A cut finite element method for a model of pressure in fractured media

We develop a robust cut finite element method for a model of diffusion in fractured media consisting of a bulk domain with embedded cracks. The crack has its own pressure field and can cut through the bulk mesh in a very general fashion. Starting from a common background bulk mesh, that covers the domain, finite element spaces are constructed for the interface and bulk subdomains leading to efficient computations of the coupling terms. The crack pressure field also uses the bulk mesh for its representation. The interface conditions are a generalized form of conditions of Robin type previously considered in the literature which allows the modeling of a range of flow regimes across the fracture. The method is robust in the following way: 1. Stability of the formulation in the full range of parameter choices; and 2. Not sensitive to the location of the interface in the background mesh. We derive an optimal order a priori error estimate and present illustrating numerical examples.

Authors

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1 Introduction

The numerical modelling of flow in fractured porous media is important both in environmental science and in industrial applications. It is therefore not surprising that it is currently receiving increasing attention from the scientific computing community. Here we are interested in models where the fractures are modelled as embedded surfaces of dimension in a dimensional bulk domain. Models on this type of geometries of mixed dimension are typically obtained by averaging the flow equations across the width of the fracture and introducing suitable coupling conditions for the modelling of the interaction with the bulk flow. Such reduced modelled have been derived for instance in [20, 24, 1]. The coupling conditions in these models typically take the form of a Robin type condition. The physical properties of the coupling enters as parameters in this interface condition. The size of these parameters can vary with several orders of magnitude depending on the physical properties of the crack and of the material in the porous matrix. This makes it challening to derive methods that both are flexible with respect to mesh geometries and robust with respect to coupling conditions. A wide variety of different strategies for the discretisation of fractured porous media flow has been proposed in the literature. One approach is to use a method that allows for nonconforming coupling between the bulk mesh and the fracture mesh [3], or even arbitrary polyhedral elements in the bulk mesh in order to be able to mesh the fractures easily. This latter approach has been developed using discontinuous Galerkin methods [2], virtual element methods [15] and high order hybridised methods [13].

Herein we will consider an unfitted approach, drawing on previous work [4, 12, 10] where flow in fractured porous media was modelled in the situation where the pressure is a globally continuous function. When using unfitted finite element methods, the bulk mesh can be created completely independently of the fractures. Instead the finite element space is modified locally to allow for discontinuities across fractures and interface conditions are typically imposed weakly, or using methods similar to Nitsche’s method. For other recent work using unfitted methods we refer to [22], where a stabilized Lagrange multiplier method is considered for the interface coupling and [14] where a mixed method is considered for the Darcy’s equations both in the bulk and on the surface.

The upshot here, compared to [10] is that the pressure in the crack has its own pressure, allowing for the accurate approximation of problems where the pressure is discontinuous between the bulk and the fracture, and that the interface conditions are imposed in a way allowing for the full range of parameter values in the Robin condition, without loss of stability. We use the variant of the interface modelling considered in [24], that was also recently applied for the numerical modelling in [2]. In these models we may obtain a wide range of parameter values in the interface condition and we therefore develop a method which handle the full range of values and produces approximations with optimal order convergence. The approach is inspired by the work of Stenberg [21] and may be viewed as a version of the Nitsche method that can handle Robin type conditions and which converges to the standard Nitsche method when the Robin parameter tends to infinity. Previous applications of this approach in the context of fitted finite elements include [18] and [28].

The finite element spaces are constructed starting from a standard mesh equipped with a finite element space. For each geometric domain (subdomains and interface) we mark all the elements intersected by the domain and then we restrict the finite element space to that set to form a finite element space for each domain. This procedure leads to cut finite elements and we use stabilization to ensure that the resulting form associated with the method is coercive and that the stiffness matrix is well conditioned. The stabilization is of face or ghost penalty type [5, 6, 23] , and is added both to the bulk and interface spaces. Previous related work work on cut finite element methods include the interface problem [17]; overlapping meshes [19]; coupled bulk-surface problems [11, 7, 10] and [16]; mixed dimensional problems [8]

, and surface partial differential equations

[6, 25]. For a general introduction to cut finite element methods we refer to [4].

