Optimal Error Analysis of Galerkin FEMs for Nonlinear Joule Heating Equations

Abstract

We study in this paper two linearized backward Euler schemes with Galerkin finite element approximations for the time-dependent nonlinear Joule heating equations. By introducing a time-discrete (elliptic) system as proposed in Li and Sun (Int J Numer Anal Model 10:622–633, 2013; SIAM J Numer Anal (to appear)), we split the error function as the temporal error function plus the spatial error function, and then we present unconditionally optimal error estimates of $r$th order Galerkin FEMs ($1 \le r \le 3$). Numerical results in two and three dimensional spaces are provided to confirm our theoretical analysis and show the unconditional stability (convergence) of the schemes.

1 Introduction

In this paper, we focus on error estimates of linearized backward Euler Galerkin finite element methods for the time-dependent nonlinear Joule heating equations defined by

$$\begin{aligned}&\frac{\partial u}{\partial t}-\Delta u=\sigma (u)|\nabla \phi |^2, \end{aligned}$$

(1.1)

$$\begin{aligned}&-\nabla \cdot (\sigma (u)\nabla \phi )=0, \end{aligned}$$

(1.2)

for $x \in \Omega$ and $t \in [0,T]$, where $\Omega$ is a bounded, smooth and convex domain in $\mathbb R ^d,\,d=2,3$. The boundary and initial conditions are taken to be

$$\begin{aligned} u(x,t)&= 0,\; \phi (x,t)=g(x,t), \quad \mathrm{for }\;x \in \partial \Omega ,\,t \in [0,T], \end{aligned}$$

(1.3)

$$\begin{aligned} u(x,0)&= u_0(x), \quad \mathrm{for}\; x \in \Omega . \end{aligned}$$

(1.4)

The above nonlinear system (1.1)–(1.4) describes the model of electric heating of a conducting body, where the first unknown $u$ is the temperature and the second unknown $\phi$ is the electric potential with $\sigma (s)$ being the temperature-dependent electric conductivity satisfying

$$\begin{aligned} \kappa \le \sigma (s)\le K, \end{aligned}$$

(1.5)

for some positive constants $\kappa$ and $K$.

Theoretical analysis for the Joule heating system can be found in [3, 5, 7, 16, 32, 33]. Among these works, existence and uniqueness of a $C^\alpha$ solution in three-dimensional space was proved in Yuan and Liu [33]. Based on this result, one can get higher regularity with suitable assumptions on the initial and boundary conditions. Numerical methods and analysis for the Joule heating problems can be found in [2, 4, 12, 31, 34, 35]. For the two dimensional problem, optimal $L^2$-norm error estimates of linearized semi-implicit schemes with Galerkin and mixed FEMs were obtained in [31] and [34] under a weak time step condition, respectively. A linearized semi-implicit Euler scheme with a linear Galerkin FEM for the three dimensional model was presented in [12] and an optimal $L^2$-norm error estimate was obtained under the time step restriction $\tau = O(h^{\frac{1}{2}})$. A more general time discretization with higher-order finite element approximations was studied in [2]. An optimal $L^2$-norm error estimate was given under the conditions $\tau = O(h^{\frac{3}{2p}})$ and $r \ge 2$, where $p$ is the order of discretization in time direction and $r$ is the degree of piecewise polynomial approximations used.

In the consideration of practical computations, linearized (semi)-implicit schemes are more efficient since at each time step, the schemes only require solving a linear system. However, the time step restriction condition of linearized schemes arising from error analysis is always a crucial issue. We refer to [1, 9–11, 13, 14, 17, 18, 20, 23, 24, 26, 28–30] for works on some typical nonlinear parabolic problems. Because of difficulties in obtaining the $L^{\infty }$ boundedness of the numerical solution, which is an essential condition for error analysis of nonlinear problems, most previous works require certain time step restriction conditions. There are some attempts to reduce the time step restriction conditions. Recently, a new approach was introduced by Li and Sun [19] (also see [21]) to get unconditional stability and optimal error analysis of a linearized backward Euler Galerkin FEM for the time-dependent Joule heating equations. The approach was based on a new splitting technique by a corresponding time-discrete system. With certain proved regularity of the solution of the time-discrete system, one can see

$$\begin{aligned} \Vert U_h^n - R_hU^n \Vert _{L^\infty } \le C h^{-d/2} \Vert U_h^n - R_hU^n \Vert _{L^2} \le C h^{-d/2} h^{r+1}. \end{aligned}$$

where $U_h^n$ is the FEM solution and $R_h$ is the Ritz projection operator. Therefore, the boundedness of $U_h^n$ in $L^{\infty }$-norm can be obtained without time step restriction. With this new approach, optimal error estimates for a linear FEM was obtained almost unconditionally in [19] (i.e., the step sizes $h,\,\tau \le s_0$ for some small positive constant $s_0$). In this paper, we present two linearized schemes with Galerkin FEMs for the time-dependent nonlinear Joule heating system (1.1)–(1.4). The first scheme is semi-decoupled and at each time step, one has to solve $\Phi _h^{n+1}$ and $U_h^{n+1}$ one by one. The second one is fully decoupled and at each time step $\Phi _h^{n+1}$ and $U_h^{n+1}$ can be solved in parallel. We apply the Li-Sun error splitting method to analyze the Galerkin FEMs. The main difficulty is that error estimates for high-order Galerkin FEMs with the splitting method require rigorous analysis of higher regularity of the solution of the corresponding time-discrete system. For instance, we have to prove the uniform boundedness of the time-discrete solution in $H^{4}$-norm for a cubic FEM. As there is no numerical experiment in [19], we present numerical examples in two and three dimensional spaces in this paper. To demonstrate the unconditional stability, we take a fixed $\tau$ with several refined spatial meshes. In our numerical tests, the errors are proportional to the temporal error $O(\tau )$ as $h/\tau \rightarrow 0$, which show clearly that no time-step condition is needed and the schemes are unconditionally stable.

The rest of the paper is organized as follows. In Sect. 2, we present two linearized schemes with Galerkin finite element methods and our main results on error estimates. We split the error function as the temporal error function plus the spatial error function by introducing a corresponding time-discrete system. In Sect. 3, we provide a priori estimates for the temporal error and suitable regularity of the solution of the time-discrete system. In Sect. 4, we provide optimal spatial error estimates for the Galerkin finite element solutions in $L^2$ and $H^1$-norm unconditionally. Numerical examples for both two and three dimensional models are given in Sect. 5 to confirm our theoretical analysis.

2 Galerkin Methods and Main Results

Before presenting the schemes, we clarify some conventional notations. For integer $k \ge 0$ and $1 \le p \le \infty$, let $W^{k,p}(\Omega )$ be the Sobolev space with the norm

$$\begin{aligned} \Vert u \Vert _{W^{k,p}}=\left\{ \begin{array}{ll} \left( \displaystyle \sum \nolimits _{|\beta |\le k}\int \nolimits _\Omega |D^\beta u|^p\,\mathrm{d}x\right) ^\frac{1}{p}, &{} \mathrm{for}\;1 \le p < \infty , \\ \displaystyle \sum \nolimits _{|\beta | \le k}\mathrm{ess\,sup}_{\Omega }|D^{\beta } u|, &{} \mathrm{for }\;p=\infty , \end{array}\right. \end{aligned}$$

where

$$\begin{aligned} D^\beta =\frac{\partial ^{|\beta |}}{\partial x_1^{\beta _1}\ldots \partial x_d^{\beta _d}}, \end{aligned}$$

for the multi-index $\beta =(\beta _1,\ldots ,\beta _d),\,\beta _1\ge 0,\,\ldots ,\,\beta _d\ge 0$, and $|\beta |=\beta _1+\cdots +\beta _d$. When $p=2$ we also note $H^k(\Omega ) := W^{k,2}(\Omega )$.

