Error Analysis of an Unfitted HDG Method for a Class of Non-linear Elliptic Problems

Abstract

We study Hibridizable Discontinuous Galerkin (HDG) discretizations for a class of non-linear interior elliptic boundary value problems posed in curved domains where both the source term and the diffusion coefficient are non-linear. We consider the cases where the non-linear diffusion coefficient depends on the solution and on the gradient of the solution. To sidestep the need for curved elements, the discrete solution is computed on a polygonal subdomain that is not assumed to interpolate the true boundary, giving rise to an unfitted computational mesh. We show that, under mild assumptions on the source term and the computational domain, the discrete systems are well posed. Furthermore, we provide a priori error estimates showing that the discrete solution will have optimal order of convergence as long as the distance between the curved boundary and the computational boundary remains of the same order of magnitude as the mesh parameter.

Appendices

Appendix A: HDG Projection

The HDG projectors introduced by [3] and their properties have been used extensively throughout the text. Here we provide a quick definition and summary of the properties used in this article.

Consider constants $l_u, l_{\varvec{q}} \in [0,k]$ and functions $(\varvec{q},u) \in H^{1+l_q}(T) \times H^{1+l_u}(T)$. Moreover, recall the discrete spaces

$$\begin{aligned} \varvec{V}_h&:= \{\varvec{v}\in \varvec{L}^2(\mathcal {T}_h) : \varvec{v}|_T \in [\mathbb {P}_k(T)]^d, \quad \forall \ T \in \mathcal {T}_h \}, \\ W_h&:= \{w\in L^2(\mathcal {T}_h) : w|_T \in \mathbb {P}_k(T), \quad \forall \ T \in \mathcal {T}_h \}, \\ M_h&:= \{\mu \in L^2(\mathcal {E}_h) : \mu |_T \in \mathbb {P}_k(e), \quad \forall \ e \in \mathcal {E}_h \}, \end{aligned}$$

defined in (2.7) in the text. We will denote by $\varvec{\Pi }(\varvec{q},u):=(\varvec{\Pi }_{\mathrm v}\varvec{q},\Pi _{\mathrm w} u)$ the projection over $\varvec{V}_h\times W_h$ defined by the unique element-wise solutions of

$$\begin{aligned} (\varvec{\Pi }_{\mathrm v}\varvec{q}, \varvec{v})_T&= (\varvec{q}, \varvec{v})_T&\forall \ \varvec{v} \in [\mathbb {P}_{k-1}(T)]^d, \end{aligned}$$

(A.1a)

$$\begin{aligned} (\Pi _{\mathrm w} u, w)_T&= (u,w)_T&\forall \ w\in \mathbb {P}_{k-1}(T), \end{aligned}$$

(A.1b)

$$\begin{aligned} \left\langle \varvec{\Pi }_{\mathrm v}\varvec{q}\cdot \varvec{n} + \tau \Pi _{\mathrm {w}} u, \mu \right\rangle _{F}&= \left\langle \varvec{q} \cdot \varvec{n} + \tau u, \mu \right\rangle _F&\forall \ \mu \in \mathbb {P}_k(F), \end{aligned}$$

(A.1c)

for every element $T\in \mathcal {T}_h$, and $F\subset \partial T$. Will will denote the $L^2$ projector into $M_h$ by $P_M$. It was proven in [3] that when the stabilization function is chosen so that $\tau _T^{\max } := \max \tau |_{\partial T}>0$, then there is a constant $C>0$ independent of T and $\tau$ such that

$$\begin{aligned} \Vert \varvec{\Pi }_{\mathrm v}\varvec{q} - \varvec{q}\Vert _T&\le C h_T^{l_{\varvec{q}}+1} |\varvec{q}|_{\varvec{H}^{l_{\varvec{q}}+1}(T)} + C h_T^{l_u+1} \tau _T^* |u|_{H^{l_u+1}(T)}, \end{aligned}$$

(A.2a)

$$\begin{aligned} \Vert \Pi _{\mathrm w} u - u\Vert _T&\le C h_T^{l_u+1} |u|_{H^{l_u+1}(T)} + C \dfrac{h_T^{l_{\varvec{q}}+1}}{\tau _T^{\max }} |\nabla \cdot \varvec{q}|_{H^{l_{\varvec{q}}}(T)}. \end{aligned}$$

(A.2b)

Here $\tau _T^* := \max \tau |_{\partial T {\setminus } F^*}$ and $F^*$ is a face of T at which $\tau |_{\partial T}$ is maximum. As is customary, the symbol $|\cdot |_{H^s}$ is to be understood as the Sobolev semi norm of order $s\in \mathbb R$.

