Approximate Solution of Nonlinear Hyperbolic Equations with Homogeneous Jump Conditions

Matthew O. Adewole$^\ast $

February 12, 2019; accepted: May 28, 2019; published online: January 21, 2020.

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.

MSC. 65M60, 65M12, 65M06

Keywords. Almost optimal, nonlinear hyperbolic equation, linearized backward difference.

$^\ast $Department of Computer Science and Mathematics, Mountain Top University, Prayer City, Ogun State, Nigeria, e-mail: olamatthews@ymail.com.

1 Introduction

In this work, we study finite element solution of the nonlinear hyperbolic equation of the form

\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}

1.1

with initial and boundary conditions

\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}

1.2

and interface conditions

\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}

1.3

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 1.. 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

\[ [u]_{m\in \Gamma } := \lim _{x\rightarrow m^+} u_1(x,t) - \lim _{x\rightarrow m^-} u_2(x,t) \]

and

\[ \left[ a(x,u)\tfrac {\partial u}{\partial n} \right]_{m\in \Gamma } := \left[ \lim _{x\rightarrow m^+} a_1\nabla u_1(x,t) - \lim _{x\rightarrow m^-}a_2\nabla u_2(x,t) \right]\cdot n \]

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 $t\in [0,T]$.

\includegraphics[width=4cm]{Dom1.png} — Figure 1. A polygonal domain $\Omega = \Omega _1 \cup \Omega _2 $ with interface $\Gamma $.

We impose the following

Assumption 1.1

$\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.
$f(x,t)\in H^1(0,T;L^2(\Omega ))$. Functions $a$ and $b$ satisfy

\[ a_i(x,\xi )\geq \mu _1, \qquad b_i(x,\xi )\geq \mu _1, \qquad \| a_i(x,0)\| _{L^\infty (\Omega )} + \| b_i(x,0)\| _{L^\infty (\Omega )}\leq \mu _2, \]

\[ |a_i(x,\xi )-a_i(x,\psi )|+|b_i(x,\xi )-b_i(x,\psi )|\leq \mu _3\| \xi -\psi \| _{L^2(\Omega _i)} , \]

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 $.

Hyperbolic partial differential equations arise in many physical problems such as vibrating string, vibrating membrane, shallow water waves, etc [ 13 , 23 , 24 ] and become interface problems when medium or materials with different properties are involved [ 10 , 16 , 15 ] . 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 [ 21 ] . Thus, obtaining exact solutions or approximate solutions with higher order accuracy may be difficult.

Finite element solutions of non-interface hyperbolic problems have been extensively discussed in [ 7 , 8 , 9 , 17 , 19 , 22 , 25 ] . The convergence of finite element solutions of linear hyperbolic interface problems has been considered in [ 3 , 4 , 14 , 15 , 16 ] . In [ 16 ] , 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 [ 15 ] , 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 [ 14 ] . In [ 3 ] , 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 [ 4 ] , 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 1.1–1.3 is scarce in literature.

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 1.1–1.3 on finite elements. Both semi discrete and fully discrete schemes are analyzed. Full discretization of 1.1–1.3 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 [ 1 ] . 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.

Let $v_i$ be the restriction of $v$ to $\Omega _i$, $i=1,2$, we shall need the following spaces for the convergence analysis

\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}

1.4

equipped with the norms

\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}

1.4

\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*}

The weak form of 1.1$-$1.3 is to find $u(t)\in H^1_0(\Omega )$, $t\in (0,T]$ such that

\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}

1.4

where

\[ (\phi ,\psi ) = \int _\Omega \phi \psi \; dx \quad A(\xi :\phi ,\psi ) = \int _\Omega \left[ a(x,\xi ) \nabla \phi \cdot \nabla \psi +b(x,\xi )\phi \psi \right]\; dx . \]

For 1.4, we have the following energy estimate

Theorem 1.2

\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}

Proof â–¼

Let $v=u_t$ in 1.4. For $t\in [0,T]$, a simple calculation shows that

\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}

1.7

Let

\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}

1.8

It is clear from 1.3 that $g_1 +g_2=0$. From 1.1$-$1.3, we have

\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}

1.8

We take $v=u_{tt}$ and obtain

\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}

1.8

which implies

\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}

It follows directly that

\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}

Now, we multiply 1.1 by $-u_{tt}$, integrate over $\Omega _i$ then simplify the resulting equation and obtain

\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}

1.9

It follows from 1.7, 1.8 and 1.9 that

\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}

1.5 follows from 1.7, 1.8 and 1.10.