For $t \in (0,T]$, the weak formulation of the system (1.1)–(1.2) with the boundary conditions (1.3) is defined by

$$\begin{aligned}&( u_t, \, \xi _u) + ( \nabla u, \, \nabla \xi _u) = (\sigma (u)| \nabla \phi |^2, \, \xi _u), \quad \forall \; \xi _u \in H^1_0(\Omega ), \end{aligned}$$

(2.1)

$$\begin{aligned}&( \sigma (u) \nabla \phi , \, \nabla \xi _{\phi } ) = 0, \quad \forall \; \xi _\phi \in H^1_0(\Omega ). \end{aligned}$$

(2.2)

Let $\mathcal{T }_h$ be a regular partition of $\Omega$ into triangles $T_j,\,j=1,\ldots ,M$ in $\mathbb R ^2$ or tetrahedra in $\mathbb R ^3$, and $h=\max _{1\le j\le M}\{\text{ diam }\,T_j\}$ be the mesh size. For a triangle $T_j$ with two nodes (or a tetrahedron with three nodes) on the boundary, we use $\widetilde{T}_j$ to denote the triangle with one curved edge (or a tetrahedron with one curved face) with the same nodes as $T_j$. For interior element, we simply set $\widetilde{T}_j$ as $T_j$ itself. Following classical FEM theory [27, 36], for a given partition of $\Omega$, we define the finite element space

$$\begin{aligned}&\widehat{S}_{h}=\{v_h\in C(\overline{\Omega }_h): v_h|_{T_j} \mathrm{is~a~polynomial~of~degree}~r\}, \\&\widehat{V}_{h}=\{v_h\in C(\overline{\Omega }_h): v_h|_{T_j} \mathrm{is~a~polynomial~of~degree}~r\;\mathrm{and }\;v_h=0\mathrm{on }\;\partial \Omega _h\} \, , \end{aligned}$$

we can see that $\widehat{S}_h$ is a subspace of $H^1(\Omega _h)$ and $\widehat{V}_h$ is a subspace of $H^1_0(\Omega _h)$. Let $G:\Omega _h\rightarrow \Omega$ be a mapping such that both $G$ and $G^{-1}$ are Lipschitz continuous and, for interior element $G$ is the identity mapping, for $T_j$ at the boundary, $G$ maps $T_j$ onto $\widetilde{T}_j$ smoothly. We define an operator $\mathcal{G}:L^2(\Omega _h)\rightarrow L^2(\Omega )$ by $\mathcal{G} v(x)=v(G^{-1}(x))$ for $x\in \Omega$. Defining

$$\begin{aligned} S_{h}=\{\mathcal{G}v_h: v_h\in \widehat{S}_{h}\}, \end{aligned}$$

it is easy to see that $S_h$ is a finite element subspace of $H^1(\Omega )$. For any $v \in H^1(\Omega )$, We define $\Pi _h v = \mathcal{G}\widehat{\Pi }_h\mathcal{G}^{-1} v$, where $\widehat{\Pi }_h:C_0(\overline{\Omega }_h)\rightarrow \widehat{S}_h$ is the Lagrange interpolation operator of degree $r$, then $\forall \, v \in W^{r+1,p}(\Omega )$

$$\begin{aligned} \Vert v-\Pi _hv\Vert _{L^p}+h\Vert v-\Pi _hv\Vert _{W^{1,p}}\le Ch^{r+1}\Vert v\Vert _{W^{r+1,p}}, \quad \mathrm{for }\;1\le p\le \infty . \end{aligned}$$

(2.3)

We set

$$\begin{aligned} V_{h}=\{\mathcal{G}v_h: v_h\in \widehat{V}_{h}\}, \end{aligned}$$

and it is easy to verify that $V_h$ is a finite element subspace of $H^1_0(\Omega )$. We define $R_h:H_0^1(\Omega )\rightarrow V_h$ to be a Ritz projection operator by

$$\begin{aligned} \left( \nabla (v-R_hv),\nabla w \right) =0, \quad {\forall }\, w \in V_h. \end{aligned}$$

By the standard theory of finite element methods to elliptic equations [6, 27],

$$\begin{aligned} \Vert v-R_hv\Vert _{L^2} + h\Vert v-R_hv\Vert _{H^1}\le C h^{r+1} \Vert v\Vert _{H^{r+1}}. \end{aligned}$$

(2.4)

Moreover, let $\{ t_n \}_{n=0}^N$ be a partition in time direction with $t_n = n \tau ,\,T=N\tau$ and

$$\begin{aligned} u^n = u(x,t_n), \quad \phi ^n = \phi (x,t_n). \end{aligned}$$

For any sequence of functions $\{ f^n \}_{n=0}^{N-1}$, we define

$$\begin{aligned} D_\tau f^{n+1}=\frac{f^{n+1}-f^n}{\tau }. \end{aligned}$$

Now we introduce two linearized schemes to solve the time-dependent nonlinear Joule heating Eqs. (1.1)–(1.4).

The first linearized backward Euler Galerkin finite element method is to find $U_h^{n+1} \in V_h,\,\Phi _h^{n+1} \in S_h$ such that

$$\begin{aligned}&\left( D_\tau U_h^{n+1}, \, \xi _u \right) +\left( \nabla U^{n+1}_h, \, \nabla \xi _u \right) =\left( \sigma (U^n_h)|\nabla \Phi ^{n+1}_h|^2, \, \xi _u \right) , \quad \forall \, \xi _u \in V_h, \end{aligned}$$

(2.5)

$$\begin{aligned}&\left( \sigma (U^n_h) \nabla \Phi ^{n+1}_h,\, \nabla \xi _{\phi } \right) =0, \quad \forall \, \xi _{\phi } \in V_h, \end{aligned}$$

(2.6)

with the initial and boundary conditions $U_h^0 = \Pi _h u^0$ and $\Phi _h^{n+1} |_{\partial \Omega } = \Pi _{h}g^{n+1} |_{\partial \Omega }$.

The second one is the fully decoupled linearized backward Euler Galerkin FEMs, which is to find $U_h^{n+1} \in V_h,\,\Phi _h^{n+1} \in S_h$ such that

$$\begin{aligned}&\left( D_\tau U_h^{n+1}, \, \xi _u \right) +\left( \nabla U^{n+1}_h, \, \nabla \xi _u \right) =\left( \sigma (U^n_h)|\nabla \Phi ^{n}_h|^2, \, \xi _u \right) , \quad \forall \, \xi _u \in V_h \, , \end{aligned}$$

(2.7)

$$\begin{aligned}&\left( \sigma (U^n_h) \nabla \Phi ^{n+1}_h, \, \nabla \xi _{\phi } \right) =0, \quad \forall \, \xi _{\phi } \in V_h \, , \end{aligned}$$

(2.8)

with boundary conditions $\Phi _h^{n+1} |_{\partial \Omega } = \Pi _{h}g^{n+1} |_{\partial \Omega }$ and initial conditions $U_h^0 = \Pi _h u^0$ and $\Phi ^{0}_h$, which is the solution of

$$\begin{aligned} \left( \sigma (u^0) \nabla \Phi ^{0}_h, \, \nabla \xi _{\phi } \right) =0, \quad \forall \, \xi _{\phi } \in V_h, \end{aligned}$$

with boundary condition $\Phi _h^0|_{\partial \Omega } = \Pi _h g^0|_{\partial \Omega }$.

The scheme (2.5)–(2.6) is semi-decoupled. At each time step, one has to solve (2.6) for $\Phi _h^{n+1}$ and then (2.5) for $U_h^{n+1}$. A similar semi-decoupled scheme was given in [19], where $\Phi _h^{0}$ was obtained by solving an elliptic PDE. The scheme (2.7)–(2.8) is fully decoupled. At each time step, one only needs to solve two systems of $U_h^{n+1}$ and $\Phi _h^{n+1}$ in parallel.

In this paper, we only present error analysis for the linearized scheme (2.5)–(2.6). The analysis presented here can be easily extended to the second linearized scheme (2.7)–(2.8), which will be confirmed numerically in Sect. 5.

In the rest part of this paper, we assume that $\sigma (s) \in C^r(\mathbb R )$ and the solution to the initial boundary value problem (1.1)–(1.4) exists and satisfies

$$\begin{aligned} \left\{ \begin{array}{l} \Vert u\Vert _{L^\infty ((0,T);H^{r+1})}+\Vert u_t\Vert _{L^2((0,T);H^{r^{*}})} +\Vert u_t\Vert _{L^\infty ((0,T);H^{1})}+\Vert u_{tt}\Vert _{L^2((0,T);H^{1})} \le C, \\ \Vert \phi \Vert _{L^\infty ((0,T);W^{{r+1},4})} + \Vert \phi _{tt}\Vert _{L^2((0,T);H^{1})} + \Vert g\Vert _{L^\infty ((0,T);W^{r+1,4})} \le C . \end{array}\right. \nonumber \\ \end{aligned}$$

(2.9)

where $r^{*}=\max (r,2)$.

We present our main results on error estimates in the following theorem.

Theorem 2.1

Suppose that the system (1.1)–(1.2) with the boundary conditions (1.3) and initial condition (1.4) has a unique solution $(u, \phi )$ satisfying (2.9). Then the finite element system (2.5)–(2.6) with $U_h^0 = \Pi u^0$ (for $r \le 3$) admits a unique solution $(U^{n+1}_h, \,\Phi ^{n+1}_h)$, and there exist two positive constants $\tau _0$ and $h_0$ such that when $\tau < \tau _0$ and $h \le h_0$

$$\begin{aligned} \max _{1\le n\le N}\Vert U^n_h - u^n\Vert _{L^2} +\max _{1\le n\le N}\Vert \Phi ^n_h -\phi ^n\Vert _{L^2} \le C_0(\tau +h^{r+1}), \end{aligned}$$

(2.10)

and

$$\begin{aligned} \max _{1\le n\le N}\Vert U^n_h - u^n\Vert _{H^1} +\max _{1\le n\le N}\Vert \Phi ^n_h - \phi ^n\Vert _{H^1} \le C_0(\tau +h^r), \end{aligned}$$

(2.11)

where $C_0$ is a positive constant, independent of $n,\,h$ and $\tau$.

