Low Regularity Estimates for CutFEM Approximations of an Elliptic Problem with Mixed Boundary Conditions

07/06/2020
by   Erik Burman, et al.
0

We show error estimates for a cut finite element approximation of a second order elliptic problem with mixed boundary conditions. The error estimates are of low regularity type where we consider the case when the exact solution u ∈ H^s with s∈ (1,3/2]. For Nitsche type methods this case requires special handling of the terms involving the normal flux of the exact solution at the the boundary. For Dirichlet boundary conditions the estimates are optimal, whereas in the case of mixed Dirichlet-Neumann boundary conditions they are suboptimal by a logarithmic factor.

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1 Introduction

In this paper we will consider the finite element approximation of the Poisson problem with mixed boundary conditions under minimal regularity assumptions. Let be a domain in with smooth boundary , which is decomposed into two subdomains and such that and . Consider the problem: find such that

in (1)
on (2)
on (3)

where , and satisfy the following bound for ,

(4)

Here and below we used the notation for , with a positive constant.

For the approximation of the problem we apply a Cut Finite Element Method (CutFEM). In CutFEM the boundary is allowed to cut through the computational cells in an (almost) arbitrary way and stabilization terms are added in the vicinity of the boundary to ensure that the method is coercive and that the resulting linear system of equations is invertible.

In previous work on fictitious domain finite element methods see [BH12, BCHLM15], error estimate were shown under the assumption that with . The objective of the present work is to relax this regularity requirement. Indeed, we show an a priori error estimates in the energy norm, requiring only that , where , and is in on some arbitrarily thin neighborhood of the Dirichlet boundary . Since the test functions in the Nitsche formulation of the Dirichlet condition are not zero on , we will also have to choose the Neumann data in a slightly smaller space than . We focus our attention on the effects of rough data in CutFEM. We assume that the boundary of the domain is smooth and that we can evaluate integrals on the intersection of simplices and the domain and its boundary, exactly. Estimation of the error resulting from approximation of the domain can be handled using the techniques in [BHLZ16].

The study of the convergence of nonconforming methods for the approximation of solution with low regularity has received increasing interest since the seminal paper by Gudi [Gudi10]. In that work optimal convergence for low regularity solutions were obtained using ideas from a posteriori error analysis, where the error is upper bounded by certain residuals of the discrete solution. These residuals are then shown to lead to optimal upper bounds using the discrete local efficiency bounds. A similar approach was used by Lüthen et al. [LJS17] for a generalised Nitsche’s method on fitted meshes. This approach does however not seem to be suitable for the case of cut finite element method since for cut elements the local efficiency bounds are not robust with respect to the mesh boundary intersection. Instead, in the spirit of [EG18]

, we use a version of duality pairing to handle the term involving the normal flux of the interpolation error. This is made more delicate by the presence of mixed boundary conditions. Indeed to include this case in the analysis we introduce a regularized bilinear form and use the solution to the regularized problem as pivot in the error estimate. The regularization gives rise to a logarithmic factor. Observe that this is due to the mixed boundary conditions. For pure Dirichlet conditions or pure Neumann conditions the analysis results in optimal error bounds for

.

The paper is organized as follows: In Section 2 we introduce the functional framework for the model problem and formulate the finite element method and in Section 3 we derive the error estimates.

2 Weak Formulation and the Finite Element Method

Since we consider low regularity solutions of a problem with mixed boundary condition we must be careful with the fractional Sobolev spaces for the traces of the functions. In this section we first introduce the notations and definitions for the functional analytical framework, leading to the weak formulation of (1)–(3). Then we introduce the cut finite element method for the approximation of the weak solutions.

2.1 Function Spaces

Let and let denote the usual Sobolev spaces on . Define

(5)
(6)

and for , define

(7)
(8)

and the subspace

(9)

Then and . Next define the dual spaces

(10)
(11)
(12)

consisting of functionals with duality pairing and norm

(13)

with . We will use the simplified notation for the duality pairing. Note that for each we may define by and thus .

2.2 Weak Formulation

The problem (1)–(3) can be cast on weak form: find such that

(14)

where

(15)

and, for each ,

(16)

For , , and , there exists a unique weak solution to (14) and the following elliptic regularity estimate holds, ,

(17)

We refer to Savaré [Sav97] for a precise characterization of the regularity for the mixed problem.

2.3 The Normal Flux

The normal flux , where is the exterior unit normal, plays an important role in what follows. For , with , it can be defined by the identity

(18)

Observe that in the finite element method we work with weakly enforced boundary conditions and therefore we will not have test functions that vanish on , i.e. the test functions are not in , and herefore we will consider boundary data such that

(19)

where the Neumann data is chosen in the smaller space , compared to the strong formulation and corresponding weak form (14). We will also assume that the source term is square integrable over some (arbitrary thin) neighbourhood of the boundary , see (59) below.

