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.


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 d − 1 in a d 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.

Governing Equations
Let Ω be a convex polygonal domain in R d , d = 2 or 3, with boundary ∂Ω and exterior unit normal n. Let Γ be a smooth embedded interface in Ω, which partitions Ω into two subdomains Ω 1 and Ω 2 with exterior unit normals n 1 and n 2 . We assume that Γ is a closed surface without boundary residing in the interior of Ω, more precisely we assume that there is δ 0 > 0 such that the distance between Γ and ∂Ω is larger than δ. We consider for simplicity the case with homogeneous Dirichlet conditions on ∂Ω.
The problem takes the form: find u i : Here the jump (or sum) of the normal fluxes is defined by In the interface condition (2.3), B is a 2 × 2 symmetric matrix with eigenvalues λ i such that λ i ∈ [0, ∞) and we used the notation and thus in component form (2.3) reads The coefficients A 1 , A 2 , are smooth uniformly positive definite symmetric d × d matrices, A Γ is smooth tangential to Γ and uniformly positive definite on the tangent space of Γ, so that where denotes less or equal up to a constant. Finally we assume f i ∈ L 2 (Ω i ) and f Γ ∈ L 2 (Γ).
Remark 2.1 Several generalizations are possible on the external boundary. For instance, we may let the interface intersect the boundary of Ω. In this case we let ν denote the unit exterior conormal to Γ ∩ ∂Ω, i.e. ν is tangent to Γ and normal to ∂Ω ∩ Γ, and we assume that ν · n ≥ c > 0 for some constant c so that the interface is transversal to ∂Ω. We may then enforce the Dirichlet condition u Γ = g Γ on ∂Ω ∩ Γ (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 t takes the form where n Γ is a unit normal vector field to Γ, U t/2 (Γ) is the set of points with distance less than t/2 to Γ, v e denotes the extension of a function v on Γ that is constant in the normal direction, A Γ is the tangential tangential permeability tensor, and finally a Γ,n is the permeability across the interface. Also assuming that f = f e Γ and u = u e in U t/2 (Γ), the equation on the interface (2.2) may be modelled as follows Note that the last term on the right hand side does not scale with t 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 where ξ and α are parameters. The parameter α is related to physical properties of the interface as follows where a Γ,n is the permeability coefficient across the interface Γ and d is the thickness of the interface, see (3.8) in [24] In matrix form we obtain which leads to We note that we have the eigen pairs with the corresponding eigen vectors defined by and thus B is positive definite for ξ > 1/2, singular for ξ = 1/2, and indefinite for ξ < 1/2. It is therefore natural to consider the case when α > 0 and ξ > 1/2. We remark that when α tends to infinity both eigenvalues tend to infinity and when ξ tends to 1/2 from above one eigenvalue tends to infinity. It is therefore important to construct a method which is robust in the full range λ i ∈ (0, ∞) 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 where the jumps and averages of the bulk fields across the the interface are defined by 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 which are precisely the conditions used in [2].

Weak Form
Define the function spaces Thus we arrive at the weak problem: where the forms are defined by

Existence and Uniqueness
Introducing the energy norm on V , we directly find using a Poincaré inequality and the Cauchy-Schwarz inequality that the form A is coercive and continuous Furthermore, L is a continuous functional on V and it follows from the Lax-Milgram Lemma that there is a unique solution in V to (2.26).
In the case considered here where Γ is a smooth, closed interface the model problem (2.26) satisfies the elliptic regularity estimate This follows in a straightforward manner from standard regularity theory. First note that since This means that the right hand side of (2.2) is in L 2 and hence u Γ ∈ H 2 (Γ) 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 H 1 2 (Γ) on Γ. By the elliptic regularity of the Robin problem we can then conclude that (2.31) holds.

The Mesh and Finite Element Spaces
To formulate the finite element method we introduce the following notation: • Let T h,0 be a quasiuniform mesh on Ω with mesh parameter h ∈ (0, h 0 ].
Define the active meshes associated with the bulk domains Ω i , i = 1, 2, and interface Γ, and the domains covered by the meshes • Let V h,0 be the space of continuous piecewise linear functions on T h,0 and define and

Standard Formulation
The standard finite element method takes the form: Here the form A S h is defined by where s h is a stabilization term of the form where ζ(X) denotes the maximum eigenvalue of the matrix X,

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 v ∈ H The stability properties are well known and we collect them in the following Lemma.
Lemma 3.1 There are constants such that and 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 A i and A Γ .

