Stability and convergence of fully discrete Galerkin FEMs for the nonlinear thermistor equations in a nonconvex polygon

Abstract

In this paper, we establish the unconditional stability and optimal error estimates of a linearized backward Euler–Galerkin finite element method (FEM) for the time-dependent nonlinear thermistor equations in a two-dimensional nonconvex polygon. Due to the nonlinearity of the equations and the non-smoothness of the solution in a nonconvex polygon, the analysis is not straightforward, while most previous efforts for problems in nonconvex polygons mainly focused on linear models. Our theoretical analysis is based on an error splitting proposed in [30, 31] together with rigorous regularity analysis of the nonlinear thermistor equations and the corresponding iterated (time-discrete) elliptic system in a nonconvex polygon. With the proved regularity, we establish the stability in $l^\infty (L^\infty )$ and the convergence in $l^\infty (L^2)$ for the fully discrete finite element solution without any restriction on the time-step size. The approach used in this paper may also be applied to other nonlinear parabolic systems in nonconvex polygons. Numerical results confirm our theoretical analysis and show clearly that no time-step condition is needed.

Appendix: Proof of Lemma 1

The following lemmas are consequences of [24] (which can be extended to Hölder continuous coefficients via a basic perturbation argument) and [41].

Lemma A.1

Let $\Omega$ be a Lipschitz domain in $\mathbb {R}^2$. Suppose that $\sigma (u)$ is Hölder continuous and satisfies (1.5). Then there exists a positive constant $p_1>4$ (depending on the domain $\Omega$) such that the solution v of the equation

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\nabla \cdot (\sigma (u)\nabla v)=\nabla \cdot \mathbf{b} &{} \quad \text{ in } \;\Omega ,\\ v=0 &{} \quad \text{ on } \;\partial \Omega , \end{array}\right. \end{aligned}$$

satisfies that

$$\begin{aligned} {\left\| v \right\| _{{W^{1,q}}}} \le {C_q}{\left\| \mathbf{{b}} \right\| _{{L^q}}} \quad for\;{p_{\mathrm{1}}}{\mathrm{/}}({p_{\mathrm{1}}}{\mathrm{- 1}}) \le q \le {p_{\mathrm{1}}}. \end{aligned}$$

Lemma A.2

Let $\Omega$ be a Lipschitz domain in $\mathbb {R}^2$. Then the solution of the inhomogeneous heat equation

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \frac{\partial u}{\partial t}-\varDelta u= \nabla \cdot f &{} \quad \mathrm{in } \; \Omega , \\ u=0 &{} \quad \mathrm{on } \; \partial \Omega , \\ u(x,0)=u_0(x) &{} \quad \mathrm{for } \; x\in \Omega , \end{array}\right. \end{aligned}$$

satisfies that

$$\begin{aligned} {\left\| u \right\| _{{L^2}((0,T);{W^{1,4}})}} \le C\sum \limits _{j = 1}^2 {{{\left\| {{f_j}} \right\| }_{{L^2}((0,T);{L^4})}}} + C{\left\| {{u_0}} \right\| _{{H^{1/2}}}}. \end{aligned}$$

From [7] we know that the regularity of g given in Lemma 1 implies that g can be extended to the interior of the domain $\Omega$ with $g\in L^\infty ((0,T);H^{1+\beta })$ and $g_t\in L^\infty ((0,T);W^{1,4})$.

Based on Yuan and Liu’s results [44, 45], the solution of (1.2)–(1.4) satisfies that

$$\begin{aligned} \Vert u\Vert _{C^\alpha (\overline{\Omega })}\le C. \end{aligned}$$

(A.1)

By Lemma A.1, the Eq. (1.3) with the Hölder continuity of u implies that

$$\begin{aligned} \Vert \phi \Vert _{L^\infty ((0,T);W^{1,p_1})}\le C\Vert g\Vert _{L^\infty ((0,T);W^{1,p_1})}\le C, \quad \mathrm{for~some }\;p_1>4. \end{aligned}$$

(A.2)

Then (1.2) implies that

$$\begin{aligned} \Vert u_t\Vert _{L^2((0,T);L^2)} \le C\Vert \sigma (u)|\nabla \phi |^2\Vert _{L^2((0,T);L^2)}+C\Vert u_0\Vert _{H^1} \le C. \end{aligned}$$

(A.3)

Let $w=u_t$. Differentiating (1.2)–(1.3) with respect to t, we obtain

$$\begin{aligned}&\partial _tw-\varDelta w=\nabla \cdot ( \sigma '(u)w\phi \nabla \phi ) +\nabla \cdot (\sigma (u)\phi _t\nabla \phi ) +\nabla \cdot (\sigma (u)\phi \nabla \phi _t), \end{aligned}$$

(A.4)

$$\begin{aligned}&-\nabla \cdot (\sigma (u)\nabla \phi _t)=\nabla \cdot (\sigma '(u)w\nabla \phi ). \end{aligned}$$