The outline of the paper is as follows: In Section 2 we introduce the model problem and discuss the relation between our formulation of the interface conditions and previous work; in Section 3 we formulate the cut finite element method; in Section 4 we prove the basic properties of the formulation and in particular an optimal order a priori error estimate which is uniform in the full range of interface parameters; and in Section 5 we present numerical results.

2 The Model Problem

2.1 Governing Equations

Let be a convex polygonal domain in , or , with boundary and exterior unit normal . Let be a smooth embedded interface in , which partitions into two subdomains and with exterior unit normals and . We assume that is a closed surface without boundary residing in the interior of , more precisely we assume that there is such that the distance between and is larger than . We consider for simplicity the case with homogeneous Dirichlet conditions on .

The problem takes the form: find and such that

 −∇⋅Ai∇ui =fi in Ωi (2.1) −∇Γ⋅AΓ∇ΓuΓ =fΓ+⟦n⋅A∇u⟧ on Γ (2.2) n⋅A∇u+B(u−uΓ) =0 on Γ (2.3) u =0 on ∂Ω (2.4)

Here the jump (or sum) of the normal fluxes is defined by

 ⟦n⋅A∇v⟧=2∑i=1ni⋅Ai∇vi, (2.5)

In the interface condition (2.3), is a

symmetric matrix with eigenvalues

such that and we used the notation

 n⋅A∇v=[n1⋅A1∇v1n2⋅A2∇v2],v−vΓ=[v1−vΓv2−vΓ] (2.6)

and thus in component form (2.3) reads

 [n1⋅A1∇u1n2⋅A2∇u2]+B[u1−uΓu2−uΓ]=[00] (2.7)

The coefficients , , are smooth uniformly positive definite symmetric matrices, is smooth tangential to and uniformly positive definite on the tangent space of , so that

 2∑i=1∥∇vi∥2Ωi+∥∇ΓvΓ∥2Γ≲2∑i=1(Ai∇vi,∇vi)Ωi+(AΓ∇ΓvΓ,∇ΓvΓ) (2.8)

where denotes less or equal up to a constant. Finally we assume and .

Remark 2.1

Several generalizations are possible on the external boundary. For instance, we may let the interface intersect the boundary of . In this case we let denote the unit exterior conormal to , i.e. is tangent to and normal to , and we assume that for some constant so that the interface is transversal to . We may then enforce the Dirichlet condition on (see [9]) or some other standard boundary condition.

Remark 2.2

In practical modeling we may want to take the thickness of the interface inte account. Assuming that the permeability matrix in an interface of thickness takes the form

 A|Ut/2(Γ)=AeΓ+aeΓnΓ⊗nΓ (2.9)

where

is a unit normal vector field to

, is the set of points with distance less than to , denotes the extension of a function on that is constant in the normal direction,

is the tangential tangential permeability tensor, and finally

is the permeability across the interface. Also assuming that and in , the equation on the interface (2.2) may be modelled as follows

 −∇Γ⋅tAΓ∇ΓuΓ =tfΓ+⟦n⋅A∇u⟧ on Γ (2.10)

Note that the last term on the right hand side does not scale with since it accounts for flow into the crack from the bulk domains.