For simplicity, through out this paper, we denote by $C$ a generic positive constant and $\epsilon$ a generic small positive constant, which are independent of $n,\,h,\,\tau$ and $C_0$ in the above theorem.

For $n=0,\,1,\,\ldots ,\,N-1$, we define the $U^{n+1}$ and $\Phi ^{n+1}$ to be the solutions of the following elliptic system (or time discrete parabolic equations)

$$\begin{aligned}&D_\tau U^{n+1} - \Delta U^{n+1}=\sigma (U^n)|\nabla \Phi ^{n+1}|^2, \end{aligned}$$

(2.12)

$$\begin{aligned}&-\nabla \cdot (\sigma (U^{n})\nabla \Phi ^{n+1})=0, \end{aligned}$$

(2.13)

with $U^0=u_0$ and boundary conditions

$$\begin{aligned} U^{n+1}(x)=0,\quad \Phi ^{n+1}(x)=g(x,t_{n+1}), \quad \mathrm{for }\;x\in \partial \Omega . \end{aligned}$$

(2.14)

In terms of the LS-splitting proposed in [19, 20], the error functions under certain norm $\Vert \cdot \Vert$ can be written by

$$\begin{aligned}&\Vert U^n_h- u^n\Vert \le \Vert e^n \Vert + \Vert e_h^n \Vert + \Vert U^n - R_h U^n \Vert , \\&\Vert \Phi _h^n- \phi ^n\Vert \le \Vert \eta ^n \Vert + \Vert \eta _h^n \Vert + \Vert \Phi ^n - \Pi _h \Phi ^n \Vert , \end{aligned}$$

with

$$\begin{aligned}&e^n = U^n - u^n, \quad e_h^n = U_h^n - R_h U^n, \\&\eta ^n = \Phi ^n - \phi ^n, \quad \eta _h^n = \Phi _h^n - \Pi _h \Phi ^n. \end{aligned}$$

Here we can see both $e^n$ and $\eta ^n$ have zero trace

$$\begin{aligned} e^n = \eta ^n = 0, \quad \mathrm{for } \,\,x \in \partial \Omega . \end{aligned}$$

To prove our main results in Theorem 2.1, we will analyze the temporal error functions $(e^n, \eta ^n)$ in Sect. 3 and the spatial error functions $(e^n_h, \eta ^n_h)$ in Sect. 4, respectively. We present the Gagliardo–Nirenberg inequality and discrete Gronwall’s inequality in the following two lemmas which will be frequently used in our proofs.

Lemma 2.1

Gagliardo–Nirenberg inequality (see [25]): Let $u$ be a function defined on $\Omega$ and $\partial ^{s} u$ be any partial derivative of $u$ of order $s$, then

$$\begin{aligned} \Vert \partial ^{j} u\Vert _{L^p} \le C \Vert \partial ^{m} u\Vert _{L^r}^{a} \, \Vert u\Vert _{L^q}^{1-a} + C \Vert u\Vert _{L^q}, \end{aligned}$$

for $0 \le j < m$ and $\frac{j}{m} \le a \le 1$ with

$$\begin{aligned} \frac{1}{p} = \frac{j}{n} + a \left( \frac{1}{r} - \frac{m}{n}\right) +(1-a) \frac{1}{q}, \end{aligned}$$

except $1 < r < \infty$ and $m-j-\frac{n}{r}$ is a non-negative integer, in which case the above estimate holds only for $\frac{j}{m} \le a < 1$.

Lemma 2.2

Discrete Gronwall’s inequality [15]: Let $\tau ,\,B$ and $a_{k},\,b_{k},\,c_{k},\,\gamma _{k}$, for integers $k \ge 0$, be non-negative numbers such that

$$\begin{aligned} a_{n} + \tau \sum _{k=0}^{n} b_{k} \le \tau \sum _{k=0}^{n} \gamma _{k} a_{k} + \tau \sum _{k=0}^{n} c_{k} + B , \quad \mathrm{for } \quad n \ge 0, \end{aligned}$$

suppose that $\tau \gamma _{k} < 1$, for all $k$, and set $\sigma _{k}=(1-\tau \gamma _{k})^{-1}$. Then

$$\begin{aligned} a_{n} + \tau \sum _{k=0}^{n} b_{k} \le \exp \left( \tau \sum _{k=0}^{n} \gamma _{k} \sigma _{k}\right) \left( \tau \sum _{k=0}^{n} c_{k} + B\right) , \quad \mathrm{for } \quad n \ge 0. \end{aligned}$$

3 Temporal Error Estimates

Theorem 3.1

Suppose that the time-dependent nonlinear Joule heating system (1.1)–(1.4) has a unique solution $(u, \phi )$ satisfying (2.9). Then the elliptic system (2.12)–(2.14) with $U^0 = u^0$ admits a unique solution $(U^{n+1}, \Phi ^{n+1})$ such that

$$\begin{aligned} \max _{0 \le n \le N} \Vert e^{n}\Vert _{H^1} + \max _{1 \le n \le N} \Vert \eta ^{n}\Vert _{H^1} \le C \tau , \end{aligned}$$

(3.1)

and

$$\begin{aligned} \max _{0 \le n \le N}\Vert U^n\Vert _{H^{r+1}} + \sum _{n=1}^{N}\Vert D_\tau U^n\Vert _{H^{r^{*}}}^2 \, \tau \le C , \quad \max _{1 \le n \le N} \Vert \Phi ^n\Vert _{W^{r+1, 4}} \le C. \end{aligned}$$

(3.2)

Proof

We first prove the temporal error estimate (3.1) and then prove the uniform bound (3.2) for all the three cases $r=1,\,2$ and $3$. It is clear that $e^0=0$. By (1.1)–(1.2) and (2.12)–(2.14), the temporal error functions $(e^n, \eta ^n)$ satisfy

$$\begin{aligned} D_\tau e^{n+1}-\Delta e^{n+1}&= (\sigma ( U^n) - \sigma (u^n)) |\nabla \phi ^{n+1}|^2 \nonumber \\&\quad + \sigma (U^n)( \nabla \phi ^{n+1} + \nabla \Phi ^{n+1}) \cdot \nabla \eta ^{n+1} - R_u^{n+1}, \end{aligned}$$

(3.3)

and

$$\begin{aligned} -\nabla \cdot (\sigma (U^{n}) \nabla \eta ^{n+1}) = \nabla \cdot [(\sigma ( u^{n})-\sigma (U^{n})) \nabla \phi ^{n+1}] - \nabla \cdot R_{\phi }^{n+1} \end{aligned}$$

(3.4)

where $R_u^{n+1}$ and $R_\phi ^{n+1}$ are the truncation errors. With the regularity given in (2.9), we have

$$\begin{aligned} \Vert R_u^{n+1}\Vert _{H^1} \le C \tau , \quad \Vert R_\phi ^{n+1}\Vert _{H^{1}} \le C \tau . \end{aligned}$$

(3.5)

Using classical energy method as done in [19], we can derive that there exists a small positive constant $\tau _0$ such that when $\tau <\tau _0$,

$$\begin{aligned} \max _{1 \le n \le N}\Vert e^{n}\Vert _{L^2}^2 + \sum _{m=0}^{N-1}\Vert e^{m+1}\Vert _{H^1}^2 \tau + \max _{1 \le n \le N}\Vert \eta ^{n} \Vert _{H^1} \tau \le C \tau ^2. \end{aligned}$$

(3.6)

It follows from (3.6) that, for $1\le n \le N$

$$\begin{aligned} \Vert D_\tau e^n \Vert _{L^2}, \quad \Vert D_\tau \eta ^n \Vert _{H^1}, \quad \Vert D_\tau U^n \Vert _{L^2}, \quad \Vert D_\tau \Phi ^n \Vert _{H^1} \le C, \quad \Vert e^n\Vert _{H^1} \le C \tau ^{1/2}.\quad \end{aligned}$$

(3.7)

To obtain the $H^s$-norm estimates, $s=2,\,3$ and $4$, we need the following lemma and we refer to [8] for the details of the proof. $\square$

Lemma 3.1

Suppose that $\Omega \in \mathbb R ^3$ be a bounded and smooth domain and $v \in H^k(\Omega )$ is a solution of

$$\begin{aligned} -\Delta v = f, \quad x \in \Omega , \end{aligned}$$

satisfying $v|_{\partial \Omega } = g$, where $g$ can be extended to a function on $\Omega$ such that $g \in W^{k+1,p}(\Omega )$. Then

$$\begin{aligned} \Vert v \Vert _{W^{k+1,p}} \le C \Vert f \Vert _{W^{k-1,p}} + C \Vert g \Vert _{W^{k+1,p}}, \quad \mathrm{for }\;2 \le p < \infty . \end{aligned}$$

We rewrite (3.4) by

$$\begin{aligned} - \Delta \eta ^{n+1}&= \frac{1}{\sigma (U^{n})} \left( \nabla \cdot [(\sigma ( u^{n})-\sigma (U^{n})) \nabla \phi ^{n+1}] - \nabla \cdot R_{\phi }^{n+1} \right) \\&\quad + \frac{\sigma ^{\prime }(U^{n})}{\sigma (U^{n})} \nabla U^n \cdot \nabla \eta ^{n+1}. \end{aligned}$$

