Abstract
We develop a stabilized cut finite element method for the convection problem on a surface based on continuous piecewise linear approximation and gradient jump stabilization terms. The discrete piecewise linear surface cuts through a background mesh consisting of tetrahedra in an arbitrary way and the finite element space consists of piecewise linear continuous functions defined on the background mesh. The variational form involves integrals on the surface and the gradient jump stabilization term is defined on the full faces of the tetrahedra. The stabilization term serves two purposes: first the method is stabilized and secondly the resulting linear system of equations is algebraically stable. We establish stability results that are analogous to the standard meshed flat case and prove \(h^{3/2}\) order convergence in the natural norm associated with the method and that the full gradient enjoys \(h^{3/4}\) order of convergence in \(L^2\). We also show that the condition number of the stiffness matrix is bounded by \(h^{-2}\). Finally, our results are verified by numerical examples.
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1 Introduction
In this contribution we develop a stabilized cut finite element for stationary convection on a surface embedded in \(\mathbb {R}^3\). The method is based on a three dimensional background mesh consisting of tetrahedra and a piecewise linear approximation of the surface. The finite element space is the continuous piecewise linear functions on the background mesh and the bilinear form defining the method only involves integrals on the surface. In addition we add a consistent stabilization term which involves the normal gradient jump on the full faces of the background mesh. In the case of the Laplace–Beltrami operator the idea of using the restriction of a finite element space to the surface was developed in [23], and a stabilized version was proposed and analyzed in [6].
We show that for the convection problem the properties of cut finite element method completely reflects the properties of the corresponding method on standard triangles or tetrahedra, see the analysis for the latter in [3]. In particular, we prove discrete stability estimates in the natural energy norm, involving the \(L^2\) norm of the solution and \(h^{1/2}\) times the \(L^2\) norm of the streamline derivative where h is the meshsize, and corresponding optimal a priori error estimates of order \(h^{3/2}\). Furthermore, we also show an error estimate of order \(h^{3/4}\) for the error in the full gradient which is also in line with [3]. The stabilization term is key to the proof of the discrete stability estimates and enables us to work in the natural norms corresponding to those used in the standard analysis on triangles or tetrahedra. The analysis utilizes a covering argument first developed in [6], which essentially localizes the analysis to sets of elements, with a uniformly bounded number of elements, that together has properties similar to standard finite elements. The stabilization term also leads to an algebraically stable linear system of equations and we prove that the condition number is bounded by \(h^{-2}\).
We note that similar stabilization terms have recently been used for stabilization of cut finite element methods for time dependent problems in [20], bulk domain problems involving standard boundary and interface conditions [4, 5, 18, 21], and for coupled bulk–surface problems involving the Laplace–Beltrami operator on the surface in [8]. We also mention [7] where a discontinuous cut finite element method for the Laplace–Beltrami operator was developed. None of these references consider the convection problem on the surface. For convection problems streamline diffusion stabilization was used in [9, 25]. Methods on evolving surfaces were studied in [20, 22, 24].
An advantage of the proposed stabilization method is that it is straightforward to extend the method to a time dependent problem on a stationary surface. Indeed any A-stable finite difference discretization of the time derivative leads to a stable scheme with the accuracy of the truncation error [2]. Runge–Kutta methods of second and third orders are also stable and accurate and also explicit up to the inversion of the mass matrix [1]. In the explicit case the mass matrix is stabilized using a scaled version of the normal gradient jump term. For time-dependent domains on the other hand it may be more convenient to use the aforementioned space–time finite elements for a consistent tracking of the surface displacement [20, 22] or a combination with the characteristic approach developed in [19].
Finally, we refer to [10, 13,14,15] for general background on finite element methods for partial differential equations on surfaces.
The outline of the remainder of this paper is as follows: In Sect. 2 we formulate the model problem; in Sect. 3 we define the discrete surface, its approximation properties, and the finite element method; in Sect. 4 we summarize some preliminary results involving lifting of functions from the discrete surface to the continuous surface; in Sect. 5 we first derive some technical lemmas essentially quantifying the stability induced by the stabilization term, and then we derive the key discrete stability estimate; in Sect. 6 we prove a priori estimates; in Sect. 7 we prove an estimate of the condition number; and finally in Sect. 8, we present some numerical examples illustrating the theoretical results.
2 The convection problem on a surface
2.1 The surface
Let \(\Gamma \) be a smooth surface embedded in \(\mathbb {R}^3\) with signed distance function \(\rho \) such that the exterior unit normal to the surface is given by \(n = \nabla \rho \). We let \(p:\mathbb {R}^3 \rightarrow \Gamma \) be the closest point mapping. Then there is a \(\delta _0>0\) such that p maps each point in \(U_{\delta _0}(\Gamma )\) to precisely one point on \(\Gamma \), where \(U_\delta (\Gamma ) = \{ x \in \mathbb {R}^3 : | \rho (x)| < \delta \}\) is the open tubular neighborhood of \(\Gamma \) of thickness \(\delta \).
2.2 Tangential calculus
For each function u on \(\Gamma \) we let the extension \(u^e\) to the neighborhood \(U_{\delta _0}(\Gamma )\) be defined by the pull back \(u^e = u \circ p\). For a function \(u: \Gamma \rightarrow \mathbb {R}\) we then define the tangential gradient
where \({P}_\Gamma = I - n \otimes n\), with \(n = n(x)\), \(x \in \Gamma \), in the projection onto the tangent plane \(T_{x}(\Gamma )\). We also define the surface divergence
where \((u^e \otimes \nabla )_{ij} = \partial _j u^e_i\). It can be shown that the tangential derivative does not depend on the particular choice of extension.
2.3 The convection problem on \(\varvec{\Gamma }\)
The strong form of the convection problem on \(\Gamma \) takes the form: find \(u:\Gamma \rightarrow \mathbb {R}\) such that
where \(\beta :\Gamma \rightarrow \mathbb {R}^3\) is a given tangential vector field, \(\alpha :\Gamma \rightarrow \mathbb {R}\) and \(f:\Gamma \rightarrow \mathbb {R}\) are given functions.
Assumption
The coefficients \(\alpha \in C(\Gamma )\) and \(\beta \in C^1(\Gamma )\) satisfy
for a positive constant C.
