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 , 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 , virtual element methods  and high order hybridised methods .
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 , where a stabilized Lagrange multiplier method is considered for the interface coupling and  where a mixed method is considered for the Darcy’s equations both in the bulk and on the surface.
The upshot here, compared to  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 , that was also recently applied for the numerical modelling in . 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  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  and .
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 ;
overlapping meshes ; coupled bulk-surface problems
[11, 7, 10] and ; mixed dimensional problems  , and surface partial
, and surface partial differential equations[6, 25]. For a general introduction to cut finite element methods we refer to .
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
Here the jump (or sum) of the normal fluxes is defined by
and thus in component form (2.3) reads
The coefficients , , are smooth uniformly positive definite symmetric matrices, is smooth tangential to and uniformly positive definite on the tangent space of , so that
where denotes less or equal up to a constant. Finally we assume and .
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 ) or some other standard boundary condition.
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
where is a unit normal vector field to is the tangential tangential permeability tensor, and finally
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 finallyis the permeability across the interface. Also assuming that and in , 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 since it accounts for flow into the crack from the bulk domains.
We comment on how our interface condition (2.3) relates to the condition in  and later reformulated, see , in terms of averages and jumps of the bulk fields across the interface. The interface conditions in , equations (3.18) and (3.19), takes the form
where and are parameters. The parameter is related to physical properties of the interface as follows
where is the permeability coefficient across the interface and is the thickness of the interface, see (3.8) in  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 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  we first note that we have the expansions
where the jumps and averages of the bulk fields across the the interface are defined by
which are precisely the conditions used in .
2.2 Weak Form
Define the function spaces
and let denote the vector . We will also use the notation for functions such that , , and , with . Using partial integration on we obtain
Thus we arrive at the weak problem: find such that
where the forms are defined by
2.3 Existence and Uniqueness
Introducing the energy norm
on , we directly find using a Poincaré inequality and the Cauchy-Schwarz inequality that the form is coercive and continuous
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
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
associated with the bulk domains , and interface , and the domains covered by the meshes
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
3.2 Standard Formulation
The standard finite element method takes the form: find such that
Here the form is defined by
where is a stabilization term of the form
where denotes the maximum eigenvalue of the matrix ,
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.
There are constants such that
where we introduced the (semi) norm .
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
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
weakly using a modified Nitsche method. This approach was originally developed in  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 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, is a stabilization parameter (a matrix) of the form
where is a positive parameter and we recall that and are the eigenvalues and eigenvectors of . The parameter is chosen so that
where is the weighted norm over .
The choice of can be further refined as follows
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
It follows by the design of that for a sufficiently smooth exact solution of the problem (2.26) there holds
As a consequence we immediately get the Galerkin orthogonality
4 Error Estimates
4.1 The Energy Norm
We introduce the energy norm
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 (
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
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  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
We define the interpolation operator by
where is the Scott-Zhang interpolant and we recall that is the domain covered by . We then have the error estimate
Proof. Using the notation we obtain
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
Observe also that since by construction, and then assuming in (4.6) we see that
To define the interpolant we first let
for . Then and there are such that and
for all , with small enough. We let and define by
where and is the Scott-Zhang interpolant. We have the error estimate
Proof. Using the trace inequality
where the hidden constant is independent of , we obtain