) and the initial data \(u_0\), \(u_1\) are determined from the choice of \(u\) with \(b=0\) and </p>
<div class="displaymath" id="a0000000053">
  \begin{equation*}  a =\left\{  \begin{array}{lll} 1+u & \mbox{in} &  \Omega _1 \times (0,T]\\[2mm] \tfrac {1}{1+u^2} &  \mbox{in} &  \Omega _2 \times (0,T] \end{array} \right. . \end{equation*}
</div>
<p>We allow \(k\) and \(h\) to vary simultaneously by choosing \(k=O(h)\). Errors in \(H^1\)-norm at \(t=1\) and convergence rates are presented in Table <a href="#tab1">4.</a>. To verify the agreement of the numerical experiment with the theoretical results, we use the formula </p>
<div class="equation" id="a0000000054">
<p>
  <div class="equation_content">
    \begin{equation} \notag \mbox{Order of convergence} = \tfrac {\ln (e_{i+1}/e_i)}{\ln (\mathfrak {h}_{i+1}/\mathfrak {h}_i)}, \end{equation}
  </div>
  <span class="equation_label">4.23</span>
</p>
</div>
<p> where \(e_i\) is the error at the \(i\)-th iteration corresponding to the mesh size \(h_i\) and \(\mathfrak {h}_i=h_i\left( 1+\tfrac {1}{|\ln h_i|} \right)^{1/2}\). <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<div class="table"  id="a0000000055">
  <div class="centered"> <small class="footnotesize"><table class="tabular">
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> \(k\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> \(h\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> Error</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> Convergence rate</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>0.008 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>\(0.1518120\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> \(2.02004\times 10^{-1}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>0.004 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>\(0.0793667\) </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> \(1.06102\times 10^{-1}\) </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.889 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>0.002 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>\(0.0403482\) </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> \(5.24888\times 10^{-2}\) </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.965</p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>0.001 </p>

    </td>
    <td  style="text-align:center; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>\(0.0206032\) </p>

    </td>
    <td  style="text-align:center; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(2.63159\times 10^{-2}\)</p>

    </td>
    <td  style="text-align:center; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> 0.973 </p>

    </td>
  </tr>
</table> </small> <figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">4.</span> 
  <span class="caption_text">Error estimates in \(H^1\)-norm for Example <a href="#res5">4.1</a>.</span> 
</figcaption> <div class="flushright"> <small class="footnotesize"><table class="tabular">
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>  &#8195;\(h\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> Error (\(k=0.001\))</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>\(0.196096\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \(2.9256627\times 10^{-1}\) </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\(0.101640\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \(1.4581576\times 10^{-1}\)</p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\(0.0519419\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \(7.8258517\times 10^{-2}\) </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>\(0.0267211\) </p>

    </td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(5.0963435\times 10^{-2}\) </p>

    </td>
  </tr>
</table> </small> </div> <div class="flushleft"> <small class="footnotesize"><table class="tabular">
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right" 
        rowspan=""
        colspan="">
      <p> \(k\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> Error (\(h= 0.0267211\)) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right" 
        rowspan=""
        colspan="">
      <p>0.0050 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \(7.5769633\times 10^{-2}\) </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:right" 
        rowspan=""
        colspan="">
      <p>0.0025 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \(5.9446024\times 10^{-2}\) </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:right" 
        rowspan=""
        colspan="">
      <p>0.0020 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \(5.6473694\times 10^{-2}\) </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:right; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>0.0010 </p>

    </td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(5.0963435\times 10^{-2}\) </p>