We introduce the Hilbert space \(V = \{ v:\Gamma \rightarrow \mathbb {R}: \Vert v \Vert ^2_V = \Vert v\Vert _\Gamma ^2 + \Vert \beta \cdot \nabla v\Vert ^2_\Gamma < \infty \}\) and the operator \(L: V\ni v \mapsto \beta \cdot \nabla _\Gamma v + \alpha v \in L^2(\Gamma )\). We note that using Green’s formula and assumption (2.4) we have the estimate
Proposition 2.1
If the coefficients \(\alpha \) and \(\beta \) satisfy assumption (2.4), then there is a unique \(u \in V\) such that \(L u = f\) for each \(f\in L^2(\Gamma )\).
Proof
The essential idea in the proof is to consider the corresponding time dependent problem with a smooth right hand side and show that the solution exists and converges to a solution to the stationary problem as time tends to infinity. Then we use a density argument to handle a right hand side in \(L^2\).
Smooth right hand side For any \(0< T <\infty \) consider the time dependent problem: find \(u:[0,T]\times \Gamma \rightarrow \mathbb {R}\), such that
Consider first a smooth right hand side g, which does not depend on time. Using characteristic coordinates we conclude that there is smooth solution u(t) to (2.6). Next taking the time derivative of Eq. (2.6) we find that the solution satisfies the equation
where we used the fact that \(\alpha \), \(\beta \), and g, do not depend on time. Multiplying (2.7) by \(u_t\) and integrating over \(\Gamma \) we get
Using (2.4) we obtain
which implies
Integrating over \([\epsilon ,T]\), \(0< \epsilon <T\), we get
Letting \(\epsilon \rightarrow 0^+\) and using the smoothness of u we find, using the Eq. (2.6), that \(u_t(\epsilon ) = g - \beta \cdot \nabla _\Gamma u(\epsilon ) - \alpha u(\epsilon ) \rightarrow g - \beta \cdot \nabla _\Gamma u(0) - \alpha u(0) = g\) since \(u(0)=0\) and therefore also \(\nabla _\Gamma u(0) = 0\). We thus conclude that
Using (2.12) we have
for \(0\le T_ 1\le T_2< \infty \). Using the time dependent Eq. (2.6) we have
and therefore, using the fact that \(\Vert \alpha \Vert _{L^\infty (\Gamma )} \lesssim 1\), we have the estimate
where we used (2.12) and (2.14) in the last step. Together, (2.14) and (2.17) leads to the estimate
Thus we conclude that for each \(\epsilon > 0 \) there is \(T_\epsilon \) such that \(\Vert u(T_1) - u(T_2) \Vert _V \le \epsilon \) for all \(T_1,T_2> T_\epsilon \). We can then pick a sequence \(u_n = u(T_n)\) with \(T_n=n, n=1,2,3,\dots \) and conclude from (2.18) that the sequence is Cauchy in V and therefore it converges to a limit \(u_{g} \in V\). We then have
and thus the limit \(u_g\) is a solution to the stationary problem in the sense of \(L^2\) and from (2.18) with \(T_1=0\), we have the stability estimate
Right hand side in\({{L}}^\mathbf{2}{({\Gamma })}\) For \(f \in L^2(\Gamma )\) we pick a sequence of smooth functions \(f_n\) that converges to f in \(L^2(\Gamma )\). Then for each \(f_n\) there is a solution \(u_n \in V\) to \(L u_n = f_n\) and we note that \(L(u_n - u_m) = f_n - f_m\) and therefore it follows from (2.20) that
and thus \(\{u_n\}\) is a Cauchy sequence since \(\{f_n\}\) is a Cauchy sequence. Denoting the limit of \(u_n\) by u we have
which tends to zero as n tends to infinity and thus \(u \in V\) is a solution to \(L u = f\) in the sense of \(L^2\). \(\square \)
3 The finite element method
3.1 The discrete surface
Let \(\Omega _0\) be a polygonal domain that contains \(U_{\delta _0}(\Gamma )\) and let \(\{\mathcal {T}_{0,h}, h \in (0,h_0]\}\) be a family of quasiuniform partitions of \(\Omega _0\) into shape regular tetrahedra with mesh parameter h. Let \(\Gamma _h\subset \Omega _0\) be a connected surface such that \(\Gamma _h \cap T\) is a subset of some hyperplane for each \(T\in \mathcal {T}_{0,h}\) and let \(n_h\) be the piecewise constant unit normal to \(\Gamma _h\).
Geometric approximation property The family \(\{\Gamma _h: h \in (0,h_0]\}\) approximates \(\Gamma \) in the following sense:
-
\(\Gamma _h \subset U_{\delta _0}(\Gamma )\), \(\forall h \in (0,h_0]\), and the closest point mapping \(p:\Gamma _h \rightarrow \Gamma \) is a bijection.
-
The following estimates hold
$$\begin{aligned} \Vert \rho \Vert _{L^\infty (\Gamma _h)} \lesssim h^2, \qquad \Vert n - n_h \Vert _{L^\infty (\Gamma _h)} \lesssim h \end{aligned}$$(3.1)
We introduce the following notation for the geometric entities involved in the mesh
We also use the notation \(\omega ^l = \{p(x)\in \Gamma : x \in \omega \subset \Gamma _h \}\), in particular, \(\mathcal {K}_h^l = \{K^l : K\in \mathcal {K}_h^l\}\) is a partition of \(\Gamma \).
Remark 3.1
The assumption that \(\mathcal {T}_h\) is quasiuniform can be relaxed to locally quasiuniform meshes since all our arguments are local in the sense that elementwise or patchwise, with patches consisting of a uniformly bounded number of elements, estimates are used.
3.2 The finite element method
We let \(V_h\) be the space of continuous piecewise linear functions defined on \(\mathcal {T}_h\). The finite element method takes the form: find \(u_h \in V_h\) such that
Here the forms are defined by
and
where \(\nabla _{\Gamma _h}v = {P}_{\Gamma _h}\nabla v = (I - n_h\otimes n_h) \nabla v\) is the elementwise defined tangent gradient on \({\Gamma _h}\), \(c_F\) is a positive stabilization parameter, \(\alpha _h\) and \(\beta _h\) are discrete approximations of \(\alpha \) and \(\beta \). The jump at a face F shared by two elements \(T^+\) and \(T^-\) is defined by
where \(n_F^{\pm }\) is the exterior unit normal of the face F and element \(T^\pm \) and \(v^\pm = v|_{T^\pm }\).