Applying Lemma 3.1 to the above equation, we have

$$\begin{aligned} \Vert \eta ^{n+1} \Vert _{H^2}&\le C \Vert \nabla e^n\nabla \phi ^{n+1} \Vert _{L^2} + C \Vert e^n \Delta \phi ^{n+1} \Vert _{L^2} + C \Vert \nabla \cdot R_{\phi }^{n+1} \Vert _{L^2} \\&\quad + C \Vert \nabla e^n \cdot \nabla \eta ^{n+1} \Vert _{L^2} + C \Vert \nabla u^n \cdot \nabla \eta ^{n+1} \Vert _{L^2} \\&\le \epsilon \Vert \nabla \eta ^{n+1} \Vert _{L^6}^2 + \epsilon ^{-1} C \Vert \nabla e^n \Vert _{L^3}^2 + C \Vert e^n \Vert _{H^1} + C \tau \\&\le \epsilon C \Vert \eta ^{n+1} \Vert _{H^2}^2 + C \epsilon ^{-1}\Vert e^n \Vert _{H^1} \Vert e^n \Vert _{H^2} + C \Vert e^n \Vert _{H^1} + C \tau , \end{aligned}$$

which with $C \epsilon \le \frac{1}{2}$ reduces to

$$\begin{aligned} \Vert \eta ^{n+1} \Vert _{H^2} \le C \Vert e^n \Vert _{H^1} \Vert e^n \Vert _{H^2} + C \Vert e^n \Vert _{H^1} + C \tau . \end{aligned}$$

(3.8)

Now we prove a primary estimate by mathematical induction

$$\begin{aligned} \Vert \eta ^{n+1} \Vert _{H^2} \le 1, \quad \mathrm{for }\;0 \le n \le N-1. \end{aligned}$$

(3.9)

From (3.8), it is clear that $\Vert \eta ^{1} \Vert _{H^2} \le C \tau$, (3.9) holds for $n=0$ if we require $C \tau \le 1$. We assume that (3.9) holds for $n \le k-1$.

By applying Lemma 3.1 to (3.3), with estimates (3.6), (3.7) and the above assumption (3.9), we can derive that

$$\begin{aligned} \Vert e^{k} \Vert _{H^2}&\le \Vert D_\tau e^{k}\Vert _{L^2} + \big \Vert (\sigma ( U^{k-1} ) - \sigma (u^{k-1})) |\nabla \phi ^{k}|^2 \big \Vert _{L^2} \\&\quad + \big \Vert \sigma (U^{k-1})( \nabla \phi ^{k} + \nabla \Phi ^{k}) \cdot \nabla \eta ^{k} \big \Vert _{L^2} + \Vert R_u^{k} \Vert _{L^2} \\&\le C \Vert \nabla \eta ^{k} \Vert _{L^4}^2 + C + C \tau \\&\le C \Vert \eta ^{k} \Vert _{H^2}^{2} + C \\&\le C. \end{aligned}$$

With (3.7), substituting the above estimate into (3.8) gives

$$\begin{aligned} \Vert \eta ^{k+1} \Vert _{H^2} \le C \Vert e^k \Vert _{H^1} + C \tau \le C \tau ^{1/2} + C \tau , \end{aligned}$$

and therefore, $\Vert \eta ^{k+1} \Vert _{H^2} \le 1$ when $C \tau ^{1/2} + C \tau \le 1$.

Thus, we complete the induction and obtain

$$\begin{aligned} \Vert \eta ^{n+1} \Vert _{H^2} \le C \Vert e^n \Vert _{H^1} + C \tau , \end{aligned}$$

(3.10)

and

$$\begin{aligned} \Vert e^{n}\Vert _{H^2} \le C, \quad \Vert U^{n}\Vert _{H^2} \le C, \quad \Vert e^{n}\Vert _{L^\infty } \le C, \quad \Vert U^{n}\Vert _{L^\infty } \le C. \end{aligned}$$

(3.11)

Again, we rewrite Eqs. (2.12) and (2.13) by

$$\begin{aligned} - \Delta U^{n+1}= \sigma (U^n)|\nabla \Phi ^{n+1}|^2 - D_\tau U^{n+1}, \end{aligned}$$

(3.12)

and

$$\begin{aligned} - \Delta \Phi ^{n+1} = \frac{\sigma ^{\prime }(U^n)}{\sigma (U^n)} \nabla U^n \cdot \nabla \Phi ^{n+1}. \end{aligned}$$

(3.13)

Applying Lemma 3.1 to Eq. (3.13) gives

$$\begin{aligned} \Vert \Phi ^{n+1} \Vert _{W^{2,4}}&\le C \big \Vert \frac{\sigma ^{\prime }(U^n)}{\sigma (U^n)} \nabla U^n \cdot \nabla \Phi ^{n+1} \big \Vert _{L^4} + C \Vert g^{n+1} \Vert _{W^{2,4}} \nonumber \\&\le C \Vert \nabla U^n \Vert _{L^6} \, \Vert \nabla \Phi ^{n+1} \Vert _{L^{12}} + C \nonumber \\&\le C \big \Vert \Phi ^{n+1} \big \Vert _{H^{1}}^{\frac{5}{7}} \, \big \Vert \Phi ^{n+1} \big \Vert _{W^{2,4}}^{\frac{2}{7}} + C \nonumber \\&\le \frac{2}{7}\Vert \Phi ^{n+1} \Vert _{W^{2,4}} + C , \end{aligned}$$

(3.14)

where we have used the Gagliardo–Nirenberg inequality in Lemma 2.1. It follows that $\Vert \Phi ^{n+1} \Vert _{W^{2,4}} \le C$.

With (3.11) and the above uniform bound for $\Phi ^{n+1}$, multiplying the Eq. (3.3) by $-\Delta e^{n+1}$ yields further

$$\begin{aligned}&D_\tau (\Vert e^{n+1}\Vert _{H^1}^2) + \Vert \Delta e^{n+1}\Vert _{L^2}^2 \nonumber \\&\quad \le C \big \Vert (\sigma ( U^n) - \sigma (u^n)) |\nabla \phi ^{n+1}|^2 \big \Vert _{L^2}^2 \nonumber \\&\quad \quad + C \big \Vert \sigma (U^n)( \nabla \phi ^{n+1} + \nabla \Phi ^{n+1}) \cdot \nabla \eta ^{n+1} \big \Vert _{L^2}^2 + C \Vert R_u^{n+1} \Vert _{L^2}^2 \nonumber \\&\quad \le C \Vert \sigma (U^n)\Vert _{L^\infty }^2 \, (\Vert \nabla \phi ^{n+1}\Vert _{L^\infty }^2 + \Vert \nabla \Phi ^{n+1}\Vert _{L^{\infty }}^2) \, \Vert \nabla \eta ^{n+1} \Vert _{L^{2}}^2 \nonumber \\&\quad \quad + C \Vert e^n\Vert _{L^2}^2 \Vert \nabla \phi ^{n+1} \Vert _{L^\infty }^4 + C \tau ^2 \nonumber \\&\quad \le C \tau ^2. \end{aligned}$$

(3.15)

Thanks to Gronwall’s inequality, there exists a small constant $\tau _0$, such that when $\tau \le \tau _0$

$$\begin{aligned} \max _{0 \le n \le N} \Vert e^{n}\Vert _{H^1}^2 + \sum _{m=0}^{N-1} \Vert e^{m+1}\Vert _{H^2}^2 \tau \le C \tau ^2, \end{aligned}$$

(3.16)

where we have noted the fact that $\Vert e^{n+1}\Vert _{H^2} \le C \Vert \Delta e^{n+1}\Vert _{L^2}$. The estimate (3.16) also implies that $\Vert D_{\tau } U^{n+1}\Vert _{H^1} \le C$. Substituting the above results into (3.10) gives

$$\begin{aligned} \max _{1 \le n \le N} \Vert \eta ^{n} \Vert _{H^2} \le C\tau . \end{aligned}$$

(3.17)

Thus, we complete the proof of the temporal error estimate (3.1) by combining estimates (3.6) and (3.16).

Next, we prove the estimate (3.2) for $r=1,\,2,\,3$. From (3.16) we can also derive that

$$\begin{aligned} \sum _{m=0}^{N-1} \Vert D_\tau U^{m+1} \Vert _{H^2}^2 \tau&\le \sum _{m=0}^{N-1} \left( \Vert D_\tau e^{m+1} \Vert _{H^2}^2 \tau + \Vert D_\tau u^{m+1} \Vert _{H^2}^2 \tau \right) \nonumber \\&\le C \tau ^{-2}\sum _{m=0}^{N-1} \Vert e^{m+1} \Vert _{H^2}^2 \tau + C \nonumber \\&\le C. \end{aligned}$$

(3.18)

By combining (3.11), (3.14) and (3.18), we see that the uniform bound (3.2) holds for $r=1$.

