Discontinuous Galerkin Approximation of Linear Parabolic Problems with Dynamic Boundary Conditions

Abstract

In this paper we propose and analyze a Discontinuous Galerkin method for a linear parabolic problem with dynamic boundary conditions. We present the formulation and prove stability and optimal a priori error estimates for the fully discrete scheme. More precisely, using polynomials of degree $p\ge 1$ on meshes with granularity h along with a backward Euler time-stepping scheme with time-step $\Delta t$, we prove that the fully-discrete solution is bounded by the data and it converges, in a suitable (mesh-dependent) energy norm, to the exact solution with optimal order $h^p + \Delta t$. The sharpness of the theoretical estimates are verified through several numerical experiments.

Appendix: Proof of Theorem 4.1

Proof of Theorem 4.1

As the proof follows is based on standard arguments (see, e.g., [12, Chapter 7.1]), we only sketch the main steps.

1. Construction of the discrete space Let $\{e_i\}_{i\ge 1}$ be an orthonormal basis of $L^2(\Omega )$ such that

$$\begin{aligned} \int _{\Omega } \nabla e_i \cdot \nabla z = \lambda _i \int _{\Omega } e_i z \quad \forall z\in H^1(\Omega ),\ i\ge 1, \end{aligned}$$

i.e., $\lambda _i$ and $e_i$ are respectively the eigenvalues and eigenfunctions of the weak form of eigenvalue problem $-\Delta e = \lambda e$ with homogeneous Neumann and periodic boundary conditions on $\Gamma _1$ and $\Gamma _2$, respectively. Reordering $\{e_i\}_{i\ge 1}$ such that $\lambda _1=0$, it is easy to see that there holds

$$\begin{aligned} \int _{\Omega }\nabla e_i \cdot \nabla e_j = 0,\quad \text {for }\, i\ne j\quad \text {and }\quad \int _{\Omega }|\nabla e_i|^2=\lambda _i>0,\ \ \ \text { for } i>1. \end{aligned}$$

Let $V^n=\text {span}\{e_i : i=1,\ldots ,n\}$, $n\ge 1$, and let $u_0^n$ be the $L^2(\Omega )$- projection of $u_0$ on $V^n$. Since the domain is regular, the eigenfunctions $e_i$ belong to $H^2(\Omega )$.

2. Finite-dimensional approximation of (17) We introduce the following finite dimensional problem: find $u^n\in H^1(0,T;V^n)$ such that, for $t\in (0,T)$,

$$\begin{aligned} {\left\{ \begin{array}{ll} (\partial _t u^n,z)_{L^2(\Omega )}+\lambda (\partial _t u^n,z)_{L^2(\Gamma _1)}+ a(u^n,z)=(f,z)_{L^2(\Omega )} +(g,z)_{L^2(\Gamma _1)},\\ {u^n}_{|t=0}=u_0^n, \end{array}\right. } \end{aligned}$$

(34)

for all $z \in V^n$, In the sequel we prove that problem (34) admits a unique solution in $H^1(0,T; V^n)$. We write

$$\begin{aligned} u^n(t)=\sum _{j=1}^n u_j(t)e_j. \end{aligned}$$

The problem (34) is equivalent to find $\mathbf u =(u_1, \ldots , u_n)^T\in H^1(0,T;\mathbb {R}^n)$ such that, for each $t\in (0,T)$,

$$\begin{aligned} {\left\{ \begin{array}{ll} M \dot{\mathbf {u}}(t) + A \mathbf {u}(t) = \mathbf {F}(t),\\ \mathbf {u}(0)=(u_{0,1}, \ldots , u_{0,n})^T, \end{array}\right. } \end{aligned}$$

where, for $i,j=1,\ldots ,n$,

$$\begin{aligned} M_{ij}= & {} M^\Omega + \lambda M^{\Gamma _1}:=\delta _{ij}+ \lambda (e_i,e_j)_{L^2(\Gamma _1)},\\ A_{ij}= & {} a(e_i,e_j),\qquad F_{i}=(f,e_i)_{L^2(\Omega )} + (g,e_i)_{L^2(\Gamma _1)}, \qquad u_{0,i}=(u_0, e_i)_{L^2(\Omega )}. \end{aligned}$$

Since the matrix $M^{\Gamma _1}$ is semi-positive definite, we see that M is positive definite. In addition, $\mathbf {F}\in L^2(0, T; \mathbb {R}^{n})$ and $A:\mathbb {R}^n\rightarrow \mathbb {R}^n$ is Lipschitz continuous. Therefore, by standard existence theory of ordinary differential equations, there exists a unique solution $\mathbf {u}(t)$ for a.e. $0\le t\le T$.

