Abstract
We consider the P1/P1 or P1b/P1 finite element approximations to the Stokes equations in a bounded smooth domain subject to the slip boundary condition. A penalty method is applied to address the essential boundary condition \(u\cdot n = g\) on \(\partial \Omega \), which avoids a variational crime and simultaneously facilitates the numerical implementation. We give \(O(h^{1/2} + \epsilon ^{1/2} + h/\epsilon ^{1/2})\)-error estimate for velocity and pressure in the energy norm, where h and \(\epsilon \) denote the discretization parameter and the penalty parameter, respectively. In the two-dimensional case, it is improved to \(O(h + \epsilon ^{1/2} + h^2/\epsilon ^{1/2})\) by applying reduced-order numerical integration to the penalty term. The theoretical results are confirmed by numerical experiments.
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Acknowledgments
The authors appreciate the suggestions from the anonymous referees that have improved the quality of this paper. They thank Professor Fumio Kikuchi, Professor Norikazu Saito, and Professor Masahisa Tabata for valuable comments to the results of this paper. They also thank Professor Xuefeng Liu for giving them a program which converts matrices described by the PETSc-format into those by the CRS-format.
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The first author was supported by JST, CREST. The second author was supported by JSPS KAKENHI Grant Numbers 24224004, 26800089. The third author was supported by JST, CREST and by JSPS KAKENHI Grant Number 23340023.
Appendices
Appendix 1: Transformation between \(\varGamma \) and \(\varGamma _h\) and related estimates
Let \(\Omega \) be a bounded domain in \(\mathbb R^N\). Its boundary \(\varGamma \) is assumed to be \(C^{1,1}\)-smooth, namely, there exist a system of local coordinates \(\{(U_r, y_r, \varphi _r)\}_{r=1}^M\) and positive numbers \(\alpha , \beta \) such that: 1) \(\{U_r\}_{r=1}^M\) forms an open covering of \(\varGamma \); 2) \(y_r = (y_{r1}, \dots , y_{rN-1}, y_{rN}) = (y_r', y_{rN})\) is a rotated coordinate of the original one x, that is, \(y_r = A_rx\) for some orthogonal transformation \(A_r\); 3) \(\varphi _r\in C^{1,1}(\varDelta _r)\) gives a graph representation of \(\varGamma \cap U_r\), where \(\varDelta _r := \{y_r' \in \mathbb R^{N-1} :\; |y_r'| < \alpha \}\), that is,
Since \(\varGamma \) is compact, there exists \(h_0>0\) such that for any \(x\in \varGamma \) the open ball \(B(x; h_0)\) is contained in some local coordinate neighborhood \(U_r\). According to the fact that \(C^{1,1}(\varDelta _r) = W^{2,\infty }(\varDelta _r)\), the derivatives of \(\varphi _r\) are bounded up to second order, i.e.,
where \(C_0,\,C_1,\,C_2\) are constants independent of r, and \(\nabla '\) means \(\nabla _{y'}\).
The smoothness of \(\varGamma \) is connected with that of the signed distance function d(x) defined by
We collect several known properties on d(x) below. For the details, see e.g. [13, Section 14.6] or [8, Section 7.8]. Let \(\varGamma (\delta ) := \{ x\in \mathbb R^N :\; |d(x)| < \delta \}\) be a tubular neighborhood of \(\varGamma \) with width \(2\delta \). Then there exists \(\delta \) depending only on the curvature of \(\varGamma \) such that for arbitrary \(x\in \varGamma (\delta )\) the decomposition
is uniquely determined. Here, n is the outer unit normal field defined on \(\varGamma \), which coincides with \(\nabla d|_{\varGamma }\). We extend n from \(\varGamma \) to \(\varGamma (\delta )\) by \(n(x) = \nabla d(x)\), which also agrees with \(n(\pi (x))\). The fact that \(\varGamma \) is \(C^{1,1}\)-smooth implies that \(d\in C^{1,1}(\varGamma (\delta ))\), \(n\in C^{0,1}(\varGamma (\delta ))\), and \(\pi \in C^{0,1}(\varGamma (\delta ))\). We call \(\pi : \varGamma (\delta )\rightarrow \varGamma \) the orthogonal projection onto \(\varGamma \), in view of its geometrical meaning. We may assume that
where we re-choose the constants \(C_0,\,C_1,\,C_2\) if necessary.