For the case $r=2$, we apply Lemma 3.1 to the Eqs. (3.12) and (3.13) again to deduce

$$\begin{aligned} \Vert U^{n+1} \Vert _{H^3}&\le C \big \Vert \sigma (U^n)|\nabla \Phi ^{n+1}|^2 - D_\tau U^{n+1} \big \Vert _{H^1} \nonumber \\&\le C\Vert \sigma (U^n) \Vert _{L^\infty } \, \Vert \nabla \Phi ^{n+1}\Vert _{L^\infty } \, \Vert \Phi ^{n+1}\Vert _{H^2} \nonumber \\&\quad + C \Vert \sigma (U^n) \Vert _{H^1} \, \Vert \nabla \Phi ^{n+1}\Vert _{L^\infty }^2 + C \nonumber \\&\le C, \end{aligned}$$

(3.19)

and

$$\begin{aligned} \Vert \Phi ^{n+1} \Vert _{W^{3,4}}&\le C \big \Vert \frac{\sigma ^{\prime }(U^n)}{\sigma (U^n)} \nabla U^n \cdot \nabla \Phi ^{n+1} \big \Vert _{W^{1,4}} + C \Vert g \Vert _{W^{3,4}} \nonumber \\&\le \big \Vert \frac{\sigma ^{\prime }(U^n)}{\sigma (U^n)} \big \Vert _{W^{1,\infty }} \left( \Vert U^n \Vert _{W^{2,4}} \Vert \Phi ^{n+1} \Vert _{W^{1,\infty }} + \Vert U^n \Vert _{W^{1,\infty }} \Vert \Phi ^{n+1} \Vert _{W^{2,4}} \right) + C \nonumber \\&\le C \Vert U^n \Vert _{H^3} \Vert \Phi ^{n+1} \Vert _{W^{1,\infty }} + \Vert U^n \Vert _{H^3} \Vert \Phi ^{n+1} \Vert _{W^{2,4}} \nonumber \\&\le C, \end{aligned}$$

(3.20)

which imply the uniform bound (3.2) holds for $r=2$.

Now we turn our proof to the uniform bound (3.2) for the case $r=3$. We multiply (3.3) by $- D_\tau \Delta e^{n+1}$ to get

$$\begin{aligned}&D_\tau (\Vert \Delta e^{n+1}\Vert _{L^2}^2) + \Vert \nabla D_\tau e^{n+1}\Vert _{L^2}^2 \\&\quad \le C \Vert \nabla \left( (\sigma ( U^n) - \sigma (u^n)) |\nabla \phi ^{n+1}|^2 \right) \Vert _{L^2}^2 \\&\quad \quad - \left( \sigma (U^n)( \nabla \phi ^{n+1} + \nabla \Phi ^{n+1}) \cdot \nabla \eta ^{n+1}, \, D_\tau \Delta e^{n+1} \right) + \Vert \nabla R_u^{n+1} \Vert _{L^2}^2 \\&\quad \le - \left( \sigma (U^n)( \nabla \phi ^{n+1} + \nabla \Phi ^{n+1}) \cdot \nabla \eta ^{n+1}, \, D_\tau \Delta e^{n+1} \right) + C \Vert e^n \Vert _{H^1}^2 + C \tau ^2 \,, \end{aligned}$$

which shows further

$$\begin{aligned}&\Vert \Delta e^{n+1}\Vert _{L^2}^2 + \sum _{m=0}^{n} \tau \Vert \nabla D_\tau e^{m+1}\Vert _{L^2}^2 \nonumber \\&\quad \le - \sum _{m=0}^{n} \tau \left( \sigma (U^m)( \nabla \phi ^{m+1} + \nabla \Phi ^{m+1}) \cdot \nabla \eta ^{m+1}, \, D_\tau \Delta e^{m+1} \right) + C \tau ^2 \nonumber \\&\quad = \sum _{m=1}^{n} \tau \left( D_\tau \left( \sigma (U^m)( \nabla \phi ^{m+1} + \nabla \Phi ^{m+1}) \cdot \nabla \eta ^{m+1} \right) , \, \Delta e^{m} \right) \nonumber \\&\quad \quad - \left( \sigma (U^n)( \nabla \phi ^{n+1} + \nabla \Phi ^{n+1}) \cdot \nabla \eta ^{n+1} , \, \Delta e^{n+1} \right) + C \tau ^2 \nonumber \\&\quad \le \sum _{m=1}^{n} \tau \Vert D_\tau \left( \sigma (U^m) (\nabla \phi ^{m+1} + \nabla \Phi ^{m+1}) \cdot \nabla \eta ^{m+1} \right) \Vert _{L^2}^2 \nonumber \\&\quad \quad + \frac{1}{2} \Vert \Delta e^{n+1}\Vert _{L^2}^2 + \sum _{m=1}^{n} \tau \Vert \Delta e^{m}\Vert _{L^2}^2 + C \tau ^2 \nonumber \\&\quad \le C \sum _{m=1}^{n} \tau \Vert \nabla (D_\tau \eta ^{m+1})\Vert _{L^2}^2 + \frac{1}{2} \Vert \Delta e^{n+1}\Vert _{L^2}^2 + \sum _{m=1}^{n} \tau \Vert \Delta e^{m}\Vert _{L^2}^2 + C \tau ^2 \, , \end{aligned}$$

(3.21)

where the summation by parts is used in the temporal direction. In order to estimate $\Vert \nabla (D_\tau \eta ^{m+1})\Vert _{L^2}^2$, we take $D_\tau$ to both sides of the Eq. (3.4) and multiply the result with $D_\tau \eta ^{n+1}$ to deduce that

$$\begin{aligned} \Vert D_\tau \eta ^{n+1} \Vert _{H^1}^2&\le C \Vert (D_\tau \sigma (U^n) ) \nabla \eta ^{n+1} \Vert _{L^2}^2 + C \Vert D_\tau e^n \Vert _{L^2}^2 + C \tau ^2 \nonumber \\&\le C \Vert D_\tau e^n \Vert _{L^2}^2 + C \tau ^2 \nonumber \\&\le C \Vert \Delta e^n \Vert _{L^2}^2 + C \tau ^2 \, , \end{aligned}$$

(3.22)

where we have used the Eq. (3.3).

Substituting (3.22) into (3.21), with the help of Gronwall’s inequality, we have

$$\begin{aligned} \max _{0 \le n \le N}\Vert e^{n}\Vert _{H^2}^2 + \sum _{m=0}^{N-1}\Vert \nabla D_\tau e^{m+1}\Vert _{L^2}^2 \tau \le C \tau ^2, \end{aligned}$$

(3.23)

when $\tau$ is less than certain $\tau _0 > 0$. It follows that

$$\begin{aligned} \Vert D_\tau U^{n+1}\Vert _{H^2} \le C. \end{aligned}$$

Moreover, by applying Lemma 3.1 to (3.3), we can obtain

$$\begin{aligned} \Vert e^{n+1}\Vert _{H^3}&\le C \Vert D_\tau e^{n+1}\Vert _{H^1} + C \Vert (\sigma ( U^n) - \sigma (u^n)) |\nabla \phi ^{n+1}|^2\Vert _{H^1} \nonumber \\&\quad + C \Vert \sigma (U^n)( \nabla \phi ^{n+1} + \nabla \Phi ^{n+1}) \cdot \nabla \eta ^{n+1}\Vert _{H^1} + C \Vert R_u^{n+1} \Vert _{H^1} \nonumber \\&\le C \Vert D_\tau e^{n+1}\Vert _{H^1} + C \tau \, , \end{aligned}$$

(3.24)

and therefore, by estimate (3.23), we have

$$\begin{aligned} \sum _{m=0}^{N-1} \Vert D_\tau U^{m+1} \Vert _{H^3}^2 \tau&\le \sum _{m=0}^{N-1} (\Vert D_\tau e^{m+1} \Vert _{H^3}^2 + \Vert D_\tau u^{m+1} \Vert _{H^3}^2) \tau \nonumber \\&\le \sum _{m=0}^{N-1} \left( C \tau ^{-2} \Vert e^{m+1} \Vert _{H^3}^2 \right) \tau + \sum _{m=0}^{N-1} \Vert D_\tau u^{m+1} \Vert _{H^3}^2 \tau \nonumber \\&\le C \tau ^{-2} \left( \sum _{m=0}^{N-1} ( \Vert D_\tau e^{m+1} \Vert _{H^1}^2 + \tau ^2) \tau \right) + C \nonumber \\&\le C \, . \end{aligned}$$

(3.25)

Finally, we apply Lemma 3.1 to Eqs. (3.12) and (3.13) to get further

$$\begin{aligned} \Vert U^{n+1} \Vert _{H^4}&\le C \Vert \sigma (U^n)|\nabla \Phi ^{n+1}|^2 - D_\tau U^{n+1} \Vert _{H^2} \nonumber \\&\le C \Vert U^n \Vert _{H^{2}} C (\Vert \Phi ^{n+1}\Vert ^2_{W^{1,\infty }} + \Vert \Phi ^{n+1}\Vert _{W^{1,\infty }}\,\Vert \Phi ^{n+1}\Vert _{W^{2,\infty }}) \nonumber \\&\quad + C \Vert U^n \Vert _{L^\infty } \, \Vert \Phi ^{n+1}\Vert ^2_{W^{1,\infty }} \, \Vert \Phi ^{n+1}\Vert ^2_{W^{3,4}} + C \nonumber \\&\le C, \end{aligned}$$