Remark 2.3

We comment on how our interface condition (2.3) relates to the condition in [24] and later reformulated, see [2], in terms of averages and jumps of the bulk fields across the interface. The interface conditions in [24], equations (3.18) and (3.19), takes the form

 ξn1⋅A1∇v1−(1−ξ)n2⋅A2∇v2 =α(vΓ−v1) (2.11) ξn2⋅A2∇v2−(1−ξ)n1⋅A1∇v1 =α(vΓ−v2) (2.12)

where and are parameters. The parameter is related to physical properties of the interface as follows

 α=2aΓ,nd (2.13)

where is the permeability coefficient across the interface and is the thickness of the interface, see (3.8) in [24] In matrix form we obtain

 [ξξ−1ξ−1ξ][n1⋅A1∇v1n2⋅A2∇v2]+[α00α][v1−vΓv2−vΓ]=0 (2.14)

 B=12ξ−1[ξ1−ξ1−ξξ][α00α]=α2ξ−1[ξ1−ξ1−ξξ] (2.15)

We note that we have the eigen pairs

 Be1=α2ξ−1λ1e1,Be2=αλ2e2 (2.16)

with the corresponding eigen vectors defined by

and thus is positive definite for , singular for , and indefinite for . It is therefore natural to consider the case when and . We remark that when tends to infinity both eigenvalues tend to infinity and when tends to from above one eigenvalue tends to infinity. It is therefore important to construct a method which is robust in the full range of possible values for the two eigenvalues

To see the relation to the formulation of the interface conditions in [2] we first note that we have the expansions

 [n1⋅A1∇v1n2⋅A2∇v2]=2−1/2⟦n⋅A∇⟧ e1+21/2⟨⟨n⋅A∇v⟩⟩ e2 (2.17)
 [v1−vΓv2−vΓ]=21/2(⟨⟨v⟩⟩−vΓ) e1+2−1/2⟦v⟧ e2 (2.18)

where the jumps and averages of the bulk fields across the the interface are defined by

 ⟦n⋅A∇v⟧=2∑i=1ni⋅Ai∇vi,⟦v⟧=v1−v2 (2.19)
 ⟨⟨n⋅A∇v⟩⟩=12(n1⋅A1∇v1−n2⋅A2∇v2),⟨⟨v⟩⟩=12(v1+v2) (2.20)

Using the expansions (2.17) and (2.18) together with (2.16

) and matching the coefficients associated with each eigenvector we obtain the interface conditions

 ⟦n⋅A∇v⟧+2α2ξ−1(⟨⟨v⟩⟩−vΓ)=0 (2.21) ⟨⟨n⋅A∇v⟩⟩+α2⟦v⟧=0 (2.22)

which are precisely the conditions used in [2].

2.2 Weak Form

Define the function spaces

 V =V1⊕V2⊕VΓ (2.23) Vi ={vi∈H1(Ωi):v=0 on ∂Ω∩∂Ωi}i=1,2 (2.24) VΓ ={vΓ∈H1(Γ):v=0 on ∂Ω∩Γ} (2.25)

and let denote the vector . We will also use the notation for functions such that , , and , with . Using partial integration on we obtain

 2∑i=1(fi,vi)Ωi =2∑i=1(−∇⋅Ai∇ui,vi)Ωi =2∑i=1(Ai∇ui,∇vi)Ωi−(ni⋅Ai∇ui,vi)∂Ωi =2∑i=1(Ai∇ui,∇vi)Ωi−(ni⋅Ai∇ui,vi−vΓ)∂Ωi−(ni⋅Ai∇ui,vΓ)∂Ωi =2∑i=1(Ai∇ui,∇vi)Ωi−(n⋅A∇u,v−vΓ)Γ−(⟦n⋅A∇u⟧,vΓ)Γ =2∑i=1(Ai∇ui,∇vi)Ωi+(B(u−uΓ),v−vΓ)Γ +(AΓ∇ΓuΓ,∇ΓvΓ)Γ−(fΓ,vΓ)Γ

Thus we arrive at the weak problem: find such that

 (2.26)

where the forms are defined by

 A(u,v) =2∑i=1(Ai∇ui,∇vi)Ωi+(AΓ∇ΓuΓ,∇ΓvΓ)Γ+(B(u−uΓ),v−vΓ)Γ (2.27) L(v) =2∑i=1(fi,vi)Ωi+(fΓ,vΓ)Γ (2.28)

2.3 Existence and Uniqueness

Introducing the energy norm

 ⫴v⫴2=2∑i=1∥v∥2H1(Ωi)+∥vΓ∥2H1(Γ)+∥v−vΓ∥2Γ (2.29)

on , we directly find using a Poincaré inequality and the Cauchy-Schwarz inequality that the form is coercive and continuous

 ⫴v⫴2≲A(v,v),A(v,w)≲⫴v⫴⫴w⫴ (2.30)

Furthermore, is a continuous functional on and it follows from the Lax-Milgram Lemma that there is a unique solution in to (2.26).