3. Energy estimates Taking $z=u^n$ in (34) and using the Cauchy–Schwarz inequality, we obtain

$$\begin{aligned}&\frac{d}{dt}\left( ||u^n||_{L^2_\lambda (\Omega , \Gamma _1)}^2\right) + ||\nabla u^n||^2_{L^2(\Omega )} + \alpha ||u^n||^2_{L^2(\Gamma _1)} + \beta ||\nabla _\Gamma u^n||^2_{L^2(\Gamma _1)}\nonumber \\&\quad \lesssim ||u^n||_{L^2_\lambda (\Omega , \Gamma _1)}^2 +||f||^2_{L^2(\Omega )} + ||g||^2_{L^2(\Gamma _1)} \end{aligned}$$

(35)

for a.e. $t \in [0,T]$. Using the differential form of the Gronwall’s inequality, data regularity and Lemma 6.1 we obtain

$$\begin{aligned} \max _{0\le t\le T}||u^n(t)||_{L^2_\lambda (\Omega ,\Gamma _1)} \lesssim ||u_0||_{L^2_\lambda (\Omega , \Gamma _1)}^2 +||f||^2_{L^2(0,T;L^2(\Omega ))} + ||g||^2_{L^2(0,T;L^2(\Gamma _1))}\le C. \end{aligned}$$

Integrating (35) in [0, T] and employing the above inequality together with data regularity and Lemma 6.1 we get

$$\begin{aligned} ||u^n||_{L^2(0,T;H^1_\lambda (\Omega ,\Gamma _1))}\lesssim ||u_0||_{L^2_\lambda (\Omega , \Gamma _1)}^2 +||f||^2_{L^2(0,T;L^2(\Omega ))} + ||g||^2_{L^2(0,T;L^2(\Gamma _1))}\le C. \end{aligned}$$

On the other hand, taking $z=\partial _t u^n$ in (34), integrating in t and using the Cauchy–Schwarz inequality, we obtain, for every $\tau \in (0,T]$,

$$\begin{aligned}&\frac{1}{2} \int _0^\tau ||\partial _t u^n||^2_{L^2_\lambda (\Omega ,\Gamma _1)} + \frac{1}{2}||\nabla u^n(\tau )||^2_{L^2(\Omega )} + \frac{\alpha }{2}||u^n(\tau )||^2_{L^2(\Gamma _1)} + \frac{\beta }{2}||\nabla _\Gamma u^n(\tau )||^2_{L^2(\Gamma _1)}\\&\quad \le \frac{1}{2}||\nabla u^n_0||^2_{L^2(\Omega )} + \frac{\alpha }{2}|| u^n_0||^2_{L^2(\Gamma _1)}+\frac{\beta }{2}||\nabla _\Gamma u^n_0||^2_{L^2(\Omega )}\\&\quad +\, \frac{1}{2} \int _0^\tau ||f||^2_{L^2(\Omega )} + \frac{1}{2\lambda } \int _0^\tau ||g||^2_{L^2(\Gamma _1)}, \end{aligned}$$

where the right-hand side of the above inequality can be bounded using Lemma 6.1 and data regularity.

Moreover, differentiating (34) with respect to t and setting $\tilde{u}^n:= \partial _t u^n$ we get for any $t\in [0,T]$

$$\begin{aligned} (\partial _t \tilde{u}^n,z)_{L^2(\Omega )}+\lambda (\partial _t \tilde{u}^n,z)_{L^2(\Gamma _1)}+ a(\tilde{u}^n,z)=(\partial _t f,z)_{L^2(\Omega )} +(\partial _t g,z)_{L^2(\Gamma _1)}, \end{aligned}$$

(36)

for all $z\in V^n$. Testing (36) with $z=\tilde{u}^n$, it is easy to show that it holds

$$\begin{aligned}&\Vert \partial _t u^n\Vert ^2_{L_\lambda ^2(\Omega ,\Gamma _1)} + \int _0^t \Vert \partial _t u^n (s)\Vert ^2_{H_\lambda ^1(\Omega ,\Gamma _1)}\, ds \lesssim \int _0^t \Vert \partial _t f (s) \Vert ^2_{L^2(\Omega )}~ds \nonumber \\&\qquad + \int _0^t \Vert \partial _t g (s) \Vert ^2_{L^2(\Gamma _1) }\, ds + \Vert \partial _t u^n (0)\Vert ^2_{L_\lambda ^2(\Omega ,\Gamma _1)}. \end{aligned}$$

(37)

Taking $t=0$ in (34), testing with $z=\partial _t u^n (0)$, integrating by parts and employing the Cauchy–Schwarz inequality once more, we obtain

$$\begin{aligned} \Vert \partial _t u^n(0)\Vert ^2_{L^2_\lambda (\Omega ,\Gamma _1)} \lesssim \Vert u^n(0)\Vert ^2_{H^2_\lambda (\Omega ,\Gamma _1)} + \Vert f(0,\cdot ) \Vert ^2_{L^2(\Omega )} + \Vert g(0,\cdot ) \Vert ^2_{L^2(\Gamma _1)}, \end{aligned}$$

whose right-hand side can be bounded by resorting to compatibility conditions, Lemma 6.1 and data regularity assumptions.