Remark 1.3

Estimate 1.5 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 $u_1\in H^m(\Omega )$ [ 18 , Theorems 5 and 6, pages 389-391 ] . 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 [ 6 , 21 ] .

This paper is organized as follows. In Section 2, 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 3, we give the discrete versions of 1.4 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 4. 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.

2 Finite Element Discretization

$\mathcal{T}_h$ 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 2.);

\[ \mathcal{T}^\star _h = \{ K\in \mathcal{T}_h:K\cap \Gamma \neq \emptyset \} \]

$K\in \mathcal{T}^\star _h$ is called an interface element and we write $\Omega ^\star _h = \bigcup _{K\in \mathcal{T}^\star _h} K$.

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

$\displaystyle \bar{\Omega }=\bigcup _{K\in \mathcal{T}_h}\bar{K}$
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.
Each $K\in \mathcal{T}_h$ is either in $\Omega ^h_1$ or $\Omega ^h_2$, and has at most two vertices lying on $\Gamma _h$.
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

\[ C_0r_K \leq h \leq C_1 \bar{r}_K \quad \forall \; h\in (0,h_0) \]

\includegraphics[width=5cm]{work121.png} — Figure 2. A typical interface element.

Let $S_h \subset H^1_0(\Omega )$ denote the space of continuous piecewise linear functions on $\mathcal{T}_h$ vanishing on $\partial \Omega $.
The FE solution $u_h(x,t)\in S_h$ is represented as

\[ u_h(x,t)=\sum ^{N_h}_{j=1}\alpha _j(t)\phi _j(x) \; , \]

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 $S_h$.

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

\begin{equation} \label{eq22} A(u:\nu -P_h\nu , \phi ) = 0 \quad \forall \; \phi \in S_h, \; t\in [0,T]. \end{equation}

2.12

For this projection, we have

Lemma 2.1

Let $a=a(x,u)$, $b=b(x,u)$ satisfy Assumption 1.1 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 2.12, then

\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*}

for $n=0,1,2$.

Proof â–¼

It can be proved using the interpolation error estimate [ 2 , Lemma 2.1 ] and a similar argument to the proof of [ 5 , Lemmas 2.4 and 2.5 ] but with little modification due to different assumptions on $a(x,u)$ and $b(x,u)$.

Remark 2.2

The term $|\ln h|$ in Lemma 2.1 is due to the fact that the mesh in Section 2 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 [ 16 ] 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 [ 12 ] .â–¡

3 Error Estimates

3.1 Continuous-in-Time Approximation

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

\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}

3.13

Below is the main results concerning the convergence of 3.13 to the exact solution in the $L^\infty (0,T;H^1(\Omega ))$-norm.

Theorem 3.1

Suppose that the conditions of Assumption 1.1 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 1.4 and 3.13 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

\[ \max _{0\leq t\leq T}\| u-u_h\| _{H^1(\Omega )}\; \leq \; Ch\left( 1+\tfrac {1}{|\ln h|} \right)^{1/2} \]

Proof â–¼

Subtract 3.13 from 1.4

\[ (u_t-u_{h,tt},v_h)+A(u:u,v_h)-A(u_h:u_h,v_h)=0\qquad \forall \; v_h\in S_h \]

