An Eulerian spacetime finite element method for diffusion problems on evolving surfaces
Abstract
In this paper, we study numerical methods for the solution of partial differential equations on evolving surfaces. The evolving hypersurface in defines a dimensional spacetime manifold in the spacetime continuum . We derive and analyze a variational formulation for a class of diffusion problems on the spacetime manifold. For this variational formulation new wellposedness and stability results are derived. The analysis is based on an infsup condition and involves some natural, but nonstandard, (anisotropic) function spaces. Based on this formulation a discrete in time variational formulation is introduced that is very suitable as a starting point for a discontinuous Galerkin (DG) spacetime finite element discretization. This DG spacetime method is explained and results of numerical experiments are presented that illustrate its properties.
1 Introduction
Partial differential equations (PDEs) posed on evolving surfaces arise in many applications. In fluid dynamics, the concentration of surface active agents attached to an interface between two phases of immiscible fluids is governed by a transportdiffusion equation on the interface [13]. Another example is the diffusion of transmembrane receptors in the membrane of a deforming and moving cell, which is typically modeled by a parabolic PDE posed on an evolving surface [2].
Recently, several approaches for solving PDEs on evolving surfaces numerically have been introduced. The finite element method of Dziuk and Elliott [6] is based on the Lagrangian description of a surface evolution and benefits from a special invariance property of test functions along material trajectories. If one considers the Eulerian description of a surface evolution, e.g., based on the level set method [19], then the surface is usually defined implicitly. In this case, regular surface triangulations and material trajectories of points on the surface are not easily available. Hence, Eulerian numerical techniques for the discretization of PDEs on surfaces have been studied in the literature. In [1, 21] numerical approaches were introduced that are based on extensions of PDEs off a twodimensional surface to a threedimensional neighbourhood of the surface. Then one can apply a standard finite element or (as was done in [1, 21]) finite difference disretization to treat the extended equation in . The extension, however, leads to degenerate parabolic PDEs and requires the solution of equations in a higher dimensional domain. For a detailed discussion of this extension approach we refer to [12, 7, 3]. A related approach was developed in [8], where advectiondiffusion equations are numerically solved on evolving diffuse interfaces.
A different Eulerian technique for the numerical solution of an elliptic PDE posed on a hypersurface in was introduced in [17, 15]. The main idea of this method is to use finite element spaces that are induced by the volume triangulations (tetrahedral decompositions) of a bulk domain in order to discretize a partial differential equation on the embedded surface. This method does not use an extension of the surface partial differential equation. It is instead based on a restriction (trace) of the outer finite element spaces to the (approximated) surface. This leads to discrete problems for which the number of degrees of freedom corresponds to the twodimensional nature of the surface problem, similar to the Lagrangian approach. At the same time, the method is essentially Eulerian as the surface is not tracked by a surface mesh and may be defined implicitly as the zero level of a level set function. For the discretization of the PDE on the surface, this zero level then has to be reconstructed. Optimal discretization error bounds were proved in [17]. The approach was further developed in [4, 18], where adaptive and streamline diffusion variants of this surface finite element were introduced and analysed. These papers [17, 15, 4, 18], however, treated elliptic and parabolic equations on stationary surfaces.
The goal of this paper is to extend the approach from [17] to parabolic equations on evolving surfaces. An evolving surface defines a threedimensional spacetime manifold in the spacetime continuum . The surface finite element method that we introduce is based on the traces of outer spacetime finite element functions on this manifold. The finite element functions are piecewise polynomials with respect to a volume mesh, consisting of fourdimensional prisms (4D prism = 3D tetrahedron time interval). For this discretization technique, it is natural to start with a variational formulation of the differential problem on the spacetime manifold. To our knowledge such a formulation has not been studied in the literature, yet. One new result of this paper is the derivation and analysis of a variational formulation for a class of diffusion problems on the spacetime manifold. For this formulation we prove wellposedness and stability results. The analysis is based on an infsup condition and involves some natural, but nonstandard, anisotropic function spaces. A second important result is the formulation and analysis of a discrete in time variational formulation that is very suitable as a starting point for a discontinuous Galerkin spacetime finite element discretization. Further, we present a finite element method, which then results in discretization (in space and time) of a parabolic equation on an evolving surface.