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 B. Our aim is to design a formulation which is robust in the case when the eigenvalues of B degenerate. Indeed as we saw above as ξ approaches 1/2, λ 1 blows up. For clarity we recall the abstract boundary condition where we now assume that the matrix B is a positive definite symmetric 2 × 2 matrix with eigenvalues λ i and eigenvectors e i , such that λ i ∈ (0, ∞) and thus one or both eigenvalues may become very large or small. To handle this situation we instead enforce 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 where we introduced the bilinear form A 1 for brevity. To enforce the interface conditions we proceed as follows where the last two terms are zero due to the interface condition and the resulting form on the right hand side is symmetric. Furthermore, τ is a stabilization parameter (a 2 × 2 matrix) of the form where β is a positive parameter and we recall that λ i and e i are the eigenvalues and eigenvectors of B. The parameter β is chosen so that The choice of τ i can be further refined as follows where e i = [e i1 e i2 ] T . This approach is beneficial in situations where the components of e i are very different and there is a large difference between the n j 2 Aj ,∞,Γ with j = 1 and j = 2.
It follows by the design of A R that for a sufficiently smooth exact solution u ∈Ṽ of the problem (2.26) there holds As a consequence we immediately get the Galerkin orthogonality Lemma 3.2 Let u ∈Ṽ be the solution of (2.26) and u h ∈ V h the solution of (3.18) then there holds Proof. The proof follows by combining (3.23) and (3.18).

The Energy Norm
We introduce the energy norm where w 2 ψ,ω = ω ψw 2 is the ψ weighted L 2 norm over the set ω.

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. where N (T ) is the set of all elements in T h,i that share a node with T . 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 s ≥ 0, which is stable in the sense that We define the interpolation operator π h,i : where π h,i,SZ : Proof. Using the notation where we used the fact that Ω i ⊂ O h,i , the interpolation error estimate (4.2), and finally the stability (4.3) of the extension operator E i .
Interface Field. Let p Γ : U δ (Γ) → Γ be the closest point mapping from the tubular neighborhood U δ (Γ) := {x : dist(x, Γ) < δ} to Γ, which is well defined for all δ ∈ (0, δ 0 ] for some δ 0 > 0. Define the extension operator E Γ : Since Γ is smooth we have the stability estimate Observe also that since n Γ · ∇E Γ v Γ = 0 by construction, and then assuming s > 3/2 in (4.6) we see that To define the interpolant we first let for all h ∈ (0, h 0 ], with h 0 small enough. We let V h,δ,Γ = V h,0 | O h,δ,Γ and define π h,Γ : where δ ∼ h and π h,δ,Γ,SZ : Proof. Using the trace inequality where the hidden constant is independent of δ, we obtain where we used (4.9), the interpolation error estimate (4.2), the stability (4.6) of the extension operator E Γ , and the fact that δ ∼ δ ∼ h.
We define the interpolation operator π h : V → V h as follows Lemma 4.1 There is a constant not dependent on the matrix B, in the inter- Proof.
Let v − π h v = ρ be the interpolation error. Using the triangle inequality and (4.20), with δ ∼ h and the desired estimate follows. Here we used the bounds To prove (4.15) we employ the trace inequality with δ ∼ h, to estimate the interface terms involving ρ i as follows For (4.16) we apply the elementwise trace inequality In a similar way we prove (4.17), see [6] and [23] for details.

Continuity and Coercivity
We start with a lemma collecting some useful estimates for expressions involving the stabilization parameter τ and then we prove continuity and coercivity of the form A h .

Lemma 4.2
The following estimates related to the stabilization parameter τ hold Proof. First we recall that for any symmetric matrix D it holds where γ i are the eigenvalues of D. To prove (4.18) we write B in terms of its eigenvalues λ i and eigenvectors e i , 4.22) and using the definition (3.14) of τ we obtain the identity Here we have the following estimate of the eigenvalues which in view of (4.21) completes the verification of (4.18). Next, for (4.19) we have and which proves (4.19). The final bound (4.20) is a direct consequence of the definition of τ and the estimate There is a constant independent of the eigenvalues of B, such that for all v, w ∈Ṽ + V h , There is a constant independent of the eigenvalues of B, such that for Proof. (4.28). Starting from the definition (3.19), expanding the terms in A R , and using Cauchy-Schwarz we obtain Using the estimates (4.18)-(4.19) we obtain where we used the bound β −1 n A,∞,Γ 1, see (3.15). By the Cauchy-Schwarz inequality we have s h (v, w) v h w h .
(4.29). To prove the coercivity we have the identity We conclude the argument as usual by estimating the negative terms as follows We conclude that which completes the proof.