(A.5)

with the initial condition $w(x,0)=w_0(x)$, where $w_0=\varDelta u_0+\sigma (u_0)|\nabla \phi _0|^2$ and $\phi _0$ is the solution of the elliptic PDE

$$\begin{aligned} \left\{ \begin{array}{ll} -\nabla \cdot (\sigma (u_0)\nabla \phi _0)=0, &{} \quad \mathrm{in } \; \Omega \\ \phi _0(x)=g_0 &{} \quad \mathrm{for } \; x\in \partial \Omega . \end{array}\right. \end{aligned}$$

It follows that $w_0\in H^{1}_0$. Since $w\in L^2((0,T); L^2)$ and $\nabla \phi \in L^\infty ((0,T); L^4)$, applying Lemma A.1–(A.5) gives

$$\begin{aligned} \Vert \phi _t\Vert _{L^2((0,T);W^{1,4/3})}&\le C\Vert w\nabla \phi \Vert _{L^2((0,T);L^{4/3})} +C\Vert g_t\Vert _{L^2((0,T);W^{1,4/3})} \nonumber \\&\le C\Vert w\Vert _{L^2((0,T);L^2)}\Vert \nabla \phi \Vert _{L^2((0,T);L^4)} +C\Vert g_t\Vert _{L^2((0,T);W^{1,4/3})} \nonumber \\&\le C. \end{aligned}$$

(A.6)

and by the Sobolev embedding theorem, we obtain,

$$\begin{aligned} \Vert \phi _t\Vert _{L^2((0,T);L^4)}&\le C\Vert \phi _t\Vert _{L^2((0,T);W^{1,4/3})} \le C. \end{aligned}$$

(A.7)

Again applying Lemma A.2 to (A.4) shows

$$\begin{aligned} \Vert w\Vert _{L^2((0,T);W^{1,4/3})}&\le C\Vert w \nabla \phi \Vert _{L^2((0,T);L^{4/3})} + C\Vert \phi _t\nabla \phi \Vert _{L^2((0,T);L^{4/3})} \nonumber \\&\quad +\;C\Vert \nabla \phi _t\Vert _{L^2((0,T);L^{4/3})} +\Vert w_0\Vert _{H^{1/2}} \nonumber \\&\le C \end{aligned}$$

(A.8)

and with the Sobolev embedding theorem, we have

$$\begin{aligned} \Vert w\Vert _{L^2((0,T);L^4)} \le C\Vert w\Vert _{L^2((0,T);W^{1,4/3})} \le C. \end{aligned}$$

(A.9)

From (A.5) and (A.4), we see that

$$\begin{aligned} \Vert \phi _t\Vert _{L^2((0,T);H^1)}&\le C\Vert w\nabla \phi \Vert _{L^2((0,T);L^2)} +C\Vert g_t\Vert _{L^2((0,T);H^1)} \nonumber \\&\le C\Vert w\Vert _{L^2((0,T);L^4)} \Vert \nabla \phi \Vert _{L^\infty ((0,T);L^4)}+C\Vert g_t\Vert _{L^2((0,T);H^1)} \nonumber \\&\le C \end{aligned}$$

(A.10)

and

$$\begin{aligned} \Vert w\Vert _{L^2((0,T);H^1)}&\le C\Vert w \nabla \phi \Vert _{L^2((0,T);L^2)} + C\Vert \phi _t\nabla \phi \Vert _{L^2((0,T);L^2)} \\&\quad +\; \Vert \nabla \phi _t\Vert _{L^2((0,T);L^2)}+\Vert w_0\Vert _{L^2}) \\&\le C. \end{aligned}$$

By the Sobolev embedding theorem, we have further

$$\begin{aligned} \Vert w\Vert _{L^2((0,T);L^{p_3})}+\Vert \phi _t\Vert _{L^2((0,T);L^{p_3})} \le C(\Vert w\Vert _{L^2((0,T); H^1)}+\Vert \phi _t\Vert _{L^2((0,T); H^1)}) \le C, \end{aligned}$$

(A.11)

where $p_3$ is determined by $1/p_3+1/p_1=1/4$. Then

$$\begin{aligned} \Vert w\nabla \phi \Vert _{L^2((0,T);L^4)} \le C\Vert w\Vert _{L^2((0,T);L^{p_3})} \Vert \nabla \phi \Vert _{L^\infty ((0,T);L^{p_1})} \le C, \end{aligned}$$

(A.12)

$$\begin{aligned} \Vert \phi _t\nabla \phi \Vert _{L^2((0,T);L^4)} \le C\Vert \phi _t\Vert _{L^2((0,T);L^{p_3})} \Vert \nabla \phi \Vert _{L^\infty ((0,T);L^{p_1})}\le C. \end{aligned}$$

(A.13)