    </td>
  </tr>
</table> </small> </div>  <figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">4.</span> 
  <span class="caption_text">Error estimates in \(H^1\)-norm for Example <a href="#res6">4.2</a>.</span> 
</figcaption>  </div> 
</div>
<p>The next example demonstrates that the error estimates apply even when the domain is not polygonal. </p>
<p><div class="exa_thmwrapper theorem-style-definition" id="res6">
  <div class="exa_thmheading">
    <span class="exa_thmcaption">
    Example
    </span>
    <span class="exa_thmlabel">4.2</span>
  </div>
  <div class="exa_thmcontent">
  <p> We consider <a href="#eq1" class="eqref">1.1</a>\(-\)<a href="#eq2" class="eqref">1.3</a> on the domain \(\Omega = \{ (x,y)\in \mathbb {R}^2:x^2+y^2{\lt}1\} \) where \(\Omega _1 = \{ (x,y)\in \mathbb {R}^2:x^2+y^2{\lt}\frac{1}{4}\} \), \(\Omega _2=\Omega \setminus \Omega _1\) and the interface \(\Gamma \) is the circle \(x^2+y^2=\frac{1}{4}\).<br />For the exact solution, we choose </p>
<div class="displaymath" id="a0000000056">
  \begin{equation*}  u=\left\{  \begin{array}{ll} (2-5x^2-5y^2)\sin ^2 t &  \mbox{in} \quad \Omega _1 \times (0,T]\\[2mm] (1-x^2-y^2)\sin ^2 t &  \mbox{in} \quad \Omega _2 \times (0,T] \end{array} \right. \end{equation*}
</div>
<p> The source function \(f\), interface function \(g\) and the initial data \(u_0\) are determined from the choice of \(u\) with </p>
<div class="displaymath" id="a0000000057">
  \begin{equation*}  a =\left\{  \begin{array}{cll} x^2+y^2 & \mbox{in} &  \Omega _1 \times (0,T]\\[2mm] 1+u &  \mbox{in} &  \Omega _2 \times (0,T] \end{array} \right.\quad \mbox{and}\quad b =\left\{  \begin{array}{cll} \tfrac {1}{1+u^2} & \mbox{in} &  \Omega _1 \times (0,T]\\[2mm] 1 &  \mbox{in} &  \Omega _2 \times (0,T] \end{array} \right. . \end{equation*}
</div>
<p>Figures <a href="#fig2">4.</a> and <a href="#fig2">4.</a> show the computed solution of Example <a href="#res6">4.2</a>. Errors in \(H^1\)-norm at \(t=1\) for various step size \(h\) time step \(k\) are presented in Table <a href="#tab2">4.</a>. The data show that the error is linear both in \(h\) and \(k\). </p>
<div class="displaymath" id="a0000000058">
  \begin{equation*}  \| \mbox{Error}\| _{H^1(\Omega )} \approxeq 2.262\times 10^{-2}+0.8781\, \mathfrak {h}^{1.025} \qquad \mbox{when $k$ is constant} \end{equation*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000059">
  \begin{equation*}  \| \mbox{Error}\| _{H^1(\Omega )} \approxeq 4.664\times 10^{-2}+ 15.58\times 10^{-3}k^{1.186} \qquad \mbox{when $h$ is constant} \end{equation*}
</div>
<p> where \(\mathfrak {h}=h\left( 1+ \tfrac {1}{|\ln h|}\right)^{1/2}\). </p>
<p>It can be observed that the numerical results in Tables <a href="#tab1">4.</a> and <a href="#tab2">4.</a> match the convergence rate as given in Theorem <a href="#res4">3.2</a>.<span class="qed">â–¡</span></p>
<figure id="fig2">
  <div class="centered"> <img src="img-0003.png" alt="\includegraphics[width=13cm]{NHIP_1.png}" style="width: 2160px; height: 1400px" />
 </div>
<p> <br /> </p>
<figcaption>
  <span class="caption_title">Figure</span> 
  <span class="caption_ref">4.</span> 
  <span class="caption_text">Computed solution of Example <a href="#res6">4.2</a> at \(t=1\) with \(h=0.3568\), \(k=0.001\)</span> 
</figcaption>


</figure>
<figure id="fig3">
  <div class="centered"> <img src="img-0004.png" alt="\includegraphics[width=13cm]{NHIP_2.png}" style="width: 1060px; height: 400px" />
 </div>
<p> <br /> </p>
<figcaption>
  <span class="caption_title">Figure</span> 
  <span class="caption_ref">4.</span> 
  <span class="caption_text">Computed solution of Example <a href="#res6">4.2</a> at \(t=2\), 2.5 and 3 with \(h=0.052\), \(k=0.001\)</span> 
</figcaption>


</figure>

  </div>
</div> </p>
<p><div class="acknowledgement_thmwrapper theorem-style-remark" id="a0000000060">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgement
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The author likes to thank the anonymous referees for carefully reading the manuscript and for their valuable comments and suggestions that helped to improve the original version of this manuscript. </p>

  </div>
</div> </p>
<p><br /></p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="adams1975">1</a></dt>
  <dd><p><i class="sc">R.A. Adams</i>, <i class="it">Sobolev spaces</i>, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, Pure and Applied Mathematics, <b class="bfseries">65</b> (1975). </p>
</dd>
  <dt><a name="adewole2017">2</a></dt>
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</a> </p>
</dd>
</dl>


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