Appendix B: Auxiliary Estimates

Duality argument We will consider that, given $\Theta \in L^2(\Omega )$, the solution to the auxiliary problem

$$\begin{aligned}&\kappa ^{-1}(\zeta ) \varvec{\phi }+ \nabla \psi&= 0&\text {in } \Omega , \end{aligned}$$

(B.1a)

$$\begin{aligned}&\nabla \cdot \varvec{\phi }&= \Theta&\text {in } \Omega , \end{aligned}$$

(B.1b)

$$\begin{aligned}&\psi&= 0&\text {on } \partial \Omega , \end{aligned}$$

(B.1c)

satisfies the regularity estimate

$$\begin{aligned} \Vert \varvec{\phi } \Vert _{H^1(\Omega )} + \Vert \psi \Vert _{H^2(\Omega )} \le C_{reg} \Vert \Theta \Vert _{\Omega }. \end{aligned}$$

(B.2)

where $C_{reg} >0$ depends on the domain $\Omega$. Moreover, if $\Omega _h$ is a subdomain of $\Omega$ with boundary $\Gamma _h:=\partial \Omega _h$, l(x) is the length of the transfer path connecting $\Gamma :=\partial \Omega$ to $\Gamma _h$ as defined in Sect. 2.2, $P_M$ is the $L^2-$projector onto the discrete space $M_h$ defined in (2.7), and $Id_M$ is the identity operator in $M_h$ we have

Lemma B1

[5, Lemma 5.5] Suppose Assumption (2.4b) and the elliptic regularity inequality (B.2) hold. Then, there exists a constant $\tilde{c}>0$, such that:

$$\begin{aligned} \Vert {(h^{\perp })}^{-1/2} (Id_M-P_M) \psi \Vert _{\Gamma _h}&\le \tilde{c} \, h \Vert \Theta \Vert _{\Omega }, \end{aligned}$$

(B.3a)

$$\begin{aligned} \Vert l^{1/2} (Id_M-P_M)\partial _n \psi \Vert _{\Gamma _h}&\le \tilde{c} \, R^{1/2} \, h \Vert \Theta \Vert _{\Omega }, \end{aligned}$$

(B.3b)

$$\begin{aligned} \Vert l^{-3/2} (\psi + l \partial _n \psi ) \Vert _{\Gamma _h}&\le \tilde{c} \, \Vert \Theta \Vert _{\Omega }, \end{aligned}$$

(B.3c)

$$\begin{aligned} \Vert l^{-1} \psi \Vert _{\Gamma _h}&\le \tilde{c} \, \Vert \Theta \Vert _{\Omega }. \end{aligned}$$

(B.3d)

Function Delta For any smooth enough function $\varvec{v}$ defined in $T^e\cup T_{ext}^e$ and $\varvec{x}\in \Gamma _h$ we set

$$\begin{aligned} \delta _{\varvec{v}} (\varvec{x}) := \dfrac{1}{l(\varvec{x})} \int _0^{l(\varvec{x})} [\varvec{v}(\varvec{x} + \varvec{n}s) - \varvec{v}(\varvec{x}) ] \cdot \varvec{n} ds. \end{aligned}$$

(B.4)

which hold for each $e\in \mathcal {E}_h^{\partial }$ (cf. [5, Lemma 5.2]):

$$\begin{aligned} \Vert l^{1/2} \, \delta _{\varvec{v}} \Vert _e&\le \dfrac{1}{\sqrt{3}} \, r_e^{3/2} \, C_{ext}^e \, C_{inv}^e \, \Vert \varvec{v}\Vert _{T^e}&\forall \ \varvec{v} \in [\mathbb {P}_k(T)]^d, \end{aligned}$$

(B.5a)

$$\begin{aligned} \Vert l^{1/2}\,\delta _{\varvec{v}} \Vert _e&\le \dfrac{1}{\sqrt{3}} \, r_e \, \Vert h^{\perp } \partial _n \varvec{v} \cdot \varvec{n}\Vert _{T_{ext}^e}&\forall \ \varvec{v} \in [H^1(T)]^d. \end{aligned}$$

(B.5b)

$$\begin{aligned} \Vert l^{1/2}\, \delta _{\varvec{v}}\Vert _{\infty }&\le \dfrac{1}{\sqrt{3}} \, r_e \, \sup _{\varvec{x}\in e} \Vert h_e^{\perp }\,\partial _n \varvec{v}\cdot \varvec{n} \Vert _{l(\varvec{x})}&\forall \ \varvec{v} \in [H^1(T)]^d. \end{aligned}$$

(B.5c)