In our forthcoming analysis we will need certain properties of the coefficients \(\alpha _h\), \(\beta _h\), and the right hand side \(f_h\), in Sect. 5.2 we formulate the assumptions necessary for the stability analysis and in Sect. 6.1 we formulate the assumptions necessary for the a priori error estimates, and finally in Sect. 6.4 we provide a construction of discrete coefficients that satisfy all the assumptions.
4 Preliminary results
4.1 Norms
We let \(\Vert v \Vert _\omega \) denote the \(L^2\) norm over the set \(\omega \) equipped with the appropriate Lebesgue measure. Furthermore, we introduce the scalar products
with corresponding \(L^2\) norms denoted by \(\Vert \cdot \Vert _{\mathcal {T}_h}, \Vert \cdot \Vert _{\mathcal {K}_h}, \Vert \cdot \Vert _{\mathcal {F}_h},\) and \(\Vert \cdot \Vert _{\mathcal {E}_h}\). Note that \(\Vert \cdot \Vert _{\mathcal {K}_h} = \Vert \cdot \Vert _{{\Gamma _h}}\) and that the following scaling relations hold
Finally, we introduce the energy type norms
4.2 Inverse estimates
Let \(T \in \mathcal {T}_h\), \(K=\Gamma _h \cap T\), \(E \in \mathcal {E}_h\) and \(E\subset \partial K\), then the following inverse estimates hold
with constants independent of the position of the intersection of \({\Gamma _h}\) and T. Note that the second inequality is the standard element to face inverse inequality. Here \(V(F) = \widehat{V}\circ X_F^{-1}\) (\(W(T) = \widehat{W}\circ X_T^{-1}\)), where \(\widehat{V}\) (\(\widehat{W}\)) is a finite dimensional space on the reference triangle \(\widehat{F}\) (reference tetrahedron \(\widehat{T}\)) and \(X_F:\widehat{F}\rightarrow F\) (\(X_T:\widehat{T}\rightarrow T\)) an affine bijection.
4.3 Extension and lifting of functions
In this section we summarize basic results concerning extension and liftings of functions. We refer to [6, 11] for further details.
Extension Recalling the definition of the extension and using the chain rule we obtain the identity
where
and \(\mathcal {H}= \nabla \otimes \nabla \rho \). Here \(\mathcal {H}\) is a \(\Gamma \)-tangential tensor, which equals the curvature tensor on \(\Gamma \), and for small enough \(\delta >0\), there is a constant such that
Furthermore, \(B:T_{x}(K) \rightarrow T_{p(x)} (\Gamma )\) is invertible for \(h \in (0,h_0]\) with \(h_0\) small enough, i.e. there is \(B^{-1}: T_{p(x)} (\Gamma ) \rightarrow T_{x}(K)\) such that
See [17] for further details.
Lifting The lifting \(w^l\) of a function w defined on \(\Gamma _h\) to \(\Gamma \) is defined as the push forward
and we have the identity
Estimates related to B Using the uniform bound \(\Vert \mathcal {H}\Vert _{L^\infty (U_\delta (\Gamma ))}\lesssim 1\), for \(\delta >0\) small enough, it follows that
Next consider the surface measure \(d \Gamma = |B| d {\Gamma _h}\), where |B| is the absolute value of the determinant of \([B \xi _1 \, B \xi _2\, n^e]\) and \(\{\xi _1,\xi _2\}\) is an orthonormal basis in \(T_x(K)\). We have the following estimates
In view of these bounds and the identities (4.10) and (4.15) we obtain the following equivalences
and
4.4 Interpolation
Let \(\pi _h:L^2(\mathcal {T}_h) \rightarrow V_h\) be the Clément interpolant. Then we have the following standard estimate
where \(\mathcal {N}(T)\subset \mathcal {T}_h\) is the set of neighboring elements of T. In particular, we have the \(L^2\) stability estimate
and as a consequence \(\pi _h:L^2(\mathcal {T}_h) \rightarrow V_h\) is uniformly bounded and we have the estimate
Using the trace inequality
where the constant is independent of the position of the intersection between \({\Gamma _h}\) and T, see [16] for a proof, the interpolation inequality (4.20), and finally the stability of the extension operator
with \(\delta \sim h\), we obtain the interpolation error estimate
Using (4.25) and the definition of the energy norm (4.5) we obtain
and using a standard trace inequality on tetrahedra, the interpolation estimate (4.20), and the stability (4.24) of the extension, we have
Combining these two estimates we get
5 Stability estimates
5.1 Coverings
In this section we begin by recalling a construction of coverings of \(\mathcal {T}_h\) developed in [6], Sect. 4.1. The number of tetrahedra in the covering sets are uniformly bounded and the area of their intersection with \(\Gamma _h\) is equivalent to \(h^2\). See also [12] for related results. Then we formulate two useful lemmas.
Families of coverings of \(\mathcal {T}_h\) Let x be a point on \(\Gamma \) and let \(B_\delta (x) = \{y\in \mathbb {R}^3 : |y-x|<\delta \}\) and \(D_\delta (x) = B_\delta (x)\cap \Gamma \). We define the sets of elements
With \(\delta \sim h\) we use the notation \(\mathcal {K}_{h,x}\) and \(\mathcal {T}_{h,x}\). For each \(\mathcal {T}_h\), \(h \in (0,h_0]\) there is a set of points \(\mathcal {X}_h\) on \(\Gamma \) such that \(\{\mathcal {K}_{h,x}, x \in \mathcal {X}_h\}\) and \(\{\mathcal {T}_{h,x}, x \in \mathcal {X}_h\}\) are coverings of \(\mathcal {T}_h\) and \(\mathcal {K}_h\) with the following properties:
-
The number of sets containing a given point y is uniformly bounded
$$\begin{aligned} \#\{x\in \mathcal {X}_h:y\in \mathcal {T}_{h,x}\}\lesssim 1\quad \forall y\in \mathbb {R}^3 \end{aligned}$$(5.2)for all \(h\in (0,h_0]\) with \(h_0\) small enough.