In the case considered here where is a smooth, closed interface the model problem (2.26) satisfies the elliptic regularity estimate

 ∥u1∥H2(Ω1)+∥u2∥H2(Ω2)+∥uΓ∥H2(Γ)≲∥f1∥Ω1+∥f2∥Ω2+∥fΓ∥Γ (2.31)

This follows in a straightforward manner from standard regularity theory. First note that since , and we have and using (2.3), . This means that the right hand side of (2.2) is in and hence by elliptic regularity. Considering once again (2.3) we see that in each subdomain the solution coincides with a single domain solution with a Robin condition with data in on . By the elliptic regularity of the Robin problem we can then conclude that (2.31) holds.

3 A Robust Finite Element Method

3.1 The Mesh and Finite Element Spaces

To formulate the finite element method we introduce the following notation:

• Let be a quasiuniform mesh on with mesh parameter . Define the active meshes

 Th,i={T∈Th,0:T∩Ωi≠∅}i=1,2,Th,Γ={T∈Th,0:T∩Γ≠∅} (3.1)

associated with the bulk domains , and interface , and the domains covered by the meshes

 Oh,i=∪T∈Th,ii=1,2,Oh,Γ=∪T∈Th,Γ (3.2)
• Let and define as the set of all interior faces associated with an element in .

• Let be the set of all interior faces in and .

• Let be the space of continuous piecewise linear functions on and define

 Vh,i=Vh,0|Th,ii=1,2,Vh,Γ=Vh,0|Th,Γ (3.3)

and

 Vh=Vh,1⊕Vh,2⊕Vh,Γ (3.4)

3.2 Standard Formulation

The standard finite element method takes the form: find such that

 ASh(uh,v)=L(v)∀v∈Vh (3.5)

Here the form is defined by

 ASh=A+sh (3.6)

where is a stabilization term of the form

 sh=sh,1+sh,2+sh,Γ (3.7)

with

 sh,i(v,w)=∑F∈Fh,ihF∥ζ(Ai)∥∞,F(⟦n⋅∇v⟧,⟦n∇w⟧)F,i=1,2

where denotes the maximum eigenvalue of the matrix ,

 sh,Γ(v,w) =∑F∈Fh,ΓhF∥ζ(AΓ)∥∞,F∩Γ(⟦n⋅∇v⟧,⟦n∇w⟧)Fh,Γ +∑T∈Th,Γh2K∥ζ(AΓ)∥∞,K∩Γ(nΓ⋅∇v,nΓ⋅∇w)T∩Γ.

3.2.1 Properties of the Stabilization Terms

The rationale for the design of the stabilizing terms is that they improve the stability, while remaining consistent for sufficiently smooth solutions.

Accuracy relies on the following consistency property that is immediate from the definitions above. For any function there holds for all , . For any function , such that on there holds for all .

The stability properties are well known and we collect them in the following Lemma.

Lemma 3.1

There are constants such that

 ∥∇v∥2Ai,Oh,i≲∥∇v∥2Ai,Ωi+∥v∥2sh,ii=1,2 (3.8)

and

 ∥∇Γv∥2AΓ,Oh,Γ≲∥∇Γv∥2AΓ,Γ+∥v∥2sh,Γ (3.9)

where we introduced the (semi) norm .

Proof. See [5], [6], and [23], with minor modifications to account for the varying coefficients.

Remark 3.1

Observe that the hidden constants in Lemma 3.1 depend on the variation of the and .