Let $e(t)=u-u_h$, $v_h=(P_hu-u_h)_t$
$\displaystyle \tfrac {1}{2}\tfrac {d}{dt} \| e'(t)\| ^2_{L^2(\Omega )} + \tfrac {\mu _1}{2}\tfrac {d}{dt} \| e(t)\| ^2_{H^1(\Omega )}=$

\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}

where

\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*}

For $I_1$, we have

\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}

\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}

For $I_3$, we use Young’s inequality and obtain

\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}

We substitute 3.15$-$3.17 into 3.14 and obtain

\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*}

It is obvious that

\[ h^2\left( 1+\tfrac {1}{|\ln h|} \right) \leq h\left( 1+\tfrac {1}{|\ln h|} \right)^{1/2} \quad \Leftrightarrow \quad 0{\lt}h{\lt}0.58857838891. \]

Therefore using Lemma 2.1 for $0{\lt}h{\lt}0.58857838891$, it follows that

\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*}

After a simple calculation, we have

\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*}

The result follows by taking $u_{0,h}=P_hu_0$ and $u_{1,h}=P_hu_1$.

3.2 Discrete-in-Time Approximation

In this section, we propose a linearized scheme for the solution of 1.4 due to the presence of the nonlinear terms. An almost optimal order error estimate in the $H^1(\Omega )$-norm is analyzed.

The interval $[0,T]$ is divided into $M$ equally spaced (for simplicity) subintervals:

\[ 0=t_0{\lt}t_1{\lt}\ldots {\lt}t_M=T \]

with $t_n=nk$, $k=T/M$ being the time step. Let

\[ u^n=u(x,t_n)\quad \mbox{and}\quad f^n=f(x,t_n) \; . \]

For a given sequence $\{ w_n\} ^M_{n=0}\subset L^2(\Omega )$, we have the backward difference quotient defined by

\[ \partial ^2 w^{n} \; =\; \tfrac {w^{n}-2w^{n-1}+w^{n-2}}{k^2}\; ,\quad n=2,3,\ldots , M. \]

The fully discrete finite element approximation to 1.4 is to find $U^n_h\in S_h$, such that

\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}

3.18

Scheme 3.18 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 1.4 is to find $U^n_h\in S_h$, such that

\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}

3.19

For the analysis of linearized schemes for nonlinear parabolic interface problems, see [ 5 , 26 ] .

The result below establishes the convergence of the scheme 3.19 to the exact solution in $H^1(\Omega )$-norm.

Theorem 3.2

Let $u^n$ and $U^n_h$ be the solutions of 1.4 and 3.19 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 1.1 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

\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}

3.20

Proof â–¼

Let $z^n=P_hu^n - U^n_h$. From 1.4 and 3.19 using 2.12, we have

\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*}

After a simple calculation using Young’s inequality with $v_h=\partial ^2 z^n$, we have

\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}

where

\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*}

\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}

By Taylor’s expansion, there exists $\lambda {\gt}0$, such that

\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}

Substitute 3.21 and 3.22 into 3.20, and use inverse estimate [ 11 , Theorem 4.5.11 ] ,

\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*}

We used Lemma 2.1 to obtain the last inequality. Therefore,

\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*}

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

\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*}

By iteration on $n$, we have

\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*}

Using the discrete version of Gronwall’s inequality and simplifying the resulting expression, we obtain

\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*}

By triangle inequality and Lemma 2.1,

\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*}

It is obvious that

\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*}

The result follows taking $U^0_h=P_h u_0$ and $U^1_h=U^0_h+ k P_hu_1 $.

Remark 3.3

In the proof of Theorem 3.2, we used $v_h=\partial ^2 z^n$. If we choose $v_h= z^n$, by a similar argument, one can obtain

\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}

3.23

4 Examples

Here, we present examples to verify Theorem 3.2. Globally continuous piecewise linear finite element functions based on triangulation described in Section 2 are used. The mesh generation and computation are done with FreeFEM++ [ 20 ] .