-
The number of elements in the sets \(\mathcal {T}_{h,x}\) is uniformly bounded
$$\begin{aligned} \# \mathcal {T}_{h,x} \lesssim 1\quad \forall x \in \mathcal {X}_h \end{aligned}$$(5.3)for all \(h\in (0,h_0]\) with \(h_0\) small enough, and each element in \(\mathcal {T}_{h,x}\) share at least one face with another element in \(\mathcal {T}_{h,x}\).
-
\(\forall h \in (0,h_0]\) and \(\forall x \in \mathcal {X}_h\), \(\exists T_x \in \mathcal {T}_{h,x}\) that has a large intersection with \({\Gamma _h}\) in the sense that
$$\begin{aligned} |T_x \cap {\Gamma _h}| = |K_x| \sim h^2\quad \forall x \in \mathcal {X}_h, \end{aligned}$$(5.4)for all \(h\in (0,h_0]\) with \(h_0\) small enough.
We first recall a Lemma from [6] and then we prove a lemma tailored to the particular demands of this paper.
Lemma 5.1
It holds
for all \(h\in (0,h_0]\) with \(h_0\) small enough.
Proof
See Lemma 4.5 in [6]. \(\square \)
Lemma 5.2
It holds
for all \(h\in (0,h_0]\) with \(h_0\) small enough.
Proof
Consider an arbitrary set in the covering described above. Then we shall prove that we have the estimate
where \(\mathcal {F}_{h,x}\) is the set of interior faces in \(\mathcal {T}_{h,x}\). Let \(v_x : \mathcal {T}_{h,x} \rightarrow \mathbb {R}\) be the first order polynomial that satisfies \(v_x = v|_{T_x}\), where \(T_x\) is the element with a large intersection \(K_x\). Adding and subtracting \(v_x\) we get
Term I We have
where we used the inverse estimates (4.7) and (4.8) to pass from \(\mathcal {E}_h\) to \(\mathcal {T}_h\), the inequality
with \(w=v-v_x=0\) on \(T_{x}\), and finally the fact that \([n_F \cdot \nabla v_x] = 0\).
Verification of (5.10) Considering a pair of elements \(T_1,T_2\in \mathcal {T}_{h,x}\) that share a face F, with center of gravity \(x_F\), we have the identity
where w is a continuous piecewise linear polynomial on \(T_1 \cup T_2\) and \(w_i\) the linear polynomial on \(T_1 \cup T_2\) such that \(w_i|_{T_i} = w|_{T_i}\), \(i=1,2.\) Integrating over \(T_2\) gives
To estimate \(I_1\) we used the inverse inequality
which we may prove by letting \(G_1:\mathbb {R}^3 \rightarrow \mathbb {R}^3\) be the affine mapping which maps the reference tetrahedron \(\widehat{T}\) onto \(T_1\). We note using shape regularity that there is a ball \(B_R(\widehat{x}_{\widehat{T}})\) of radius \(R\lesssim 1\) centered at the center of gravity \(\widehat{x}_{\widehat{T}}\) of \(\widehat{T}\) such that \(G^{-1}_1 (T_2) \subset B_R(\widehat{x}_{\widehat{T}})\). Changing domain of integration and using an inverse bound on the reference configuration we obtain
where \(|DG_1|\) is the absolute value of the determinant of the derivative of \(G_1\) which is constant since \(G_1\) is affine.
To estimate \(I_2\) we used the facts that \([n_F \cdot \nabla v]\) is constant on F and that \( z = (x - x_F)\cdot n_F\) is the signed distance from x to F which satisfies \(|z| \le Ch\). We then have
Finally, iterating (5.13) and using the fact that the number of elements in \(\mathcal {T}_{h,x}\) is uniformly bounded (5.10) follows.
Term II We first split \(v_x\) into one term \(v_{x,c}\) which is constant in the direction normal to \(K_x\) and a reminder term \(v_x - v_{x,c} = t n_x \cdot \nabla v_x\) where \(n_x = n_h|_{K_x}\) and t is the signed distance to the hyperplane in which \(K_x\) is contained, as follows
Using the triangle inequality we obtain
Term \(II_1\) We have
where we used the inverse estimates (4.7) and (4.8) to pass from \(\mathcal {E}_h\) to \(\mathcal {T}_h\), an inverse estimate using the fact that \(v_{x,c}\) is a polynomial on \(\mathcal {T}_{h,x}\), finally an inverse inequality which holds since \(v_{x,c}\) is constant in the normal direction.
Term \(II_2\) We have
where we used Hölder’s inequality, the bound
the inverse estimates (4.7) and (4.8) to pass from \(\mathcal {E}_h\) to \(\mathcal {T}_h\), an inverse inequality to pass from \(H^1\) to \(L^2\), and finally the fact that \(v_x = v\) on \(T_x \in \mathcal {T}_{h,x}\).
Verification of (5.22) We note that each point \(y \in \mathcal {K}_{h,x}\) can be connected to a point \(z\in K_x\) using a piecewise linear curve in \(\mathcal {K}_{h,x}\) consisting of a finite number of segments, each residing in a facet \(K_i \in \mathcal {K}_{h,x}\), of the form
where \(0\le s_i \lesssim h\) is the arclength parameter of each segment, \(a_i \in T(K_i)\) is a unit direction vector in the tangent space \(T(K_i)\) to \(K_i\), and N is uniformly bounded. Then \(t(y)= n_x \cdot (y - z )\) and we have the estimate
where \(n_{h,i}\) is the normal to the facet in which the i:th segment reside and we used the estimate \(|n_x-n_{h,i}|\lesssim h\), which follows from the fact that at each edge E shared by two facets \(K_1,K_2\in \mathcal {K}_{h,x}\) with normals \(n_{h,1}\) and \(n_{h,2}\) we have the estimate
and thus we obtain the bound since the number of elements in \(\mathcal {K}_{h,x}\) is uniformly bounded.
Conclusion Collecting the estimates we obtain
Summing over the covering and using Lemma 5.1 gives
which concludes the proof. \(\square \)
5.2 Assumptions on the coefficients for the stability estimates
In order to prove the stability estimates we make the following assumptions on the coefficients.