3.3 Robust Formulation

The stabilizing terms ensure robustness irrespective of the intersection of the fracture and the mesh. They do not counter instabilities due to degenerate . Our aim is to design a formulation which is robust in the case when the eigenvalues of degenerate. Indeed as we saw above as approaches , blows up. For clarity we recall the abstract boundary condition

 n⋅A∇v+B(v−vΓ)=0 (3.10)

where we now assume that the matrix is a positive definite symmetric matrix with eigenvalues and eigenvectors , such that and thus one or both eigenvalues may become very large or small. To handle this situation we instead enforce

 B−1n⋅A∇v+(v−vΓ)=0 (3.11)

weakly using a modified Nitsche method. This approach was originally developed in [21] where fitted finite element approximation of Robin conditions were considered.

Derivation of an Alternative Weak Form.

As before we have the identity

 L(v) =2∑i=1(Ai∇ui,∇vi)Ωi+(AΓ∇ΓuΓ,∇ΓvΓ)Γ=:A1(u,v)−(n⋅A∇u,v−vΓ)Γ (3.12) =A1(u,v)−(n⋅A∇u,v−vΓ)Γ (3.13)

where we introduced the bilinear form for brevity. To enforce the interface conditions we proceed as follows

 L(v) =A1(u,v)−(n⋅A∇u,v−vΓ)Γ =A1(u,v)+(n⋅A∇u,B−1(n⋅A∇v))Γ −(n⋅A∇u,B−1(n⋅A∇v)+(v−vΓ))Γ =A1(u,v)+(n⋅A∇u,B−1(n⋅A∇v))Γ −(n⋅A∇u,B−1(n⋅A∇v)+(v−vΓ))Γ −(B−1(n⋅A∇u)+(u−uΓ),n⋅A∇v)Γ +(B−1(n⋅A∇u)+(u−uΓ),τ(B−1(n⋅A∇v)+(v−vΓ)))Γ

where the last two terms are zero due to the interface condition and the resulting form on the right hand side is symmetric. Furthermore, is a stabilization parameter (a matrix) of the form

 τ=2∑i=1τiei⊗ei,τi=λiβλih+βi=1,2 (3.14)

where is a positive parameter and we recall that and are the eigenvalues and eigenvectors of . The parameter is chosen so that

 ∥n∥2A,∞,Γ:=2∑i=1∥ni∥2Ai,∞,Γ≲β (3.15)

where is the weighted norm over .

Remark 3.2

The choice of can be further refined as follows

 τi=λiβiλih+βii=1,2 (3.16)

with

 2∑j=1∥nj∥2Aj,∞,Γ|eij|2≲βi (3.17)

where . This approach is beneficial in situations where the components of are very different and there is a large difference between the with and .

The Robust Finite Element Method.

Find such that

 ARh(uh,v):=AR(uh,v)+sh(uh,v)=L(v)∀v∈Vh (3.18)

where

 AR(v,w) =A1(v,w)+(n⋅A∇v,B−1(n⋅A∇w))Γ (3.19) −(n⋅A∇v,B−1(n⋅A∇w)+(w−wΓ))Γ (3.20) −(B−1(n⋅A∇v)+(v−vΓ),n⋅A∇w)Γ (3.21) +(B−1(n⋅A∇v)+(v−vΓ),τ(B−1(n⋅A∇w)+(w−wΓ)))Γ. (3.22)

It follows by the design of that for a sufficiently smooth exact solution of the problem (2.26) there holds

 A(u,v)=AR(u,v)=L(v),∀v∈(V∩H2(Ω1∪Ω2∪Γ)+Vh. (3.23)

As a consequence we immediately get the Galerkin orthogonality

Lemma 3.2

Let be the solution of (2.26) and the solution of (3.18) then there holds

 AR(u−uh,v)=sh(uh,v)∀v∈Vh. (3.24)

Proof. The proof follows by combining (3.23) and (3.18).