Hence, collecting all the above results, we get

$$\begin{aligned} u^n \in C([0,T]; H_\lambda ^1(\Omega ,\Gamma _1))\cap C^1(0,T; L^2_\lambda (\Omega ,\Gamma _1)) \cap H^1(0,T; H^1_\lambda (\Omega ,\Gamma _1)). \end{aligned}$$

4. Existence of the solution u Resorting to subsequences $\{u_{m_l}\}_{l=1}^{\infty }$ of $\{u_m\}_{m=1}^{\infty }$, passing to the limit for $m\rightarrow \infty$ and using standard arguments it is possible to prove that there exists a solution u to problem (17) with

$$\begin{aligned} u \in C([0,T]; H_\lambda ^1(\Omega ,\Gamma _1))\cap C^1(0,T; L^2_\lambda (\Omega ,\Gamma _1)) \cap H^1(0,T; H^1_\lambda (\Omega ,\Gamma _1)). \end{aligned}$$

5. Uniqueness of the weak solution Let $u_1$ and $u_2$ be two solutions of weak problem (17) and set $w=u_1-u_2$. By definition, taking $z=w$, we get from (17)

$$\begin{aligned} \frac{d}{dt}\left( ||w||_{L^2_\lambda (\Omega , \Gamma _1)}^2\right) + ||\nabla w||^2_{L^2(\Omega )} + \alpha ||w||^2_{L^2(\Gamma _1)} + \beta ||\nabla _\Gamma w||^2_{L^2(\Gamma _1)}=0, \end{aligned}$$

that implies $w=0$, or $u_1=u_2$ for a.e. $0\le t \le T$.

6. Improved regularity Rewriting (17) as

$$\begin{aligned} a(u,v)=(\tilde{f},v)_{L^2(\Omega )} +(\tilde{g},v)_{L^2(\Gamma _1)}, \end{aligned}$$

where $\tilde{f}= f - \partial _t u \in L^2(0,T,L^2(\Omega ))$ and $\tilde{g}= g - \partial _t u \in L^2(0,T,L^2(\Gamma _1))$. Employing Theorem 3.1 we get $u(t)\in H^2_\lambda (\Omega ,\Gamma _1)$ for a.e. $0\le t \le T$.

7. Higher regularity We prove (18) by induction. From the above discussion the result holds true for $m=1$. Assume now the validity of (18) for some $m>1$, together with the associated higher order compatibility and regularity conditions. Differentiating (16) with respect to t, it is immediate to verify that $\tilde{u}=\partial _t u$ verifies

(38)

where $\tilde{f}=\partial _t f$, $\tilde{g}=\partial _t g$, $\tilde{u}_0 = f(0,\cdot ) + \Delta u_0$ in $\Omega$ and $\tilde{u}_0|_\Gamma = \beta \Delta _{\Gamma }u_0 - \partial _n u_0 -\alpha u_0 + g(0,\cdot )$ on $\Gamma$. Since the pair (f, g) satisfies the higher order compatibility conditions for $k=1,\ldots ,m$ then the pair $(\tilde{f},\tilde{g})$ satisfies the same type of compatibility conditions for $k=1,\ldots ,m-1$. Hence, it follows for $k=0,\ldots ,m-1$

$$\begin{aligned}&\frac{d^k \tilde{u}}{d t^k} \in C([0,T]; H^{2m-2k}(\Omega ,\Gamma _1))\cap C^1([0,T]; H^{2m-2k-2}_\lambda (\Omega ,\Gamma _1))\nonumber \\&\quad \cap ~ H^1(0,T; H^{2m-2k-1}_\lambda (\Omega ,\Gamma _1)) \end{aligned}$$

(39)

which immediately implies the validity of (18) for $k=0,\ldots ,m$. $\square$

The following result has been proof in [16, Lemmas 4.4 and 4.5].

Lemma 6.1

Let $z \in Z = \{z\in H^2(\Omega )\ |\ \partial _n z = 0 \text { on } \Gamma _1\}$. If $z_n$ is the $L^2(\Omega )$-projection of z on $V^n$, then

$$\begin{aligned} ||z_n - z||_{H^1_\lambda (\Omega ;\Gamma _1)}\rightarrow 0 \text { when } n\rightarrow \infty . \end{aligned}$$

(40)

Let $V_{\infty }=\cup _{n=1}^\infty V_n$. Moreover, Z and $V_{\infty }$ are dense in $H^1_\lambda (\Omega ;\Gamma _1)$.