Now we introduce a regular family of triangulations \(\{\mathcal T_h\}_{h\downarrow 0}\) of \(\overline{\Omega }\) in the sense of Sect. 3.1. As before, we denote by \(\mathcal S_h\) the boundary mesh inherited from \(\mathcal T_h\), and we set \(\overline{\Omega }_h = \cup _{T\in \mathcal T_h}T\) and \(\varGamma _h = \cup _{S\in \mathcal S_h}S\). In order for \(\varGamma _h\) to be compatible with the local-coordinate system \(\{(U_r, y_r, \varphi _r)\}_{r=1}^M\), we assume the following:
-
(1)
the mesh size h is less than \(\min \{h_0, 1\}\);
-
(2)
for every \(r=1,\dots ,M\), \(\varGamma _h\cap U_r\) is represented by a graph \(\{(y_r', \varphi _{rh}(y_r'))\in \mathbb R^N :\; y_r'\in \varDelta _r \}\);
-
(3)
every vertex of \(S\in \mathcal S_h\) lies on \(\varGamma \).
From these we see that every \(S\in \mathcal S_h\) is contained in some \(U_r\) and that \(\varphi _{rh}\) is a piecewise linear interpolation of \(\varphi _r\). As a result of interpolation error estimates, for arbitrary r we obtain
where we re-choose \(C_0,\,C_1\) if necessary and the subscript E refers to “error”. In the following, h is made small enough to satisfy \(2C_{0E}h^2< \min \{h_0, \delta \}\), which in particular ensures that \(\pi \) is well-defined on \(\varGamma _h\). We may assume further that \(\pi (S)\) is contained in the same \(U_r\) that contains S.
Based on the observations above, we see that the orthogonal projection \(\pi \) maps \(\varGamma _h\) into \(\varGamma \). It indeed gives a homeomorphism between \(\varGamma _h\) and \(\varGamma \), and is element-wisely a diffeomorphism, as shown below. The representation of a function f(x) in each local coordinate \((U_r, y_r, \varphi _r)\) is defined as \(\tilde{f}(y_r) := f(A_r^{-1}y_r)\). However, with some abuse of notation, we denote it simply by \(f(y_r)\). Then, the local representation of the outer unit normal n associated to \(\varGamma \) is given by
where \(K_r(y_r') := \sqrt{1 + |\nabla '\varphi _r(y_r')|^2}\). We are ready to state the following.
Proposition 8.1
If \(h>0\) is sufficiently small, then \(\pi |_{\varGamma _h}\) is a homeomorphism between \(\varGamma _h\) and \(\varGamma \).
Proof
Let us construct an inverse map \(\pi ^*: \varGamma \rightarrow \varGamma _h\) of \(\pi |_{\varGamma _h}\) which is continuous. To this end, we fix arbitrary \(x\in \varGamma \) and choose a local coordinate \((U_r, y_r, \varphi _r)\) of x such that \(B(x; h_0)\subset U_r\). For simplicity, we omit the subscript r in the following. In view of the definition of \(\pi (x)\) by (8.1), we introduce a segment given by
To each point on this segment we associate its height H(t) with respect to the graph of \(\varphi _h\), that is,
Then we assert that \(\frac{d}{dt}H(t) < 0\) and that \(H(-2C_{0E}h^2) > 0\), \(H(2C_{0E}h^2) < 0\).
To prove the first assertion, letting
we have \(\frac{d}{dt}H(t) = -\frac{1}{K(y')}(1 + \nabla '\varphi (y')\cdot \nabla '\varphi _h(Y'))\). One sees that
where \(I_1\) and \(I_2\) satisfy
Then it follows that
From this we have \(\frac{d}{dt}H(t) < 0\) provided \(C_1(2C_2C_{0E}h^2 + C_{1E}h) < 1/2\). For the second assertion, one finds that
where \(Y'\) is \(y' - \frac{2C_{0E}h^2}{K(y')} \nabla '\varphi (y')\). By Taylor’s theorem, there exists some \(\theta \in (0,1)\) such that
By the definition of K(y) and by \(|\varphi (Y') - \varphi _h(Y')| \le C_{0E}h^2\), we obtain
which implies that \(H(-2C_{0E}h^2) > 0\) provided \(C_2C_{0E}h^2 \le 1/4\). In the same way, the last assertion \(H(2C_{0E}h^2) < 0\) can be proved.