Example 4.1

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\} $, $\Omega _2=\Omega \setminus \overline{\Omega }_1$.
On $\Omega \times (0,50]$, we consider the nonlinear problem 1.1$-$1.3 whose exact solution is

\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*}

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

\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*}

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 4.. To verify the agreement of the numerical experiment with the theoretical results, we use the formula

\begin{equation} \notag \mbox{Order of convergence} = \tfrac {\ln (e_{i+1}/e_i)}{\ln (\mathfrak {h}_{i+1}/\mathfrak {h}_i)}, \end{equation}

4.23

where $e_i$ is the error at the $i$-th iteration corresponding to the mesh size $h_i$ and $\mathfrak {h}_i=h_i\left( 1+\tfrac {1}{|\ln h_i|} \right)^{1/2}$. â–¡

$k$	$h$	Error	Convergence rate
0.008	$0.1518120$	$2.02004\times 10^{-1}$
0.004	$0.0793667$	$1.06102\times 10^{-1}$	0.889
0.002	$0.0403482$	$5.24888\times 10^{-2}$	0.965
0.001	$0.0206032$	$2.63159\times 10^{-2}$	0.973

Table 4. Error estimates in $H^1$-norm for Example 4.1.

$h$	Error ($k=0.001$)
$0.196096$	$2.9256627\times 10^{-1}$
$0.101640$	$1.4581576\times 10^{-1}$
$0.0519419$	$7.8258517\times 10^{-2}$
$0.0267211$	$5.0963435\times 10^{-2}$

$k$	Error ($h= 0.0267211$)
0.0050	$7.5769633\times 10^{-2}$
0.0025	$5.9446024\times 10^{-2}$
0.0020	$5.6473694\times 10^{-2}$
0.0010	$5.0963435\times 10^{-2}$

Table 4. Error estimates in $H^1$-norm for Example 4.2.

The next example demonstrates that the error estimates apply even when the domain is not polygonal.

Example 4.2

We consider 1.1$-$1.3 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 $x^2+y^2=\frac{1}{4}$.
For the exact solution, we choose

\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*}

The source function $f$, interface function $g$ and the initial data $u_0$ are determined from the choice of $u$ with

\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*}

Figures 4. and 4. show the computed solution of Example 4.2. Errors in $H^1$-norm at $t=1$ for various step size $h$ time step $k$ are presented in Table 4.. The data show that the error is linear both in $h$ and $k$.

\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*}

and

\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*}

where $\mathfrak {h}=h\left( 1+ \tfrac {1}{|\ln h|}\right)^{1/2}$.

It can be observed that the numerical results in Tables 4. and 4. match the convergence rate as given in Theorem 3.2.â–¡

\includegraphics[width=13cm]{NHIP_1.png} — Figure 4. Computed solution of Example 4.2 at $t=1$ with $h=0.3568$, $k=0.001$

\includegraphics[width=13cm]{NHIP_2.png} — Figure 4. Computed solution of Example 4.2 at $t=2$, 2.5 and 3 with $h=0.052$, $k=0.001$

Acknowledgement

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.

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\(k\)	\(h\)	Error	Convergence rate
0.008	\(0.1518120\)	\(2.02004\times 10^{-1}\)
0.004	\(0.0793667\)	\(1.06102\times 10^{-1}\)	0.889
0.002	\(0.0403482\)	\(5.24888\times 10^{-2}\)	0.965
0.001	\(0.0206032\)	\(2.63159\times 10^{-2}\)	0.973

\(h\)	Error (\(k=0.001\))
\(0.196096\)	\(2.9256627\times 10^{-1}\)
\(0.101640\)	\(1.4581576\times 10^{-1}\)
\(0.0519419\)	\(7.8258517\times 10^{-2}\)
\(0.0267211\)	\(5.0963435\times 10^{-2}\)

\(k\)	Error (\(h= 0.0267211\))
0.0050	\(7.5769633\times 10^{-2}\)
0.0025	\(5.9446024\times 10^{-2}\)
0.0020	\(5.6473694\times 10^{-2}\)
0.0010	\(5.0963435\times 10^{-2}\)