Assumption
We assume that the discrete coefficients \(\alpha _h\) and \(\beta _h\) are uniformly bounded and satisfy:
-
There is a constant such that
$$\begin{aligned} 0< C \le \inf _{x \in {\Gamma _h}} \left( \alpha _h(x) - \frac{1}{2}\text {div}_{\Gamma _h}\beta _h(x)\right) \end{aligned}$$(5.30) -
For \(x \in \mathcal {X}_h\), there is a constant vector field \(\beta _{h,x}:\mathcal {K}_{h,x} \rightarrow \mathbb {R}^3\) such that
$$\begin{aligned} \Vert \beta _h - \beta _{h,x}\Vert _{L^\infty (\mathcal {K}_{h,x})} \le C h \end{aligned}$$(5.31)where the constant is independent of \(x\in \mathcal {X}_h\).
-
There is a constant such that
$$\begin{aligned} \Vert [n_E\cdot \beta _h]\Vert _{L^\infty (\mathcal {E}_h)} \le C h^2 \end{aligned}$$(5.32)
for all \(h\in (0,h_0]\) with \(h_0\) small enough. The jump at an edge E shared by two surface elements \(K^{\pm }\) is defined by
where \(n_E^{\pm }\) is the unit exterior conormal to \(K^\pm \) associated with edge E, i.e. the unique unit vector that is normal to E and tangent and exterior to \(K^{\pm }\), and \(\beta _h^\pm = \beta _h|_{K^\pm }\).
We return to a construction of \(\alpha _h\) and \(\beta _h\) in Sect. 6.4. Anticipating the forthcoming analysis we mention that a sufficent condition for the construction of \(\alpha _h\) and \(\beta _h\) satisfying (5.30–5.32) as well as the necessary data approximation estimates, see Sect. 6.1, is that \(\alpha \in C^2(\Gamma )\) and \(\beta \in C^2(\Gamma )\).
5.3 Technical lemmas
Lemma 5.3
There is a constant such that
for all \(h\in (0,h_0]\) with \(h_0\) small enough.
Proof
We use a covering \(\{\mathcal {T}_{h,x} : x \in \mathcal {X}_h\}\) as described above and we introduced the notation
for the set of elements \(T \in \mathcal {T}_h\) that are neighbors to an element in \(\mathcal {T}_{h,x}\). i.e. share at least one vertex with an element in \(\mathcal {T}_{h,x}\) and \({\mathcal {F}}_{h}(\mathcal {N}(\mathcal {T}_{h,x}))\) for the set of interior faces in \(\mathcal {N}({\mathcal {T}}_{h,x})\). The larger set \(\mathcal {N}({\mathcal {T}}_{h,x})\) naturally occurs when we employ the \(L^2\) stability (4.21) of the interpolation operator. We then have
where we used the following estimates. (5.36): An inverse bound to pass from \(\mathcal {K}_{h,x}\) to \(\mathcal {T}_{h,x}\). (5.37): Added and subtracted the constant vector field \(\beta _{h,x}\), defined in (5.31), and used the triangle inequality. (5.38): The second term on the right hand side of (5.37) is estimated using \(L^2\) stability (4.21) of \(\pi _h\) and Hölder’s inequality and for the first term we used the following Poincaré inequality
where \(DP_0(\mathcal {N}({\mathcal {T}}_{h,x}))\) is the space of piecewise constant functions on \(\mathcal {N}({\mathcal {T}}_{h,x})\), followed by the estimate \(\Vert [\beta _{h,x} \cdot \nabla v]\Vert _F \le \Vert \beta _{h,x}\Vert _{L^\infty (F)} \Vert [\nabla v \Vert _F \lesssim \Vert [ n_F \cdot \nabla v ] \Vert _F\). (5.39): The bound (5.31) followed by an inverse estimate to remove the gradient.
Verification of (5.40) For each element \(T \in \mathcal {T}_{h,x}\), we shall prove that
Let \(w_T\) be the constant function on \(\mathcal {N}(T)\) such that \(w_T|_T = w|_T\), then we have
where we used \(L^2\) stability (4.21) of \(\pi _h\). Now for any element \(T' \in \mathcal {N}(T)\) there is finite chain of elements from T to \(T'\) that are face neighbors and we clearly have the estimate
since \(w -w_T = 0\) on T. Summing over all elements \(T' \in \mathcal {N}(T)\), estimate (5.42) gives
Finally, summing over all \(T \in \mathcal {T}_{h,x}\), estimate (5.44) gives the desired estimate (5.40).
Conclusion Finally, summing over the covering sets and using Lemma 5.1 we obtain
which concludes the proof. \(\square \)
Lemma 5.4
There is a constant such that
for all \(h\in (0,h_0]\) with \(h_0\) small enough.
Proof
We again consider an arbitrary set \(\mathcal {T}_{h,x}\) in the covering. Adding and subtracting \(\beta _{h,x}\), that satisfies the estimate (5.31), and using the triangle inequality we get
Here we used the following estimates: (5.49): Again using an argument, similar to the verification of (5.40), we conclude that
for any \(T \in \mathcal {T}_{h,x}\). Now taking \(T=T_{x}\), the element with a large intersection \(T\cap {\Gamma _h}\) we also have the inverse estimate \(\Vert w \Vert _{T_{x}}^2 \lesssim h \Vert w \Vert ^2_{K_{x}}\) since w is constant on T. (5.50): Follows directly from the fact that \(K_{x} \in {\mathcal {K}_{h,x}}\).
Summing over the sets in the covering gives
and using Lemma 5.1 we can bound the last term and arrive at
which concludes the proof. \(\square \)
5.4 Stability estimates
Lemma 5.5
There is a positive constant \(m_0\) such that
for all \(h \in (0,h_0]\) with \(h_0\) small enough.
Proof
Integrating by parts elementwise over the surface mesh \(\mathcal {K}_h\) we obtain the identity
The first term on the right hand side may be estimated as follows
where we used Hölder’s inequality, the assumption (5.32) on \(\beta _h\), and at last Lemma 5.2. Thus we arrive at the estimate
We now have
where we used the identity (5.55), the estimate (5.59), and then we collected the terms. Thus we find that
for \(c_F>0\) and \(h \in (0,h_0]\) with \(h_0\) small enough. \(\square \)
Lemma 5.6
There are positive constants \(m_1\) and \(m_2\) such that
for all \(h \in (0,h_0]\) with \(h_0\) small enough.
Proof
We have
We now have the estimate
for \(\delta >0\). Thus taking \(\delta \) small enough the desired estimate follows directly by combining (5.67) and (5.68).