From these assertions we deduce that there exists a unique \(t^*(x) \in [-2C_{0E}h^2, 2C_{0E}h^2]\) such that \(H(t^*(x)) = 0\). Consequently, the map \(\pi ^*: \varGamma \rightarrow \varGamma _h; \; x\mapsto x + t^*(x)n(x)\) is well-defined. A direct computation combined with the uniqueness of the decomposition (8.1) shows that \(\pi ^*\) is the inverse of \(\pi |_{\varGamma _h}\). The continuity of \(\pi ^*\), especially that of \(t^*\), follows from an argument similar to the proof of the implicit function theorem (see e.g. [20, Theorem 3.2.1]). \(\square \)
Remark 8.1
Proposition 8.1 enables us to define an exact triangulation of \(\varGamma \) by \(\pi (\mathcal S_h) = \{ \pi (S) \,:\, S\in \mathcal S_h \}\). In particular, we can subdivide \(\varGamma \) into disjoint sets as \(\varGamma = \bigcup _{S\in \mathcal S_h}\pi (S)\). Under assumption (3) this fact is obvious to see for the two-dimensional case, but not for the three-dimensional case. Furthermore, the above construction of an exact triangulation even applies to the situation where the vertices of \(\varGamma _h\) do not exactly belong to \(\varGamma \) but still lie closely to it so that (8.2) is fulfilled. This may be useful information and is left as a side remark.
Next we observe that S and \(\pi (S)\) admit the same domain of parametrization for each \(S\in \mathcal S_h\), which is important in the subsequent analysis. To describe this fact, we choose a local coordinate \((U_r, y_r, \varphi _r)\) such that \(U_r \supset S\cup \pi (S)\), and introduce the projection to the base set \(b_r:\mathbb R^N\rightarrow \mathbb R^{N-1}\) by \(b_r(y_r) = y_r'\). The domain of parametrization is then defined to be \(S' = b_r(\pi (S))\). We see that the mappings
are bijective and that \(\varPhi \) is smooth on \(S'\). If in addition \(\varPhi _h\), especially \(t^*\), is also smooth on \(S'\), then \(\varPhi \) and \(\varPhi _h\) may be employed as smooth parametrizations for \(\pi (S)\) and S respectively. The next proposition verifies that this is indeed the case.
Proposition 8.2
Under the setting above, we have
where \(\tilde{C}_{0E}\) and \(\tilde{C}_{1E}\) are constants depending only on N and \(\varGamma \).
Proof
Since the first relation is already obtained in Proposition 8.1 with \(\tilde{C}_{0E} = 2C_{0E}\), we focus on proving the second one. For notational simplicity, we omit the subscript r and also use the abbreviation \(\partial _i = \frac{\partial }{\partial y_i}\,(i=1,\dots ,N)\). The fact that \(t^*\) is differentiable with respect to \(y'\) can be shown in a way similar to the proof of the implicit function theorem. Thereby it remains to evaluate the supremum norm of \(\nabla ' t\) in \(S'\), which we address in the following.
Recall that \(t^*(y')\) is determined according to the equation
where we have set \(\tilde{t}(y') := t^*(y')/K(y')\). Because \(\nabla 't^* = K\nabla '\tilde{t} + \frac{\nabla '^2\varphi \nabla '\varphi }{K^2}\, t^*\), it follows that
Therefore, it suffices to prove that \(\Vert \nabla '\tilde{t}\Vert _{L^\infty (S')} \le Ch\); here and hereafter C denotes various constants which depends only on N and \(\varGamma \).
Applying \(\nabla '\) to (8.3) gives
where \(Y' := y' + \tilde{t}(y') \nabla '\varphi (y')\). By the same way as we estimated \(I_1\) and \(I_2\) in the proof of Proposition 8.1, we obtain
Also we see that
Combining these observations with (8.4), we deduce the desired estimate \(\Vert \nabla '\tilde{t}\Vert _{L^\infty (S')} \le Ch\). \(\square \)
Remark 8.2
When \(\varGamma \in C^{2,1}\), further differentiation of (8.4) gives us \(t^*\in C^{1,1}(\pi (S))\); in fact we have \(\Vert \nabla '^2 t^*\Vert _{L^\infty (S')} \le \tilde{C}_{2E}\) with some constant \(\tilde{C}_{2E}\). This implies that \(\pi |_{S}\) is a \(C^{1,1}\)-diffeomorphism between S and \(\pi (S)\). However, since \(\nabla '\phi _h\) is smooth only within b(S), \(\pi \) is not globally a diffeomorphism.