The discretization approach based on traces of an outer spacetime finite element space studied here is also investigated in the recent report [10]. In [10] there is no analysis of the corresponding continuous variational formulation, which is the main topic of this paper. On the other hand, in [10] one finds more information on implementation aspects and an extensive numerical study of properties (accuracy and stability) of this method and some of its variants. Further results of numerical experiments for the example of surfactant transport over two colliding spheres can be found in [11]. We only very briefly comment on implementation aspects and illustrate accuracy and stability properties of the discretization method by results of a few numerical experiments.
In this paper, we do not study discretization error bounds for the presented Eulerian spacetime finite element method. This is a topic of current research, first results of which are presented in the followup report [16].
The remainder of the paper is organized as follows. In Section LABEL:sec2 we review surface transportdiffusion equations and introduce a spacetime weak formulation. Some required results for surface functional spaces are proved in Section LABEL:sec3. A general spacetime variational formulation and corresponding wellposedness results are presented in Section LABEL:sec4. A semidiscrete in time method is analyzed in Section LABEL:sec5. A fully discrete spacetime finite element method is considered in Section LABEL:sec6. Section LABEL:sec7 contains results of some numerical experiments.
2 Diffusion equation on an evolving surface
Consider a surface passively advected by a velocity field , i.e. the normal velocity of is given by , with the unit normal on . We assume that for all , is a smooth hypersurface that is closed (), connected, oriented, and contained in a fixed domain . To describe the smoothness assumption concerning and its evolution more precisely, we introduce the Langrangian mapping from the spacetime cylinder , with , to the spacetime manifold
see also [8]. We assume that the velocity field and are sufficiently smooth such that for all the ODE system
has a unique solution (recall that is transported with the velocity field ). The corresponding inverse mapping is given by , . The Lagrangian mapping induces a bijection
\hb@xt@.01(2.1) 
We assume this bijection to be a diffeomorphism between these manifolds. The conservation of a scalar quantity with a diffusive flux on leads to the surface PDE (cf. [14]):
\hb@xt@.01(2.2) 
with initial condition for . Here
denotes the advective material derivative, is the surface divergence and is the LaplaceBeltrami operator, is the constant diffusion coefficient. Let be the usual Sobolev space on . The following weak formulation of (LABEL:transport) was shown to be wellposed in [6]: Find such that and for almost all
\hb@xt@.01(2.3) 
Here is the tangential gradient for . A similar weak formulation is considered in [20]. The formulation (LABEL:weakDziuk) is a natural starting point for the Lagrangian type finite element methods treated in [6, 20]. It is, however, less suitable for the Eulerian finite element method that we introduce in this paper. Our discretization method uses the framework of spacetime finite element methods. Therefore, it is natural to consider a spacetime weak formulation of (LABEL:transport) as given below. We introduce the space
endowed with the scalar product
\hb@xt@.01(2.4) 
and consider the material derivative as a linear functional on . The subspace of all functions from such that defines a bounded linear functional form the trial space . A precise definition of the space is given in Section LABEL:sectHW. We consider the following weak formulation of (LABEL:transport): Find such that
\hb@xt@.01(2.5) 
We shall derive certain density properties for the spaces and , which we need for proving the wellposedness of (LABEL:weakSpaceTime). Actually, we show wellposedness of a slightly more general problem, which includes a possibly nonzero source term and, instead of , a generic zero order term. Our finite element method will be based on (LABEL:weakSpaceTime) rather than (LABEL:weakDziuk).
3 Preliminaries
In this section, we define the trial space and prove that both the test space and the trial space are Hilbert spaces, and that smooth functions are dense in and . We also prove that a function from has a welldefined trace as an element of for all . In the setting of a spacetime manifold, the spaces and are natural ones. In the literature, however, we did not find any analysis of their properties. The necessary results are established with the help of a homeomorphisms between , and the following standard Bochner spaces and :
\hb@xt@.01(3.1) 
In the next subsection, we collect a few properties of the Bochner spaces and that we need in our analysis.
3.1 Properties of the spaces and
The spaces and are endowed with the norms
We start with the following wellknown result.
Lemma 3.1
The space is dense in .
Proof. The inclusion is trivial.
By construction of the Bochner space, the set of simple functions
is dense in ; here is any set of mutually disjoint measurable subsets of .
For there exists and
such that is arbitrary small. This completes the proof.
For we define the weak time derivative through the functional
\hb@xt@.01(3.2) 
Then iff there is a constant such that
Remark 3.1
The definition of in (LABEL:WX), based on the weak time derivative (LABEL:r1), is equivalent to the following more standard one: is an element of iff there exists such that
\hb@xt@.01(3.3) 
for all . The definition of in (LABEL:WX) is more convenient for the analysis that follows.