Verification of (5.68) The estimate follows by combining the following estimates of Terms \(I-III\).
Term I. It holds
where we used the inequality Cauchy–Schwarz, the arithmetic-geometric mean inequality with parameter \(\delta >0\), and Lemma 5.3.
Term II It holds
where we used Hölder’s inequality, the arithmetic-geometric mean inequality with parameter \(\delta >0\), the inverse estimate (4.9) to pass from \(\mathcal {K}_h\) to \(\mathcal {T}_h\), the boundedness (4.22) of \(\pi _h\) on \(L^2(\mathcal {T}_h)\), Lemma 5.4, and finally we rearranged the terms.
Term III It holds
where we used the Cauchy–Schwarz inequality, the arithmetic-geometric mean inequality with parameter \(\delta >0\), the inverse estimate (4.8) to pass from \(\mathcal {F}_h\) to \(\mathcal {T}_h\), an inverse inequality to remove the gradient, the boundedness (4.22) of \(\pi _h\) on \(L^2(\mathcal {T}_h)\), Lemma 5.4, and finally we rearranged the terms. \(\square \)
Proposition 5.1
There is a positive constant \(m_3\) such that
for all \(h\in (0,h_0]\) with \(h_0\) small enough.
Proof
Setting \(w=v + \gamma h \pi _h(\beta _h \cdot \nabla _{\Gamma _h}v)\), for some positive parameter \(\gamma \), we get
where we used Lemmas 5.5, 5.6, and choose \(0<\gamma \) small enough. Using the bound
the desired estimate follows with \(m_3 = \widetilde{m}_3/C\).
Verification of (5.90) Using the triangle inequality
where the second term takes the form
Term I It holds
where we used the inverse estimate (4.9) to pass from \(\mathcal {K}_h\) to \(\mathcal {T}_h\), the boundedness (4.22) of \(\pi _h\) on \(L^2(\mathcal {T}_h)\), and at last Lemma 5.4.
Term II It holds
where we used the inverse estimate (4.9) to pass from \(\mathcal {K}_h\) to \(\mathcal {T}_h\), an inverse estimate to remove the transport derivative, the boundedness (4.22) of \(\pi _h\) on \(L^2(\mathcal {T}_h)\), and finally we used Lemma 5.4.
Term III It holds
where we used the inverse inequality (4.8) to pass from \(\mathcal {F}_h\) to \(\mathcal {T}_h\), an inverse estimate to remove the gradient, the boundedness (4.22) of \(\pi _h\) on \(L^2(\mathcal {T}_h)\), and finally we used Lemma 5.4.
Conclusion of verification of (5.90) Combining the estimates of Terms \(I-III\) we get
and therefore, in view of (5.91), we conclude that (5.90) holds. \(\square \)
Proposition 5.2
It holds
for all \(h\in (0,h_0]\) with \(h_0\) small enough.
Proof
Using partial integration followed by Cauchy–Schwarz we have
Term I Using Lemma 5.2 we directly obtain
and we conclude that
Term II We have the estimates
Here we used the inverse inequality (4.7) to pass from \(\mathcal {E}_h\) to \(\mathcal {F}_h\), the identity \([ab] = [a]\langle b \rangle + \langle a \rangle [b]\), where \(\langle a \rangle = (a^+ + a^-)/2\) is the average of a discontinuous function, the fact that the tangent gradient is continuous at a face, the inverse inequality (4.8) to pass from \(\mathcal {F}_h\) to \(\mathcal {T}_h\) in the first term and a direct estimate for the second, Lemma 5.1 for the first term, and finally we collected the terms. We conclude that
Conclusion Combining (5.110) with the estimates (5.112) and (5.120) we obtain
which concludes the proof. \(\square \)
6 Error estimates
6.1 Assumptions on the coefficients for the error estimates
In addition to the assumptions on the coefficients used in the stability estimates, see Sect. 5.2, we here formulate further assumptions needed in the error analysis. In Sect. 6.4 we verify these assumptions.
Assumption
Let \(\beta _h\), \(\alpha _h\), and \(f_h\), be elementwise linear polynomial approximations of \(|B|B^{-1}\beta \), \(|B|\alpha \) and |B|f. Assume that there are constants such that
6.2 Strang’s Lemma
Define the forms
Then the exact solution u to the convection problem (2.3), see Proposition 2.1, satisfies
We then have the following Strang Lemma.
Lemma 6.1
Let u be the solution to (2.3), \(u_h\) the finite element approximation defined by (3.6), then the following estimate holds
Remark 6.1
We return to the construction of \(\beta _h\) that satisfies (6.2) in the following section. In fact, we will see that we need a stronger assumption on \(\beta _h\) in order to guarantee optimal order convergence.
Proof
Adding and subtracting an interpolant \(\pi _h u^e\), and then using the triangle inequality we obtain
where we used the interpolation estimate (4.26) to estimate the first term. Proceeding with the second term we employ the inf-sup estimate in Proposition 5.1 to get the bound
Adding and subtracting the exact solution, and using Galerkin orthogonality (3.6) the numerator may be written in the following form
Here terms I and IV gives rise to the second and third terms on the right hand side in (6.6) and II and III can be estimated as follows
which together with (6.9) yields (6.6). It remains to verify (6.14).
Term II This term is immediately estimated using (4.27) as follows
Term III Using Green’s formula and changing domain of integration from \(\Gamma \) to \(\Gamma _h\) we obtain
Changing domain of integration from \(\Gamma \) to \({\Gamma _h}\) and using the identity (4.15) we obtain
where we added and subtracted \(\beta _h\) and used the Cauchy–Schwarz inequality. Next using an inverse estimate followed by Lemma 5.1 we obtain
and using the interpolation estimate (4.25) and assumption (6.1) we have
To estimate \(III_2\) we change domain of integration from \(\Gamma \) to \({\Gamma _h}\) and use the interpolation estimate
Together the bounds (6.29) and (6.32) give the desired estimate
\(\square \)
6.3 Quadrature error estimates
Lemma 6.2
Proof
Changing domain of integration to \({\Gamma _h}\) and using the identity (4.15) for the tangential derivative of a lifted function we obtain
where at last we used the \(H^1\) stability of the interpolant to conclude that
Using the same approach and (6.2) we have
which together with (6.40) yield (6.34).