Now we give an error estimate for surface integrals on \(\varGamma \) and \(\varGamma _h\). Heuristically speaking, the result reads \(|d\gamma - d\gamma _h| \le O(h^2)\), which may be found in the literature (see e.g. [11]). Here and hereafter, we denote the surface elements of \(\varGamma \) and \(\varGamma _h\) by \(d\gamma \) and \(d\gamma _h\), respectively.
Theorem 8.1
Let \(S\in \mathcal S_h\) and f be an integrable function on S. Then we have
where C is a constant depending only on N and \(\varGamma \).
Proof
Let \((U_r, y_r, \varphi _r)\) be a local coordinate that contains \(S\cup \pi (S)\). We omit the subscript r and use the abbreviation \(\partial _i = \frac{\partial }{\partial y_i}\,(i=1,\dots ,N)\). We represent the surface integral using the parametrization \(\varPhi \) as follows:
where \(G=(G_{ij})_{1\le i,j\le N-1}\) denotes the Riemannian metric tensor given by \(G_{ij} = \partial _i\varPhi \cdot \partial _j\varPhi \) (dot means the inner product in \(\mathbb R^N\)). Similarly, noting that \(\pi \circ \varPhi _h = \varPhi \), one obtains
where \(G_h\) is given by \(G_{h,ij} = \partial _i\varPhi _h \cdot \partial _j\varPhi _h\). Then we assert that:
To prove this, noting that \(\varPhi _h = \varPhi + t^*n\circ \varPhi \), we compute each component of \(G_h-G\) as follows:
For \(I_1\), we notice that \(\partial _i\varPhi \) is a tangent vector so that \(\partial _i\varPhi \cdot n\circ \varPhi =0\). This yields
which is estimated by \(\tilde{C}_{0E}h^2(1 + C_1)(C_2 + C_1C_2)\) thanks to Proposition 8.2. \(I_2\) can be bounded in the same manner. To estimate \(I_3\), we observe that
which is bounded by \(\tilde{C}_{1E}h + \tilde{C}_{0E}h^2(C_2 + C_1C_2) \le Ch\). Similarly one gets \(|\partial _j(t^*n\circ \varPhi )| \le Ch\), hence it follows that \(|I_3| \le Ch^2\). Therefore, \(|G_{h,ij} - G_{ij}| \le Ch^2\), which proves the assertion.
Now we use the following crude estimate for perturbation of determinants (cf. [17, equation (3.13)]): if A and B are \(N\times N\) matrices such that \(|A_{ij}| \le a\) and \(|B_{ij}|\le b\) for all i, j, then
Combining this with the assertion above and also with \(\sqrt{\alpha }- \sqrt{\beta }= (\alpha - \beta )/(\sqrt{\alpha }+ \sqrt{\beta })\), we obtain
In addition, note that \(\sqrt{\mathrm {det}\,G} = \sqrt{1 + |\nabla '\varphi |^2} \ge 1\). Consequently,
which proves the theorem. \(\square \)
Remark 8.3
Adding up the results of the theorem for all \(S\in \mathcal S_h\) yields
It also follows that \(|\int _{\varGamma _h} f\circ \pi \, d\gamma _h| \le C\int _\varGamma |f|\,d\gamma \). Choosing in particular \(|f|^p\) as the integrand gives \(\Vert f\circ \pi \Vert _{L^p(\varGamma _h)} \le C^{1/p}\Vert f\Vert _{L^p(\varGamma )}\) for \(p\in [1,\infty ]\).
Let f be a smooth function given on \(\varGamma \). Then its transformation to \(\varGamma _h\) is defined by \(f\circ \pi \). However, if f is extended to a neighborhood of \(\varGamma \), e.g. to \(\varGamma (\delta )\), then we may also consider f’s natural trace on \(\varGamma _h\). The next theorem provides error estimation of these two quantities.
Theorem 8.2
Let \(f\in W^{1,p}(\varGamma (\delta _1))\), where \(p\in [1,\infty ]\) and \(\delta _1 \in [\tilde{C}_{0E}h^2, 2\tilde{C}_{0E}h^2]\). Then,
where C is a constant depending only on \(p,\,N,\,\Omega \).