4 Error Estimates

4.1 The Energy Norm

We introduce the energy norm

 ⫴v⫴2h=2∑i=1∥∇vi∥2Ai,Ωi+h∥∇vi∥2Ai,Γ+∥v∥2sh+∥∇ΓvΓ∥2AΓ,Γ+∥v−vΓ∥2τ,Γ (4.1)

where is the weighted norm over the set .

4.2 Interpolation Error Estimates

We begin by introducing the interpolation operators and derive the basic approximation error estimates. Then collecting the estimates we show an interpolation error estimate in the energy norm (

4.1). Since the stabilization operator acts on the finite element solution outside its physical domain of definition, we must make sense of the solution it approximates also outside its physical domain of definition. We will show below show how this can be done using extensions from the physical geometry.

The Scott-Zhang Interpolant.

Given a mesh covering a domain and the space of piecewise linear continuous finite elements , the standard Scott-Zhang interpolation operator satisfies the element wise estimate

 ∥v−πh,i,SZv∥Hm(T)≲h2−m∥v∥H2(N(T)),m=0,1 (4.2)

where is the set of all elements in that share a node with . Note also that the Scott-Zhang interpolant preserves homogeneous boundary conditions exactly. See [26] for further details.

Bulk Domain Fields.

It is shown in [27, Section 2.3, Theorem 5] that there is an extension operator , not dependent on , which is stable in the sense that

 ∥Eivi∥Hs(Rd)≲∥vi∥Hs(Ωi) (4.3)

We define the interpolation operator by

 πh,ivi=πh,i,SZEiv (4.4)

where is the Scott-Zhang interpolant and we recall that is the domain covered by . We then have the error estimate

 ∥vi−πh,iv∥Hm(Ωi)≲h2−m∥vi∥H2(Ωi)m=0,1 (4.5)

Proof. Using the notation we obtain

 ∥ρi∥Hm(Ωi)≲∥ρi∥Hm(Oh,i)≲h2−m∥Eiui∥H2(Oh,i)≲h2−m∥ui∥2H2(Ωi)

where we used the fact that , the interpolation error estimate (4.2), and finally the stability (4.3) of the extension operator .

Interface Field.

Let be the closest point mapping from the tubular neighborhood to , which is well defined for all for some . Define the extension operator by . Since is smooth we have the stability estimate

 ∥EΓvΓ ∥Hs(Uδ(Γ))≲δ1/2∥vΓ∥Hs(Γ). (4.6)

Observe also that since by construction, and then assuming in (4.6) we see that

 sh,Γ(EΓvΓ,w)=0,∀w∈Vh,Γ+H32+ϵ(Oh,Γ). (4.7)

To define the interpolant we first let

 Th,δ,Γ={T∈Th,0:T∩Uδ(Γ)≠∅},Oh,δ,Γ=∪T∈Th,δ,ΓT (4.8)

for . Then and there are such that and

 Uδ(Γ)⊂Oh,δ,Γ⊂Uδ′(Γ)⊂Uδ0(Γ) (4.9)

for all , with small enough. We let and define by

 πh,ΓvΓ=(πh,δ,Γ,SZEΓvΓ)|Oh,Γ (4.10)

where and is the Scott-Zhang interpolant. We have the error estimate

 ∥v−πh,Γv∥Hm(Γ)≲h2−m∥v∥H2(Γ)m=0,1 (4.11)

Proof. Using the trace inequality

 ∥v∥2Γ≲δ−1∥v∥2Uδ(Γ)+δ∥∇v∥2Uδ(Γ)v∈H1(Uδ(Γ)) (4.12)

where the hidden constant is independent of , we obtain

 ∥∇mΓρ∥2Γ ≲δ−1∥∇mρ∥2Uδ(Γ)+δ∥∇m+1ρ∥2Uδ(Γ) ≲δ−1∥∇mρ∥2Oh,δ,Γ+δ∥∇m+1ρ∥2Oh,δ,Γ ≲δ−1h2(2−m)∥∇2EΓv∥2Oh,δ,Γ+δh2(1−m)∥∇2E