Finally, (6.35) follows using (6.3),
\(\square \)
Remark 6.2
Note that the fact that \(\pi _h u^e\) and \((\pi _h u^e)^l\) appear in the first slots in the bilinear forms in (6.34) is crucial since we get \(L^2\) control over the full tangent gradient \(\nabla _{\Gamma _h}(\pi _h u^e)\), i.e. \(\Vert \nabla _{\Gamma _h}(\pi _h u^e ) \Vert _{\Gamma _h} \lesssim \Vert u \Vert _{H^1(\Gamma )}\), using stability of the interpolant, while the corresponding control of the discrete solution \(u_h\) provided by Proposition 5.2 is only \(h^{3/4} \Vert \nabla _{\Gamma _h}u_h\Vert _{\mathcal {K}_h} \lesssim |||u_h |||_h \lesssim \sup _{v\in V_h} \frac{A_h(u_h,v)}{|||v |||_h} \lesssim \Vert f_h \Vert _{\mathcal {K}_h}\), indicating a higher demand on the accuracy of \(\beta _h\) in order to achieve optimal order of convergence.
Remark 6.3
Note that we do not have to take |B| into account in (6.1–6.3) since \(|\,|B|-1\,| \sim h^2\). Thus we could instead use the simplified assumptions
Furthermore, \(B^{-1} = P_{\Gamma _h}P_\Gamma + O(h^2)\), see (4.12) in [17], and thus we may simplify the assumption on \(\beta _h\) even further and assume that
Remark 6.4
The mapping \(\beta \mapsto |B|B^{-1}\beta \) is in fact a Piola mapping which maps tangent vectors on \(\Gamma \) onto tangent vectors on \(\Gamma _h\).
6.4 Construction of the discrete coefficients
Here we provide a concrete construction of coefficients that satisfy our assumptions. In order to handle the fact that a surface element \(K\in \mathcal {K}_h\) can be arbitrarily small and is not in general shape regular we interpolate the coefficients over a larger shape regular triangle \(\widetilde{K}\). More precisely, given \(K \in \mathcal {K}_h\) we may construct a shape regular triangle \(\widetilde{K}\) in the hyperplane defined by K such that \(K \subset \widetilde{K}\) and \(\text {diam}( \widetilde{K}) \lesssim h\). We can then define
where \(P_K\) is the tangent projection associated with K and \(\widetilde{I}_h\) is the linear Lagrange interpolant associated with the triangle \(\widetilde{K}\). Assuming that \(\beta \in C^2(\Gamma )\) we have
since \(P_K\) is constant. Similar constructions can be done for \(\alpha _h\) and \(f_h\).
Lemma 6.3
Assuming that \(\beta \), \(\alpha \), f are all in \(C^2(\Gamma )\), \(\beta _h\), \(\alpha _h\), and \(f_h\) satisfy all the assumptions (5.31), (5.30–5.32), needed for the stability analysis, and the assumptions (6.1–6.3) needed in the quadrature error estimates, for all \(h \in (0,h_0]\) with \(h_0\) small enough.
Proof
(5.30). Adding and subtracting \((\alpha - \frac{1}{2}\text {div}_\Gamma \beta )^e\) we obtain
where we used (2.4) and the bounds
To verify the second bound in (6.56) we first note that
which holds since \(\text {div}_{\Gamma _h}(n_h ( n_h \cdot \beta ^e )) = ( n_h \cdot \beta ^e ) \text {tr}( n_h \otimes \nabla _{\Gamma _h}) + n_h \cdot \nabla _{\Gamma _h}(n_h \cdot \beta ^e) = 0\) and we used the facts that \(n_h\) is piecewise constant and orthogonal to \(\nabla _{\Gamma _h}(n_h \cdot \beta ^e)\). Next subtracting \(\text {div}_{\Gamma _h}(P_{\Gamma _h}\beta ^e )\) and adding \(\text {div}_{\Gamma _h}\beta ^e\) and using the triangle inequality we obtain
Term I For \(K \in \mathcal {K}_h\),
where we used (6.52).
Term II Using the definition of the surface divergence operators we have
Here, for \(x\in \Gamma _h\), we interpret \(P_\Gamma \) as an operator
i.e. the operator that first projects onto the tangent plane \(T_{p(x)}(\Gamma )\) and then translates the tangent vector to a vector at x. We also recall that the transpose of B is an operator \(B^T: T_{p(x)} (\Gamma ) \ni \xi \mapsto B^T \xi \in T_x (\Gamma _h) \subset T_{x}(\mathbb {R}^3)\). Thus we can write
We then have the identities
where \(Q_\Gamma =I - P_\Gamma \) and \( Q_{\Gamma _h} = I - P_{\Gamma _h}\). We thus have the estimate
where we at last used (3.1) and (4.12).
(5.31). Adding and subtracting \(P_{\Gamma _h} \beta ^e\) we have
where we used (6.52) to estimate the first term on the right hand side of (6.80) and for the second term we used the fact that \(\beta = {P}_\Gamma \beta \) since \(\beta \) is tangential, and the third term is estimated using Taylor’s formula.
(5.32). Splitting the jump by adding and subtracting \(\beta \) we get
Terms I and II We consider I since these terms are of the same form. Using the fact that \(n^+_E\) is a tangent vector to K and definition of \(\beta _h\) we obtain
Term III Since \(\beta \) is a smooth tangent vector field on \(\Gamma \) we have
where \(t_E \in T_{p(x)}(\Gamma )\) is the unit tangent vector at p(x) to the exact surface that is orthogonal to edge E. Thus it remains to estimate \([n_E] \cdot t^e_E\). We have the identity
where \(e_E\) is the unit vector along E directed in such a way that \(n_{K^+} \times e_E = n^+_{E}\), \(n_K = n_h |_K\) , \([n_K]=n_{K^+} - n_{K^-}\), and we used the identity \(n\cdot n_h = 1 + O(h^2)\).
(6.1–6.3). We obtain (6.1) by combining Remark 6.3 and (6.52). (6.2–6.3) follows in the same way. \(\square \)
6.5 Error estimates
Theorem 6.1
Let u be the solution to (2.3) and \(u_h\) the finite element approximation defined by (3.6), then the following estimate holds
for all \(h\in (0,h_0]\) with \(h_0\) small enough.