Moreover, by applying Lemma A.1 to the Eq. (A.5), we see that

$$\begin{aligned} \Vert \phi _t\Vert _{L^2((0,T);W^{1,4})} \le C\Vert w\nabla \phi \Vert _{L^2((0,T);L^4)} +C\Vert g_t\Vert _{L^2((0,T);W^{1,4})} \le C \end{aligned}$$

(A.14)

and by Lemma A.2, (A.4) implies that

$$\begin{aligned} \Vert w\Vert _{L^2((0,T);W^{1,4})}&\le C \Vert w \nabla \phi \Vert _{L^2((0,T);L^4)} +C\Vert \phi _t\nabla \phi \Vert _{L^2((0,T);L^4)} \nonumber \\&\quad +\;C\Vert \nabla \phi _t\Vert _{L^2((0,T);L^4)}+C\Vert w_0\Vert _{H^{1/2}}) \nonumber \\&\le C. \end{aligned}$$

(A.15)

Using the Sobolev embedding theorem again, we arrive at

$$\begin{aligned} \Vert w\Vert _{L^2((0,T);L^\infty )}&\le C\Vert w\Vert _{L^2((0,T);W^{1,4})} \le C. \end{aligned}$$

(A.16)

Differentiating (1.2) with respect to t, we get

$$\begin{aligned} \partial _tw-\varDelta w=\sigma '(u)w|\nabla \phi |^2 +2\sigma (u)\nabla \phi \cdot \nabla \phi _t. \end{aligned}$$

(A.17)

and the above equation times $\varDelta w$ gives

$$\begin{aligned}&\Vert w\Vert _{L^\infty ((0,T);H^1)} +\Vert \partial _tw\Vert _{L^2((0,T);L^2)} + \Vert \varDelta w\Vert _{L^2((0,T);L^2)} \\&\quad \le C(\Vert w|\nabla \phi |^2\Vert _{L^2((0,T);L^2)} +\Vert \nabla \phi \cdot \nabla \phi _t \Vert _{L^2((0,T);L^2)}+\Vert w_0\Vert _{H^1} ) \\&\quad \le C, \end{aligned}$$

which further shows that

$$\begin{aligned} \Vert w\Vert _{L^\infty ((0,T);L^{p_3})}+\Vert u_{tt}\Vert _{L^2((0,T);L^2)} + \Vert u_t\Vert _{L^2((0,T);H^{1+s})} \le C. \end{aligned}$$

(A.18)

It follows that

$$\begin{aligned} \Vert u\Vert _{C([0,T];H^{1+s})}\le C\Vert u\Vert _{L^2((0,T);H^{1+s})}+C\Vert u_t\Vert _{L^2((0,T);H^{1+s})} \le C. \end{aligned}$$

(A.19)

From (A.5) we see that

$$\begin{aligned} \Vert \phi _t\Vert _{L^\infty ((0,T);W^{1,4})}&\le C\Vert w\nabla \phi \Vert _{L^\infty ((0,T);L^4)} +C\Vert g_t\Vert _{L^\infty ((0,T);W^{1,4})} \nonumber \\&\le C. \end{aligned}$$

(A.20)

Furthermore, we rewrite (1.3) by

$$\begin{aligned} -\varDelta \phi =\frac{\sigma '(u)}{\sigma (u)} \nabla u\cdot \nabla \phi . \end{aligned}$$

(A.21)

With (), we obtain

$$\begin{aligned} \Vert \phi \Vert _{L^\infty ((0,T);H^{1+s})}&\le C \Vert \varDelta \phi \Vert _{L^\infty ((0,T);L^2)} +C\Vert g\Vert _{L^\infty ((0,T);H^2)} \nonumber \\&\le C\Vert \nabla u\cdot \nabla \phi \Vert _{L^\infty ((0,T);L^2)} +C\le C_2. \end{aligned}$$

(A.22)

Since $H^{1+s_1}\hookrightarrow \hookrightarrow H^{1+s}\hookrightarrow W^{1,4}$, we have the following estimate (see Lemma 1.1, pp. 106 of [35])

$$\begin{aligned}&\Vert \phi (\cdot ,t_1)-\phi (\cdot ,t_2)\Vert _{H^{1+s}} \\&\quad \le \frac{1}{2}C_2^{-1}\epsilon \Vert \phi (\cdot ,t_1)-\phi (\cdot ,t_2)\Vert _{H^{1+s_1}} +C_\epsilon \Vert \phi (\cdot ,t_1)-\phi (\cdot ,t_2)\Vert _{W^{1,4}} \\&\quad \le \epsilon +C_\epsilon |t_1-t_2|\Vert \phi _t\Vert _{L^\infty ((0,T);W^{1,4})} \\&\quad \le \epsilon +C_\epsilon |t_1-t_2|, \quad \forall \; \epsilon >0, \end{aligned}$$

which implies that $\phi \in C([0,T];H^{1+s})$.

The proof of Lemma 1 is completed. $\square$