A Higher Order Isoparametric Fictitious Domain Method for Level Set Domains

Abstract

We consider a new fictitious domain approach of higher order accuracy. To implement Dirichlet conditions we apply the classical Nitsche method combined with a facet-based stabilization (ghost penalty). Both techniques are combined with a higher order isoparametric finite element space which is based on a special mesh transformation. The mesh transformation is build upon a higher order accurate level set representation and allows to reduce the problem of numerical integration to problems on domains which are described by piecewise linear level set functions. The combination of this strategy for the numerical integration and the stabilized Nitsche formulation results in an accurate and robust method. We introduce and analyze it and give numerical examples.

Appendix: Selected Proofs

We give the proofs of some of the more technical results in Sect. 5.

Proof of Lemma 2

The first estimate has been proven in [20] for $T \in \mathscr {T}_h^\varGamma$. With the extension operator applied in the projection step $P_h^2$ this property carries over to every element $T \in \mathscr {T}_h$, cf. the analysis in [30]. The proof for the inverse transformation is based around the following estimate from [13]:

$$\displaystyle \begin{aligned} \vert \varTheta_h^{-1} \vert_{l,\infty,\varTheta_h(T)} & \lesssim | \varTheta_h^{-1} |{}_{1,\infty,\varTheta_h(T)} \sum_{m=2}^l | \varTheta_h |{}_{m,\infty,T} \sum_{j \in \mathscr{I}(m,l)} \prod_{n=1}^{l-1} \vert \varTheta_h^{-1} \vert_{n,\infty,\varTheta_h(T)}^{j_n} \\ \text{ with } \quad \mathscr{I}(m,l) & := \{j = (j_1,..,j_{l-1}) \mid \sum_{n=1}^{l-1} j_n = m, \sum_{n=1}^{l-1} n j_n = l\}. \end{aligned} $$

Starting with $\vert \varTheta _h^{-1} \vert _{1,\infty ,\varTheta _h(T)} \lesssim 1$ which follows from Lemma 1 the claim follows by induction.

Proof of Lemma 6

We define $\hat {v} := v \circ \varTheta _h$ with $\hat {v}|{ }_T \in \mathscr {P}^k(T),~ T \in \mathscr {T}_h^{\varGamma ,+}$. There holds the following estimate due to a higher order chain rule for multivariate functions, cf. [13],

$$\displaystyle \begin{aligned} \vert \hat{v} \circ \varTheta_h^{-1} \vert_{l,\infty,\varTheta_h(T)} & \lesssim \sum_{m=1}^l | \hat{v} |{}_{m,\infty,T} \sum_{j \in \mathscr{J}(m,l)} \prod_{n=1}^l \vert \varTheta_h^{-1} \vert_{n,\infty,\varTheta_h(T)}^{j_n} \\ \text{ with } \quad \mathscr{J}(m,l) & := \{j = (j_1,..,j_l) \mid \sum_{n=1}^l j_n = m, \sum_{n=1}^l n j_n = l\}. \end{aligned} $$

There holds the finite element inverse inequality $\vert \hat {v} \vert _{j,\infty ,T} \lesssim h^{-j} \Vert \hat {v} \Vert _{\infty ,T},~j\geq 0.$ Now, with Lemma 2 we have $\vert \varTheta _h^{-1} \vert _{j,\infty ,\varTheta _h(T)} \lesssim 1$ and $\vert \varTheta _h \vert _{j,\infty ,T} \lesssim 1$ and $D^{k+1} \hat {v} = 0$ which completes the proof.

Proof of Lemma 7

We show (20a). The proof of estimate (20b) follows similar lines.

We mimic the proof of [33, Theorem 5.1], but need a few more technical steps due to F being curved and D ^k+1 v ≠ 0. First, we introduce simply connected domains B ₁ ⊂ T ₁, C ₂ ⊂ T ₂ and F ^∗⊂ F with diam(B ₁), diam(C ₂), diam(F ^∗) ≳ h. For such domains standard finite element estimates give (with $\hat {v} = v \circ \varTheta _h^{-1}$)

$$\displaystyle \begin{aligned}\Vert v \Vert_{T_1}^2 \simeq \Vert \hat{v} \Vert_{\hat{T}_1}^2 \simeq \Vert \hat{v} \Vert_{\varTheta_h^{-1}(B_1)}^2 \simeq \Vert v \Vert_{B_1}^2, \end{aligned}$$

and a similar result for C ₂. For a simply connected domain F ^∗⊂ F we define $T_i^\ast (F^\ast ) := \{ x \in T_i \mid x = x_F + \gamma n_F(x_F), x_F \in F, \gamma \in \mathbb {R} \}$. For x = x _F  + γn _F (x _F ) in $T_i^\ast$ we define the mirror point M(x) = x _F  − γn _F (x _F ). Now, for h sufficiently small we find domains F ^∗⊂ F, a ball $B_1 \subset T_1^\ast (F^\ast )$ and C ₂ := M(B ₁) = {x = M(y), y ∈ B ₁} which fulfil the aforementioned requirements.

To each point x ₁ = x _F  + γn _F (x _F ) in B ₁ we have a corresponding point x ₂ = M(x ₁) in C ₂. We develop $v_i:=v|{ }_{T_i}$ around x _F and obtain (for a ξ _i  = x _F  ± γ _ξ n _F  ∈ T _i , γ _ξ  ∈ [0, γ])

$$\displaystyle \begin{aligned}v_i(x_i) = v_i(x_F) + \sum_{l=1}^{k} \frac{\gamma^l}{l!} \frac{\partial^l v_i}{\partial n^l}(x_F) + \frac{\gamma_{\xi}^{k+1}}{(k+1)!} \frac{\partial^{k+1} v_i}{\partial n^{k+1}}(\xi). \end{aligned}$$

Subtracting and integrating over B ₁ then gives

$$\displaystyle \begin{aligned} \Vert v_1 \Vert_{B_1}^2 \lesssim \Vert v_2\circ M \Vert_{B_1}^2 + k \sum_{l=1}^k \frac{h^{2l+1}}{l!^2} \Vert [\hspace{-0.05cm}[ \partial_n^l v ]\hspace{-0.05cm}] \Vert_F^2 + 2 |B_1| \frac{h^{2k+2}}{(k+1)!^2} \Vert D^{k+1} v \Vert_{\infty,T_1 \cup T_2}^2. \end{aligned} $$

(30)

Exploiting the properties of M, and Lemma 6, we get

$$\displaystyle \begin{aligned} \Vert v \Vert_{T_1}^2 & \leq c \Vert v \Vert_{T_2}^2 + J_F(v,v) + c h^2 \left( \Vert v \Vert_{T_1}^2 + \Vert v \Vert_{T_2}^2 \right). \end{aligned} $$

Now, for h sufficiently small the last term can be absorbed by the others and the claim holds true.