(3.26)

and

$$\begin{aligned} \Vert \Phi ^{n+1} \Vert _{W^{4,4}}&\le C\Vert \frac{\sigma ^{\prime }(U^n)}{\sigma (U^n)} \nabla U^n \cdot \nabla \Phi ^{n+1} \Vert _{W^{2,4}} + C \Vert g \Vert _{W^{4,4}} \nonumber \\&\le C\Vert U^n \Vert _{W^{3,4}} \Vert \Phi ^{n+1} \Vert _{W^{1,\infty }} \Vert U^n \Vert _{W^{1,\infty }} + C\Vert U^n \Vert _{W^{2,\infty }} \Vert \Phi ^{n+1} \Vert _{W^{2,4}} \nonumber \\&\quad + C\Vert U^n \Vert _{W^{1,\infty }} \Vert \Phi ^{n+1} \Vert _{W^{3,4}} + C \nonumber \\&\le C \, , \end{aligned}$$

(3.27)

where we have used estimates (3.19) and (3.20).

By combining (3.25), (3.26) and (3.27) , we have proved that (3.2) holds in the case $r=3$. Thus, we obtain the uniform boundedness of the solution to the elliptic system for all the three cases.

We complete the proof of Theorem 3.1. $\square$

4 Spatial Error Estimates

Theorem 4.1

Suppose that the time-dependent nonlinear Joule heating system (1.1)–(1.4) has a unique solution $(u, \phi )$ satisfying (2.9). Then the fully-discrete system (2.5)–(2.6) with $U_h^0=\Pi _h u^0$ for $r \le 3$ admits a unique solution $(U_h^{n+1}, \Phi ^{n+1}_h)$, such that

$$\begin{aligned}&\max _{0 \le n \le N}\Vert e_h^n \Vert _{L^2} + \max _{1 \le n \le N}\Vert \eta _h^n \Vert _{L^{2}} \le Ch^{r+1} \, , \end{aligned}$$

(4.1)

$$\begin{aligned}&\max _{0 \le n \le N}\Vert \nabla e_h^n \Vert _{L^2} + \max _{1 \le n \le N}\Vert \nabla \eta _h^n \Vert _{L^{2}} \le Ch^r \,. \end{aligned}$$

(4.2)

Proof

The proof for linear FEM has been given in [19], here we only analyze the quadratic and cubic FEMs. Since the coefficient matrices for $U_h^{n+1}$ and $\Phi _h^{n+1}$ are symmetric positive definite, it is clear that the FEM system (2.5)–(2.6) is uniquely solvable. By using the inverse inequality, it is easy to verify that the $L^2$-norm estimate (4.1) implies the $H^1$-norm estimate (4.2). Thus, we only need to prove (4.1). We first prove

$$\begin{aligned} \Vert e^n_h\Vert _{L^2}^2 \le C_1 h^{2r+2}, \quad 0 \le n \le N, \end{aligned}$$

(4.3)

by mathematical induction, where $C_1$ is a positive constant independent of $n,\,h,\,\tau$ and the general constant $C$. As $u^0=U^0$, from the Lagrange interpolation error estimate (2.3) and the Ritz projection error estimate (2.4), we can easily obtain

$$\begin{aligned} \Vert e^0_h\Vert ^2 = \Vert \Pi _hu^0-R_h u^0\Vert ^2 \le C_2 h^{2r+2}, \end{aligned}$$

where $C_2$ is a positive constant independent of $\tau ,\,h$ and $n$. Therefore, if we require $C_1 \ge C_2$, (4.3) holds for $n=0$. We assume that (4.3) holds for $n \le k-1$. We need to find $C_1$, which is independent of $n,\,h,\,\tau$ and the general constant $C$, such that (4.3) also holds for $n \le k$.

With our assumption, by inverse inequality we have

$$\begin{aligned} \Vert e_h^n\Vert _{L^{\infty }} \le C h^{-d/2} \Vert e_h^n\Vert _{L^{2}} \le C C_1 h^{r+1-d/2}. \end{aligned}$$

It is clear that when $C C_1 h^{r+1-d/2} \le 1$ we get $\Vert e_h^n\Vert _{L^{\infty }} \le 1$, which implies $\Vert U_h^n\Vert _{L^{\infty }} \le C$ for $n \le k-1$.

The weak formulation of the time-discrete elliptic system (2.12)–(2.14) is

$$\begin{aligned}&\left( D_\tau U^{n+1}, \, \xi _u \right) +\left( \nabla U^{n+1}, \, \nabla \xi _u \right) =\left( \sigma (U^n)|\nabla \Phi ^{n+1}|^2, \, \xi _u \right) , \quad \forall \, \xi _u \in H_0^1 \, , \end{aligned}$$

(4.4)

$$\begin{aligned}&\left( \sigma (U^{n}) \nabla \Phi ^{n+1}, \, \nabla \xi _{\phi } \right) =0, \quad \forall \, \xi _{\phi } \in H_0^1 \, . \end{aligned}$$

(4.5)

Then, the spatial error functions $(e_h^n, \eta _h^n)$ satisfy

$$\begin{aligned}&\left( D_\tau e^{n+1}_h,\, \xi _u \right) +\left( \nabla e^{n+1}_h,\, \nabla \xi _u \right) \nonumber \\&\quad =\left( D_\tau (U^{n+1}-R_h U^{n+1}), \, \xi _u \right) + \left( (\sigma (U^n_h) - \sigma (U^n))|\nabla \Phi ^{n+1}|^2, \, \xi _u \right) \nonumber \\&\quad \quad +2\left( (\sigma (U^n_h)-\sigma (U^n)) \nabla \Phi ^{n+1} \cdot \nabla (\Phi ^{n+1}_h-\Phi ^{n+1}), \, \xi _u \right) \nonumber \\&\quad \quad +\left( \sigma (U^n_h)|\nabla (\Phi ^{n+1}_h-\Phi ^{n+1})|^2, \, \xi _u \right) \nonumber \\&\quad \quad +2\left( \sigma (U^n)\nabla \Phi ^{n+1} \cdot \nabla (\Phi ^{n+1}_h-\Phi ^{n+1}), \, \xi _{u} \right) \nonumber \\&\quad := \sum _{j=1}^5 I_j^{n+1}(\xi _u) , \quad \forall \, \xi _u \in V_h \, , \end{aligned}$$

(4.6)

and

$$\begin{aligned} \left( \sigma (U^{n})\nabla \eta ^{n+1}_h, \, \nabla \xi _{\phi } \right)&= \left( (\sigma (U^{n})-\sigma (U^{n}_h))\nabla \Phi _h^{n+1}, \, \nabla \xi _{\phi } \right) \nonumber \\&\quad +\left( \sigma (U^n) \nabla (\Phi ^{n+1}- \Pi _h\Phi ^{n+1}), \, \nabla \xi _{\phi } \right) \, , \quad \forall \, \xi _{\phi } \in V_h \, . \end{aligned}$$

(4.7)

Taking $\xi _u=e^{n+1}_h$ into (4.6), now we estimate the five residual terms of the right-hand side of (4.6). The first two terms are bounded by

$$\begin{aligned} I_1^{n+1}(e^{n+1}_h)&\le \epsilon \Vert e^{n+1}_h\Vert _{H^1}^2 + \epsilon ^{-1} C \Vert D_\tau U^{n+1}-R_h D_\tau U^{n+1}\Vert _{H^{-1}}^2 \\&\le \epsilon \Vert e^{n+1}_h\Vert _{H^{1}}^2 + \epsilon ^{-1} C \Vert D_\tau U^{n+1}\Vert _{H^{r^{*}}}^2 h^{2r+2} \, , \end{aligned}$$

and

$$\begin{aligned} I_2^{n+1}(e^{n+1}_h)&\le \Vert \sigma (U_h^n)-\sigma (U^n)\Vert _{L^2} \Vert \nabla \Phi ^{n+1}\Vert _{L^{\infty }}^2 \Vert e^{n+1}_h\Vert _{L^2} \\&\le C \Vert e^{n+1}_h\Vert _{L^2}^2 + C \Vert e^{n}_h \Vert _{L^2}^2 + C h^{2r+2}, \end{aligned}$$

where we have used the embedding inequality $\Vert \nabla \Phi ^{n+1}\Vert _{L^{\infty }} \le C \Vert \Phi ^{n+1}\Vert _{W^{2,4}}$ in Lemma 2.1 and noted $\Vert \Phi ^{n+1}\Vert _{W^{2,4}} \le C$ and $\Vert U\Vert _{H^{r+1}} \le C$ which have been proved in Theorem 3.1.