We recall a few results for the space .
Lemma 3.2
The set
is dense in . For the function is continuous from into . There is a constant such that
\hb@xt@.01(3.4) 
Proof. Proofs are given in standard textbooks, e.g., [22] Proposition 23.23. The density result is usually proved with replaced by in the definition of . The result with follows from the density of in .
3.2 The spaces and
We assume that the spacetime surface is sufficiently smooth, cf. Section LABEL:sec2. Due to the identity
\hb@xt@.01(3.5) 
the scalar product induces a norm that is equivalent to the standard norm on . Therefore, one can equivalently define the norm on by
\hb@xt@.01(3.6) 
The space is a Hilbert space, and forms a Gelfand triple (cf. Lemma LABEL:PropH below).
Recall the Leibniz formula
\hb@xt@.01(3.7) 
which implies the integration by parts identity:
\hb@xt@.01(3.8) 
Based on (LABEL:ByParts) we define the material derivative for as the functional :
\hb@xt@.01(3.9) 
Assume that for some the norm
is bounded. In Lemma LABEL:PropH we prove that is dense in and therefore can be extended to a bounded linear functional on . In this case, we write and define the space
In Section LABEL:secttHW we prove that is a Hilbert space and is dense in . Note that the space is larger than the standard Sobolev space . Spaces similar to and are introduced and analyzed in [20]. A difference between our aproach and the one used in that paper is that we define and directly on the spacetime manifold , whereas in [20] these are defined using a pull back operator to the manifold . We use such a pull back operator in the analysis of the spaces and in the next section, but not in their definition.
Remark 3.2
From the definition of the weak material derivative in (LABEL:weakmaterial) and the density result of Lemma LABEL:PropH it follows that for we have
3.3 Homeomorphism between {, } and {, }
Based on the diffeomorphism in (LABEL:defF), for a function defined on we define on :
Vice versa, for a function defined on we define on :
\hb@xt@.01(3.10) 
By construction we have
\hb@xt@.01(3.11) 
Now we prove that the mapping defines a linear homeomorphism between and , and also between and .
Lemma 3.3
The linear mapping from (LABEL:homo) defines a homeomorphism between and .
Proof. For any fixed , we obtain iff . Let for all . Due to the smoothness assumptions on , there are constants , independent of and , such that
\hb@xt@.01(3.12) 
Hence, iff holds, and the linear mapping is a homeomorphism between and .
For the further analysis, we need a surface integral transformation formula. For this we consider a local parametrization of , denoted by , which is at least smooth. Then, defines a smooth parametrization of . For the surface measures and on and , respectively, we have the relation
\hb@xt@.01(3.13) 
with , and similarly for . Recall that denotes the surface gradient of a scalar function defined on for any fixed . Using this integral transformation formula, for and we obtain
\hb@xt@.01(3.14) 
Lemma 3.4
The linear mapping from (LABEL:homo) defines a homeomorphism between and .
Proof. The proof makes use of the formula (LABEL:transf2). Take with , and . We use the notation if the constant that occurs in the inequality is independent of and , and if such an inequality holds in two directions. Due to the smoothness assumption on (and thus ) the function defined in (LABEL:fromtrans1) is smooth on . Define . Due to Lemma LABEL:homoH we have . Therefore, we can estimate
Hence, and holds. With similar arguments one can show that if , then and holds. For this, instead of the surface transformation formula (LABEL:fromtrans1) one starts with the formula
\hb@xt@.01(3.15) 
with , and similarly for . For and we have
Now we note that is smooth on . To check this, due to the diffeomorphism property of it is sufficient to show that the denominator in (LABEL:wg) is uniformly bounded away from zero on . For and with one can rewrite the denominator as
\hb@xt@.01(3.16) 
From the fact that is a smooth parametrization of it follows that the quantity on the righthand side of (LABEL:aux1) is uniformly bounded away from zero. Hence, the function is smooth and we can use the same arguments as above to derive . This implies that is a homeomorphism between and .
3.4 Properties of and
The homeomorphism established in Section LABEL:secHomo helps us to derive density results for the spaces and and a trace property similar to the one in (LABEL:trace1).
Lemma 3.5
is a Hilbert space. The space is dense in . The spaces form a Gelfand triple.
Proof. Let denote the mapping given in (LABEL:homo). Since is a linear homeomorphism between, the space is complete and so this is a Hilbert space. For we have, due to the smoothness assumptions on , that . Furthermore, from it follows that has compact support. Hence, . From this we get . Since is dense in and is a homeomorphism, this implies that is dense in . Since is also dense in , the space is dense in . Hence, is a Gelfand triple.