Proof
Adding and subtracting an interpolant
Here the first term is estimated using the interpolation error estimate (4.28) and for the second we apply the Strang Lemma 6.1 together with the quadrature error estimates in Lemma 6.2. \(\square \)
Theorem 6.2
Let u be the solution to (2.3) and \(u_h\) the finite element approximation defined by (3.6), then the following estimate holds
for all \(h\in (0,h_0]\) with \(h_0\) small enough.
Proof
We have the estimates
Here we added and subtracted an interpolant and used the triangle inequality, used the stability estimate in Proposition 5.2, added and subtracted an interpolant in the second term and used the triangle inequality, used interpolation error estimates (4.25) and (4.28) to estimate the first and the second term and finally Theorem 6.1 to estimate the third term, which conclude the proof of the desired estimate. \(\square \)
Remark 6.5
We note that these two error estimates are completely analogous to the estimates obtained in [3] for the corresponding method on standard triangular meshes in the plane.
7 Condition number estimate
Let \(\{\varphi _i\}_{i=1}^N\) be the standard piecewise linear basis functions associated with the nodes in \(\mathcal {T}_h\) and let \(\mathcal {A}\) be the stiffness matrix with elements \(a_{ij} = A_h(\varphi _i,\varphi _j)\). We recall that the condition number is defined by
Using the ideas introduced in [6], we may prove the following bound on the condition number of the matrix.
Theorem 7.1
The condition number of the stiffness matrix \(\mathcal {A}\) satisfies the estimate
for all \(h \in (0,h_0]\) with \(h_0\) small enough.
Proof
First we note that if \(v = \sum _{i=1}^N V_{i} \varphi _i\) and \(\{\varphi _i\}_{i=1}^N\) is the usual nodal basis on \(\mathcal {T}_h\) then the following well known estimates hold
It follows from the definition (7.1) of the condition number that we need to estimate \(| \mathcal {A}|_{\mathbb {R}^N}\) and \(|\mathcal {A}^{-1}|_{\mathbb {R}^N}\).
Estimate of \(| \mathcal {A}|_{\mathbb {R}^N}\) We have
Here we used the following continuity of \(A_h(\cdot ,\cdot )\)
In the last step we used the estimates
where we used the inverse estimates (4.9) and (4.8) to pass from \(\mathcal {K}_h\) and \(\mathcal {F}_h\) to \(\mathcal {T}_h\), an inverse estimate to remove the gradient, and finally the equivalence (7.3); and
where we used the same sequence of estimates. It follows that
Estimate of \(|\mathcal {A}^{-1} |_{\mathbb {R}^N}\) We note that using (7.3) and Lemma 5.1 we have
where in the last step we used the fact that \(h \in (0,h_0]\) and the definition (4.4) of \(|||\cdot |||_h\). Starting from (7.13) and using the inf-sup condition (5.84) we obtain
Here we used (7.13), \(h^{d/2} |W|_{\mathbb {R}^N} \lesssim |||w |||_h\), to replace \(|||w |||_h\) by \(h^{d/2} |W|_{\mathbb {R}^N} \) in the denominator. Setting \(V= \mathcal {A}^{-1} X\), \(X \in \mathbb {R}^N\), we obtain
Conclusion The claim follows by using the bounds (7.12) and (7.16) in the definition (7.1). \(\square \)
8 Numerical results
We consider an example where the surface \(\Gamma \) is a torus given by the zero level set of the the signed distance function \(\rho =(z^2+ ((x^2+y^2)^{1/2}-R )^2)^{1/2}-r\), with \(R=1\) and \(r=1/2\). We choose \(\alpha =1\),
and f such that the exact solution is
The vector field \(\beta \) is shown in Fig. 1. A structured mesh \(\mathcal {T}_{0,h}\) consisting of tetrahedra on the domain \([-1.6, 1.6] \times [-1.6, 1.6] \times [-0.6, 0.6]\) is generated independently of the position of the torus. The mesh size parameter is defined as \(h=h_x=h_y=h_z\). An approximate distance function \(\rho _h\) is constructed using the nodal interpolant \(\pi _h \rho \) on the background mesh and \(\Gamma _h\) is the zero levelset of \(\rho _h\) and \(n_h\) is the piecewise constant unit normal to \(\Gamma _h\). The triangulation of \(\Gamma _h\) is shown in Fig. 2. We use the proposed method with the stabilization parameter chosen as \(c_F=10^{-2}\). A direct solver was used to solve the linear systems. The solution \(u_h\) for \(h=0.2\) is shown in Fig. 3.
8.1 Convergence study
We compare our approximation \(u_h\) with the exact solution u given in Eq. (8.2). The convergence of \(u_h\) in the \(L^2\)-norm and the energy norm are shown in Fig. 4. We observe second order convergence in the \(L^2\)-norm and as expected a convergence order of 1.5 in the energy norm. The error in the gradient \(\Vert \nabla _{\Gamma _h}(u^e - u_h) \Vert _{\mathcal {K}_h}\) versus mesh size is shown in Fig. 5 and the observed convergence order is slightly better than 3 / 4 for the finest meshes. Our analysis does not cover the \(L^\infty \)-convergence but in Fig. 6 we show the \(L^\infty \)-errors versus mesh size and observe around second order convergence.
8.2 Condition number study
We compute the condition number of the stiffness matrix for a sequence of uniformly refined meshes and different values of the stabilization parameter \(c_F\). We find that the asymptotic behavior as the meshsize tend to zero is \(O(h^{-2})\) as expected from the analysis, while on coarser meshes for larger values of \(c_F\) the behavior is closer to \(O(h^{-1})\), see Fig. 7.
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This research was supported in part by the Swedish Foundation for Strategic Research Grant No. AM13-0029 (PH,MGL), the Swedish Research Council Grants Nos. 2011-4992 (PH), 2013-4708 (MGL), and 2014-4804 (SZ), the Swedish Research Programme Essence (MGL, SZ), and EPSRC, UK, Grant Nr. EP/J002313/2. (EB).
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Burman, E., Hansbo, P., Larson, M.G. et al. Stabilized CutFEM for the convection problem on surfaces. Numer. Math. 141, 103–139 (2019). https://doi.org/10.1007/s00211-018-0989-8
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DOI: https://doi.org/10.1007/s00211-018-0989-8