By inverse inequality, for the third term we have

$$\begin{aligned} I_3^{n+1}(e^{n+1}_h)&\le 2 \Vert \sigma (U_h^n)-\sigma (U^n) \Vert _{L^2} \Vert \nabla \Phi ^{n+1}\Vert _{L^{\infty }} \Vert \nabla (\Phi _h^{n+1}-\Phi ^{n+1})\Vert _{L^3} \Vert e_h^{n+1}\Vert _{L^6} \\&\le \epsilon \Vert e^{n+1}_h\Vert _{H^1}^2 + \epsilon ^{-1} C ( \Vert e^{n}_h \Vert _{L^2}^{2} + h^{2r+2}) (h^{-d/3} \Vert \nabla \eta ^{n+1}_h\Vert _{L^{2}}^2 + h^{2r+2}) \\&\le \epsilon \Vert e^{n+1}_h\Vert _{H^1}^2 + \epsilon ^{-1} C ( \Vert e^{n}_h \Vert _{L^2}^{2} + h^{2r+2}) (h^{-d/3} \Vert \nabla \eta ^{n+1}_h\Vert _{L^{2}}^2 + C h^{2r+2} ) \, . \end{aligned}$$

Moreover, for the fourth term

$$\begin{aligned} I_4^{n+1}(e^{n+1}_h)&\le \Vert \sigma (U_h^n)\Vert _{L^{\infty }} \Vert \nabla (\Phi _h^{n+1}-\Phi ^{n+1})\Vert _{L^{2}} \Vert \nabla (\Phi _h^{n+1}-\Phi ^{n+1})\Vert _{L^{3}} \Vert e^{n+1}_h\Vert _{L^6} \\&\le C \Vert e^{n+1}_h\Vert _{H^1} ( h^{-d/6} \Vert \nabla \eta ^{n+1}_h \Vert _{L^{2}}^2 + h^{r}\Vert \nabla \eta ^{n+1}_h \Vert _{L^{2}} + h^{2r}) \\&\le \epsilon \Vert e^{n+1}_h\Vert _{H^1}^2 + \epsilon ^{-1} C ( h^{-d/6} \Vert \nabla \eta ^{n+1}_h \Vert _{L^{2}}^2 + h^{r}\Vert \nabla \eta ^{n+1}_h \Vert _{L^{2}} + h^{2r})^2. \end{aligned}$$

Finally, by integration by parts and noting the fact that $-\nabla \cdot (\sigma (U^n) \nabla \Phi ^{n+1})=0$, we have

$$\begin{aligned} I_5^{n+1}(e^{n+1}_h)&= - 2 \left( \Phi _h^{n+1}-\Phi ^{n+1} , \nabla \cdot (\sigma (U^n) \nabla \Phi ^{n+1} e_h^{n+1}) \right) \\&= - 2 \left( \Phi _h^{n+1}-\Phi ^{n+1} , \sigma (U^n) \nabla \Phi ^{n+1} \cdot \nabla e_h^{n+1} \right) \\&\le C \Vert \Phi _h^{n+1}-\Phi ^{n+1} \Vert _{L^{2}} \Vert \sigma (U^n)\Vert _{L^{\infty }} \Vert \nabla \Phi ^{n+1}\Vert _{L^{\infty }} \Vert \nabla e_h^{n+1} \Vert _{L^2} \\&\le \epsilon \Vert e^{n+1}_h\Vert _{H^1}^2 + \epsilon ^{-1} C (\Vert \eta _h^{n+1}\Vert _{L^{2}}^2 + h^{2r+2}). \end{aligned}$$

On the other hand, taking $\xi _{\phi } = \eta ^{n+1}_h$ into the Eq. (4.7) gives

$$\begin{aligned} \Vert \nabla \eta ^{n+1}_h\Vert _{L^{2}}&\le C \Vert (\sigma (U^n_h)-\sigma (U^n))\nabla \Phi ^{n+1}_h \Vert _{L^{2}} + C \Vert \sigma (U^n)\nabla (\Phi ^{n+1}-\Pi _h\Phi ^{n+1}) \Vert _{L^{2}} \nonumber \\&\le C \Vert U^n_h-U^n \Vert _{L^6}\Vert \nabla \eta ^{n+1}_h\Vert _{L^{3}} + C \Vert U^n_h-U^n \Vert _{L^2} + Ch^r \nonumber \\&\le C h^{-d/6} ( \Vert e_h^n \Vert _{H^1} + h^r) \Vert \nabla \eta ^{n+1}_h\Vert _{L^{2}} + C \Vert e_h^n \Vert _{L^2} + Ch^r \, . \end{aligned}$$

(4.8)

By the assumption of the induction that (4.3) holds for $n \le k-1$ and applying inverse inequality, we have

$$\begin{aligned} \Vert e_h^n\Vert _{L^2} \le C_1 h^{r+1}, \quad \Vert e_h^n\Vert _{H^1} \le C C_1 h^{r}. \end{aligned}$$

Thus, taking the above inequalities into (4.8) results in

$$\begin{aligned} \Vert \nabla \eta ^{n+1}_h\Vert _{L^{2}}&\le (C C_1 h^{r+1-d/6} + C h^{r-d/6}) \Vert \nabla \eta ^{n+1}_h\Vert _{L^{2}} + C C_1 h^{r+1} + C h^r, \end{aligned}$$

and therefore, requiring $C C_1 h^{r+1-d/6} + C h^{r-d/6} \le 1/2$ and $C_1 h \le 1$ yields

$$\begin{aligned} \Vert \nabla \eta ^{n+1}_h\Vert _{L^{2}} \le C h^r. \end{aligned}$$

(4.9)

Moreover, we use Aubin–Nitsche technique [6] to estimate $\Vert \eta ^{n}_h\Vert _{L^{2}}$. Rewriting (4.7) by

$$\begin{aligned} \left( \sigma (U^n) \nabla (\Phi ^{n+1}- \Phi _h^{n+1}), \, \nabla \xi _{\phi } \right) \!+\! \left( (\sigma (U^{n})-\sigma (U^{n}_h))\nabla \Phi _h^{n+1}, \quad \nabla \xi _{\phi } \right) \!=\! 0, \quad \forall \, \xi _{\phi } \in V_h, \nonumber \\ \end{aligned}$$

(4.10)

and defining $\psi$ as the solution to the elliptic equation

$$\begin{aligned} -\nabla \cdot \left( \sigma (U^n) \nabla \psi \right) = \Phi ^{n+1}-\Phi _h^{n+1}, \end{aligned}$$

(4.11)

with Dirichlet boundary condition $\psi =0$ on $\partial \Omega$. With Lemma 3.1, it can be deduced that $\Vert \psi \Vert _{H^2} \le C \Vert \Phi ^{n+1}-\Phi _h^{n+1}\Vert$.

It is easy to see that taking $\xi _{\phi } = \Pi _h \psi$ into (4.10) gives

$$\begin{aligned} \left( \sigma (U^n) \nabla (\Phi ^{n+1}- \Phi _h^{n+1}), \, \nabla \Pi _h \psi \right) + \left( (\sigma (U^{n})-\sigma (U^{n}_h))\nabla \Phi _h^{n+1}, \, \nabla \Pi _h \psi \right) = 0. \end{aligned}$$

With the help of the estimate (4.9) and the above identity, by multiplying $(\Phi ^{n+1}-\Phi _h^{n+1})$ at both sides of Eq. (4.11), we have

$$\begin{aligned} \Vert \Phi ^{n+1}-\Phi _h^{n+1}\Vert _{L^{2}}^{2}&= \left( \sigma (U^n) \nabla (\Phi ^{n+1}-\Phi _h^{n+1}), \, \nabla \psi \right) \\&= \left( \sigma (U^n) \nabla (\Phi ^{n+1}-\Phi _h^{n+1}), \, \nabla (\psi -\Pi _h\psi ) \right) \\&\quad -\left( (\sigma (U^n)-\sigma (U^n_h)) \nabla \Phi _h^{n+1}, \, \nabla \Pi _h \psi \right) \\&\le C\Vert \nabla (\Phi ^{n+1}-\Phi _h^{n+1})\Vert _{L^{2}}\Vert \nabla (\psi -\Pi _h \psi )\Vert _{L^{2}}\\&\quad +C\Vert U^n- U^n_h\Vert _{L^2} (\Vert \nabla \eta _h^{n+1}\Vert _{L^3} + \Vert \nabla \Pi _h \Phi ^{n+1}\Vert _{L^3}) \Vert \nabla \Pi _h \psi \Vert _{L^6} \\&\le Ch\Vert \nabla (\Phi ^{n+1}-\Phi _h^{n+1}) \Vert _{L^{2}} \Vert \psi \Vert _{H^{2}} \\&\quad +C\Vert U^n- U^n_h\Vert _{L^2} (h^{-d/6}\Vert \nabla \eta _h^{n+1}\Vert _{L^{2}} + C ) \Vert \psi \Vert _{H^{2}} \\&\le C h^{r+1} \Vert \psi \Vert _{H^{2}} + C C_1 (h^{2r+1-d/6} + h^{r+1})\Vert \psi \Vert _{H^{2}} \, , \end{aligned}$$

which in fact implies that when $C_1(h^{r+1-d/6}+h) \le 1$

$$\begin{aligned} \Vert \Phi ^{n+1}-\Phi _h^{n+1}\Vert _{L^{2}} \le C h^{r+1}. \end{aligned}$$