For and denote by a trace operator: , . In Section LABEL:ssectDG we analyze a discontinuous Galerkin method in time. For such a method, one needs well defined traces of this type. For a smooth function defined on the cylinder , it is obvious that one can define at any time , the right limit on . Similarly a left limit function is defined for . For a sufficiently smooth function on , due to the fact that the domain where the trace has to be defined varies with , it is less straightforward to construct such left and right limit functions. To this end, for and a given we define by
\hb@xt@.01(3.17) 
Note that holds when . Right and left limits on are defined as
\hb@xt@.01(3.18) 
Below we show that for functions from the trace and onesided limits are welldefined and can be considered as elements of .
The next theorem gives several important properties for our trial space.
Theorem 3.6
is a Hilbert space and has the following properties:

(i) is dense in .

(ii) For every the trace operator can be extended to a bounded linear operator from to . Moreover, the inequality
\hb@xt@.01(3.19) holds with a constant independent of .

(iii) Take and let be sufficiently small, such that . For any the mapping defined in (LABEL:conti) is continuous from into . The same assertion is true for and suitable . For the onesided limits (LABEL:conti2) are welldefined.
Proof. Since the mapping given by (LABEL:homo) is a linear homeomorphism between and , the space is complete and so this is a Hilbert space.
(i) Let be the set as in Lemma LABEL:lem4, which is dense in . One easily checks . Since is dense in , this implies that is dense in .
(ii) Take and define . Using (LABEL:fromtrans1), Lemma LABEL:lem4 and Lemma LABEL:homoW we get
where the constant can be assumed to be independent of due to the smoothness of in (LABEL:fromtrans1). From this, the result in (LABEL:rrt) follows by a density argument.
(iii) Take a fixed and sufficiently small . Take . For we use the substitution and the integral transformation formula as in the proof of Lemma LABEL:homoH, resulting in:
with a constant independent of . Hence, the continuity of the mapping follows from the continuity result for in Lemma LABEL:lem4. Due to the continuity of the mappings, the onesided limits are welldefined.
Corollary 3.7
For all , the integration by parts identity holds:
\hb@xt@.01(3.20) 
Proof. Follows from the identity (LABEL:ByParts) and Theorem LABEL:PropW.
4 Wellposedness of weak formulation
Using the properties of and derived above, we prove a wellposedness result for the weak spacetime formulation (LABEL:weakSpaceTime) of the surface transportdiffusion equation (LABEL:transport). The analysis uses the LBB approach and is along the same lines as presented for a parabolic problem on a fixed Euclidean domain in [9] (Section 6.1). As usual, we first transform the problem (LABEL:transport) to ensure that the initial condition is homogeneous. To this end, consider the decomposition of the solution , where is chosen sufficiently smooth and such that
One can set, e.g., , with from (LABEL:fromtrans1). Since the solution of (LABEL:transport) has the mass conservation property , and by the choice of , the new unknown function satisfies on and
\hb@xt@.01(4.1) 
For this transformed function the diffusion equation takes the form
\hb@xt@.01(4.2)  
The righthand side is now nonzero: . Using (LABEL:partint) and the integration by parts over , one immediately finds for . For a more regular source function, , this implies for almost all .
In the analysis below, instead of the (transformed) diffusion problem (LABEL:transport1) we consider the following slightly more general surface PDE:
\hb@xt@.01(4.3)  
with and a generic righthand side , not necessarily satisfying the zero integral condition. We use the notation .
We define the inner product and symmetric bilinear form
This bilinear form is continuous on :
\hb@xt@.01(4.4) 
Consider the subspace of of all function vanishing for :
The space is welldefined, since functions from have welldefined traces on for any , see Theorem LABEL:PropW. The weak spacetime formulation of (LABEL:transport2) reads: Given , find such that
\hb@xt@.01(4.5) 
In the remainder of this section we prove that this variational problem is wellposed. Our analysis is based on the continuity and infsup conditions, cf. [9]. The continuity property is straightforward:
The next two lemmas are crucial for proving the wellposedness of (LABEL:weakformu).
Lemma 4.1
The infsup inequality
\hb@xt@.01(4.6) 
holds with some .
Proof. Take . In (LABEL:weakformu) we take a test function , with . We note the identity:
\hb@xt@.01(4.7) 
From (LABEL:hp), (LABEL:partint), and condition , we infer