A priori Error Estimates
In this section we prove error estimates for the approximate solution u h .
Theorem 4.1 Let u ∈Ṽ be the solution of (2.26) and u h ∈ V h be the solution of (3.18). Then there is a constant not dependent on the matrix B in the interface condition (2.3) such that Proof. First we decompose the error in the approximation error and the discrete error u − u h = u − π h u + π h u − u h and note that by the triangle inequality The first term on the right hand side is bounded by (4.14). For the second term on the right hand side, using coercivity (4.29), Galerkin orthogonality (3.24), and continuity (4.28) we obtain In the last inequality we used that if u e := (Eu 1 , Eu 2 , E Γ u Γ ) ∈Ṽ then where we used the interpolation error estimate (4.14). To conclude we apply the regularity estimate (2.31).
Proof. Using the triangle inequality we see that The second term on the right hand side is bounded by the arguments of Theorem 4.1. For the first term on the right hand side recall that by the consistency properties of s h and the construction of π h u there holds We conclude the proof by applying (4.16)-(4.17).
The following error estimate in the L 2 -norm also holds Theorem 4.2 Let u ∈Ṽ be the solution of (2.26) and u h ∈ V h be the solution of (3.18). Then there holds Proof. For ψ Ω ∈ L 2 (Ω) and ψ Γ , such that ψ Ω Ω + ψ Γ Γ = 1 let ϕ := (ϕ 1 , ϕ 2 , ϕ Γ ) ∈ V be the weak solution to Then by (2.31) we have Let e = (u 1 − u 1,h , u 2 − u 2,h , u Γ − u Γ,h ) and observe that using (3.23), (4.59) Applying now the Galerkin orthogonality (3.24) we see that By the continuity (4.28) we can bound the right hand side, Then applying the approximation (4.14) and the regularity (4.58) we have We conclude by applying Theorem 4.1 in the right hand side and taking the supremum over the functions (ψ Ω , ψ Γ ) in L 2 .

Numerical Examples
In this Section we illustrate the properties of the model and method by presenting some numerical results. In all examples we used β = 10 as a stabilization parameter.

Convergence and Robustness with Respect to Conditioning
We consider a simple example with known exact solution: the domain (0, 1) × (0, 1) is cut in half along a vertical line at x = 1/2. We take A 1 = A 2 = A Γ = I and choose a problem with exact solution u = x(1 − x)y(1 − y). This solution corresponds (without coupling) to the source terms Since the normal derivative of the exact solution is zero at x = 1/2, it does not contribute to the source term on the interface. We choose f Γ = 1/2 corresponding to u Γ = y(1 − y)/4, and thus u Γ = u at x = 1/2. We apply zero Dirichlet boundary conditions on u and on u Γ (imposed on the boundary of the band of elements intersected by (1/2, y)). This is now the solution of (2.1)-(2.4) independent of B. A sample discrete solution is shown in Fig. 1 with u Γ shown as a red line. We did not impose gradient jumps on the band (second term in s h,Γ ), normal stabilization proved sufficient in this case. In Figs. 2-5 we show convergence for different choices of parameters in different norms. The method is completely robust with optimal convergence for all choices. In Fig. 6 we show the variation of the condition number ((left) with respect to mesh refinement and choice of α. The condition number is O(h −2 ) as expected and does not grow with α. We also show (right) the effect of using the non-robust method (3.5) which shows a linear dependece on α on a fixed mesh, while no such effect is present in the robust method. This robustness is important since α physically depends on the crack width [24] which is expected to be small.
To show the effect of stabilization, we chose to scale s h,i and s h,Γ by a parameter γ. We retained β = 10 and normal stabilization on the band. In Figs. 7-9 we show the effect of the parameter γ. When γ = 0 the jump in diffusion on the interface leads to slight instabilities at y = 1/4 and y = 3/4 which are visible to the eye. These are less pronounced for γ = 10 −2 and not significant for γ = 1. The overall solution agrees with that of [24].

Physical Effect of Crack Width
Finally, we show the effect of the crack width with respect to the solution. We used a domain (0, 1) × (0, 1) with a quarter circle crack, shown on the computational mesh in Fig. 10. The data were A 1 = 5 I (inside the circle) A 2 = I (outside the circle) and a Γ = 0.1 with definitions as in Example 5.2. Dirichlet data u = 1 at x = 1 and u = 0 at x = 0 were used (also on the band) and homogeneous Neumann data on the remaining boundaries. In Figs. 11-13 we see the effect of decreasing the interface width by one order of magnitude between figures. The solution rapidly tends to a continuous state.