(4.12)

Therefore, with (4.9) and (4.12) and estimates for $I_j^{n+1},\,j=1,\,\ldots ,\,5$, by taking $\xi _u=e^{n+1}_h$ into the spatial error Eq. (4.6), we can derive

$$\begin{aligned} D_\tau \left( \Vert e^{n+1}_h\Vert _{L^2}^2 \right) + \Vert \nabla e_h^{n+1} \Vert _{L^2}^2&\le \epsilon \Vert e_h^{n+1} \Vert _{H^1}^2 + C \epsilon ^{-1} \Vert e_h^{n+1} \Vert _{L^2}^2 + C\epsilon ^{-1} \Vert e_h^{n} \Vert _{L^2}^2 \\&\quad + C (\epsilon ^{-1} + \Vert D_\tau U^{n+1}\Vert _{H^{r^{*}}}^2 )h^{2r+2}. \end{aligned}$$

Thus we can choose a small positive number $\epsilon$ and use Gronwall’s inequality with induction to obtain that there exists a $\tau _0 > 0$ such that when $\tau < \tau _0$

$$\begin{aligned} \Vert e^{n+1}_h\Vert _{L^2}^2 + \tau \sum _{m=1}^{n+1}\Vert e_h^{m} \Vert _{H^1}^2&\le \exp (\frac{TC}{1-\tau C}) (C T + C_2 )h^{2r+2} \\&\le \exp (2 TC) (C T + C_2 )h^{2r+2}, \end{aligned}$$

where we have used $\sum \nolimits _{n=1}^{N}\Vert D_\tau U^n\Vert _{H^{r^{*}}}^2 \tau \le C$ and noted the homogeneous Dirichlet boundary condition. Thus, (4.3) holds for $n=k$ if we take $C_1 \ge \exp (2 TC)(C T + C_2 )$. We complete the induction.

With the above estimates, we have the following result directly from (4.12)

$$\begin{aligned} \Vert \Phi ^{n}-\Phi _h^{n}\Vert _{L^{2}} \le C h^{r+1}. \end{aligned}$$

(4.13)

The proof of Theorem 4.1 is complete. $\square$

We complete the proof of Theorem 2.1 by combining Theorem 3.1, Theorem 4.1, the interpolation error estimate (2.3) and the projection error estimate (2.4). $\square$

5 Numerical Results

In this section, we provide some numerical examples to confirm our theoretical analysis. The computations are performed with free software FEniCS [22]. We set the final time $T=1.0$ in all the computations.

Example 5.1

$(2d)$ We rewrite the system (1.1)–(1.2) by

$$\begin{aligned}&\frac{\partial u}{\partial t} - \Delta u = \sigma (u) {|\nabla \phi |}^{2} + f_1, \end{aligned}$$

(5.1)

$$\begin{aligned}&\quad - \nabla \cdot ( \sigma (u) \nabla \phi ) = f_2 , \end{aligned}$$

(5.2)

and the electric conductivity $\sigma$ takes the form

$$\begin{aligned} \sigma (u) = \frac{1}{1+u^{2}}+1. \end{aligned}$$

The functions $f_{1},\,f_{2}$ and the initial and boundary conditions are determined correspondingly by the exact solution

$$\begin{aligned} u(x,y,t) = \exp (x+y-t), \quad \phi (x,y,t) = 1 + \sin (x+y+t). \end{aligned}$$

Here we only present convergence rate results of the scheme (2.5)–(2.6), and it should be remarked that the fully decoupled scheme (2.7)–(2.8) has similar convergence results. We test the scheme (2.5)–(2.6) on two different domains, one is the unit circle with $\Omega = \{(x,y): x^2+y^2 < 1 \}$ and another is the unit square with $\Omega = (0,1)\times (0,1)$. A regular triangulation with $M$ elements in the radial direction is made on the unit circle, and a uniform triangulation with $M+1$ nodes in both horizontal and vertical directions is made on the unit square, see Fig. 1 for the case $M=10$. Here we can see the mesh size $h=O(1/M)$. We solve Eqs. (5.1)–(5.2) by the two linearized backward Euler scheme (2.5)–(2.6) and the fully decoupled scheme (2.7)–(2.8), denoted by Scheme I and Scheme II, respectively.

Fig. 1

The FEM meshes of the unit circle and the unit square with $M=10$

To confirm our error estimates in Theorem 2.1, we choose $\tau = h^{r+1},\,r=1,\,2,\,3$, for the linear, quadratic and cubic FE methods, respectively. Thus, from our theoretical analysis the $L^2$-norm errors are of scale $O(h^{r+1}+h^{r+1}) \sim O(h^{r+1})$ and the errors in $H^1$-norm are of scale $O(h^{r+1}+h^{r}) \sim O(h^{r})$. We present the $L^2$ and $H^1$-norm errors of Scheme I in Table 1 for the unit circle and in Table 2 for the unit square, respectively. It is clear that for both unit circle and unit square the $L^2$-norm errors of $u$ and $\phi$ are proportional to $h^{r+1}$ and the $H^1$-norm errors are proportional to $h^{r},\,r=1,\,2,\,3$, which indicate the optimal convergence rates of the methods.

Table 1 $L^2$ and $H^1$ errors of Scheme I for the unit circle (Example 5.1. (2d))

Table 2 $L^2$ and $H^1$ errors of Scheme I for the unit square (Example 5.1. (2d))

To show the unconditional convergence of the two schemes, we use the linear FE method to solve (5.1)–(5.2) with three different time steps $\tau =0.01,\,0.05,\,0.25$ on gradually refined meshes with $M=10 i,\,i=1,\,2,\,\ldots ,\,6$ for both domains. The $L^2$-norm errors are given in Fig. 2 for Scheme I and in Fig. 3 for Scheme II, respectively. We should remark that the two schemes with linear FE approximations give $L^2$-norm errors of the scale $O(\tau + h^2)$. From Fig. 2 and (3), we can see that for a fixed $\tau$, when refining the mesh gradually, the $L^2$-norm errors converge to a constant, i.e., the temporal error of the scale $O(\tau )$. It is easy to see that for both domains the two proposed schemes are unconditionally convergent (stable).

Fig. 2

$L^2$-norm errors of scheme I with linear FEM

Fig. 3

$L^2$-norm errors of scheme II with linear FEM

Example 5.2

$(3d)$ We consider Eqs. (5.1)–(5.2) in three-dimensional space with exact solution

$$\begin{aligned} u(x,y,z,t)&= \exp (2x+y-z)(2t+\sin (t)), \\ \phi (x,y,z,t)&= \sin (x-2y)\cos (z)\exp (t), \end{aligned}$$

where $\Omega =\{(x,y,z): x^2+y^2+z^2 < 1 \}$ is the unit ball. We solve the system by these two schemes with quadratic FE method. We take the time steps $\tau = 0.01,\,0.05,\,0.25$ for the scheme I and $\tau = 0.005,\,0.01,\,0.05$ for the scheme II. For the spatial discretizations, We use a regular tetrahedra partition with $M$ elements in the radial direction (see Fig. 4 for the case $M=10$). We refine the mesh gradually by taking $M=5i,\,i=1,\,2,\,\ldots ,\,5$. Plots for $L^2$-norm errors against $M$ are given in Fig. 5 for scheme (2.5)–(2.6) and in Fig. 6 for the fully decoupled scheme (2.7)–(2.8), respectively. From Theorem 2.1, the $L^2$-norm errors are of scale $O(\tau + h^3)$. From Figs. 5 and 6, we can see that if we fix $\tau$ and refine the mesh gradually, the $L^2$-norm errors will asymptotically converge to a constant.

Fig. 4

The three dimensional mesh: inner structure and the surface of the partition with total 1,331 nodes and 6,000 elements $(M=10)$

Fig. 5

$L^2$-norm errors of scheme I with quadratic FEM on a unit ball

Fig. 6

$L^2$-norm errors of scheme II with quadratic FEM on a unit ball

This phenomenon also indicates the unconditional stability of the two schemes in three dimensional space. Previous error analysis for three-dimensional problems often requires a stronger time step restriction than for two-dimensional problems. Our numerical results for both two and three dimensional problems show clearly that no time step condition is needed.

6 Conclusions

We have presented two linearized backward Euler schemes for the nonlinear Joule heating equations in two and three dimensional spaces and provided unconditionally optimal error estimates for the $r$-order Galerkin FEMs $(1 \le r \le 3)$ in both $L^2$ and $H^1$ norms. Numerical results for both two and three dimensional problems confirm our theoretical analysis and show clearly the unconditional stability of the two schemes. The technique presented in this paper can be applied to analyze higher order finite element methods for other nonlinear parabolic equations.