Proof
Since \(\varGamma (\delta _1) = \bigcup _{S\in \mathcal S_h} \pi (S,\delta _1)\), where \(\pi (S, \delta _1) = \{ x\in \varGamma (\delta _1) : \pi (x)\in \pi (S) \}\) denotes a tubular neighborhood of S, it suffices to prove that
To this end, using the notation in Theorem 8.1, we estimate the left-hand side of (8.5) by
Here, for fixed \(y'\in S'\) we have
Because \(|t^*(y')| \le \tilde{C}_{0E}h^2 \le \delta _1\), it follows that
where we have used Hölder’s inequality. Consequently,
On the other hand, we observe that the N-dimensional transformation
is bijective and smooth. Application of this transformation to the right-hand side of (8.5) leads to
where \(J = (\partial _1\varPhi + t\partial _1(n\circ \varPhi ), \cdots , \partial _{N-1}\varPhi + t\partial _{N-1}(n\circ \varPhi ), n\circ \varPhi )\) denotes the Jacobi matrix of \(\Psi \). Letting \(\tilde{J} := (\partial _1\varPhi , \cdots , \partial _{N-1}\varPhi , n\circ \varPhi )\), we find that
because \(|t\partial _i(n\circ \varPhi )| = |t (\partial _in + \partial _i\varphi \partial _Nn)_{|\varPhi }| \le \delta _1(C_2 + C_1C_2)\). This implies
which combined with \(\mathrm {det}\,\tilde{J} = K(y') \ge 1\) yields \(\mathrm {det}\,J \ge 1/2\) if h is sufficiently small. Therefore,
The desired estimate (8.5) is now a consequence of (8.6) and (8.8). This completes the proof. \(\square \)
Finally, we show that the \(L^p\)-norm in a tubular neighborhood can be bounded in terms of its width. Such estimate is stated e.g. in [35, Lemma 2.1] or in [28, equation (3.6)]. However, since we could not find a full proof of this fact (especially for \(N=3\)) in the literature, we present it here.
Theorem 8.3
Under the same assumptions as in Theorem 8.2, we have
where C is a constant depending only on \(p,\,N,\,\Omega \).
Proof
We adopt the same notation as in the proofs of Theorems 8.1 and 8.2. Then it suffices to prove that
To this end, using the transformation \(\Psi \) given in (8.7) we express the left-hand side as
For \(I_1\), we see from the same argument as before that
which yields
For \(I_2\), it follows that
We have thus obtained the desired estimate, which completes the proof. \(\square \)
Appendix 2: Error of n and \(n_h\)
Let us prove that \(|n\circ \pi - n_h| \le O(h)\) on \(\varGamma _h\) and also that, when \(N=2\), it is improved to \(O(h^2)\) if the consideration is restricted to the midpoint of edges.
Lemma 9.1
Let n and \(n_h\) be the outer unit normals to \(\varGamma \) and \(\varGamma _h\) respectively. Then there holds
If in addition \(N=2\), \(\varGamma \in C^{2,1}\), and \(m_S\) denotes the midpoint of \(S\in \mathcal S_h\), then
Here, C is a constant depending only on N and \(\varGamma \).
Proof
Let \(S\in \mathcal S_h\) be arbitrary and let \((U_r, y_r, \phi _r)\) be a local coordinate that contains \(S\cup \pi (S)\). We omit the subscript r in the following. One sees that n and \(n_h\) are represented as
A direct computation gives
This combined with the observation that
proves (9.1). When \(N=2\) and \(\varGamma \in C^{2,1}\), by using Taylor expansion, we find that \(m_S\) is a point of super-convergence such that \(|\frac{d\varphi _h}{dy_1} (b(m_S)) - \frac{d\varphi }{dy_1} (b(m_S))| \le Ch^2\), which improves (9.3) to \(O(h^2)\). Thus (9.2) is proved. \(\square \)
Remark 9.1
In the case \(N=3\), it is known that the barycenter of a triangle is not a point of super-convergence for the derivative of linear interpolations; see [34, p. 1930]. For this reason, (9.2) holds only for \(N=2\).
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Kashiwabara, T., Oikawa, I. & Zhou, G. Penalty method with P1/P1 finite element approximation for the Stokes equations under the slip boundary condition. Numer. Math. 134, 705–740 (2016). https://doi.org/10.1007/s00211-016-0790-5
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DOI: https://doi.org/10.1007/s00211-016-0790-5