2.4 Finite Element Method

To define the cut finite element method let be a polygonal domain such that and let be a family of quasiuniform meshes covering with mesh parameter . For a subset , define the submesh of elements intersecting , by , and let be the so called active mesh. Let be the conforming finite element space defined on consisting of piecewise affine functions and define . Define the bilinear forms

(20)
(21)
(22)

with positive parameters and , the set of interior faces in associated with an element that intersects the boundary, and the jump in the normal flux at face shared by elements and is defined by

(23)

where and is the unit exterior normal.

Define the finite element method: find such that

(24)

3 Error Analysis

In this section we will derive the error estimates, here as usual the consistency of the method is of essence. However, for solutions with low regularity this is delicate in the case of mixed boundary conditions. Indeed, in the low regularity case, (18) is not sufficient to make sense of the term for approximation purposes, since the division on and necessarily results in a boundary integral over one of the subdomains that has to be lifted in some other fashion. This is problematic since the solution is not regular enough to allow for the usual trace inequality arguments. To handle this difficulty we introduce a regularized finite element formulation (for analysis purposes only), where a smooth weight function is introduced and the problematic term is replaced by

(25)

The regularized method has a consistency error that can be controlled by sharpening the cut off function .

3.1 Outline

We shall prove low regularity energy norm error estimates using the following approach:

  • Similarly to [Gudi10] we estimate the error in a norm which does not involve the norm of the normal trace of the gradient.

  • For the case of mixed boundary conditions, we introduce a regularized bilinear form and the corresponding (nonconsistent) finite element method. The regularization takes the form of a weight function smoothing the transition from the Dirichlet to the Neumann boundary condition in the first boundary integral of the form , see equation (20). In the regularized norm we can use a version of duality in an neighborhood of .

  • The total error is estimated using a Strang type argument. The error is divided into the approximation error, the discrete error between an interpolant and the finite element solution of the regularized formulation and finally the regularization error between the regularized and standard finite element solutions.

3.2 The Cut Off Function

Key to the regularized problem is the design of the weight function, with support in a neighbourhood of . This function takes the value on and decays smoothly to zero in an neighbourhood of and into the domain away from the boundary. This way it plays the role of a cut off, that localizes the boundary integral to , while the form remains well defined for low regularity solutions. In order to define the cut off function we introduce some notation.

Notation.

For , , let be the distance function and let be the closest point mapping. In the case we drop the subscript. For , define the -neighbourhood of ,

(26)

Then there is such that the closest point mapping maps every to precisely one point at . We also define -neighbourhood of and as follows

(27)

Let be the smooth interface separating and and let be the unit conormal to exterior to and tangent to . See Figure 1. For let

(28)
(29)
(30)

Note that is a bijection for all . Let

(31)

be the tubular neighborhood of in , and assume that with small enough to guarantee that the closest point mappings are well defined for all , and let

(32)

Define

(33)

with for and , see Figure 2. Defining, for ,

(34)

where is the closest point mapping associated with , we have . Note that , which is a subset of the dimensional normal space to the dimensional tangent space of at . In the case , consists of distinct points and in that case , for small enough. Finally, let

(35)
Figure 1: Left: the Dirichlet boundary , the Neumann boundary , the interface , and the tubular neighborhood . Right: the set .
Figure 2: Left: Close up of the set including . Right: The set and .

The Cut Off Function.

We will below take and with . Let be smooth such that

(36)

Observe that in the definition above denotes the projection of the gradient on the tangent plane of . By the construction of , is bounded and depends only on , and the regularity of .

The cut off function satisfies the following estimate

(37)

and with

(38)

for we obtain

(39)
Proof.

Using the bounds for we obtain

Estimate (39) follows directly from the definition of and . ∎

3.3 The Regularized Problem

For define the regularized form

(40)

and define . We will show that the mapping is continuous for small enough, see Lemma 3.4 below for details.

For define the regularized finite element method: find such that

(41)

This method is not consistent, but we have the identity

(42)
(43)

since using Green’s formula gives

(44)
(45)
(46)
(47)

where we used the fact that on to conclude that

(48)
(49)

3.4 Properties of the Bilinear Forms

We here summarize the basic results on the bilinear forms and conclude with a proof of existence, uniqueness, and stability of the finite element solutions.

Inverse and Trace Inequalities.

Let us recall some inverse and trace inequalities. Here denotes the set of polynomials of degree less than or equal to on the simplex .

  • Inverse inequalities (see [DiPE12, Section 1.4.3]),

    (50)

    and

    (51)
  • Trace inequalities (see [DiPE12, Section 1.4.3]),

    (52)

    and

    (53)
  • Inverse trace inequality on cut elements. For a simplex such that , there holds

    (54)

Stabilization Estimates.

For any two elements and in , sharing a face , we have the estimate

(55)

Repeated use of (55) leads to

(56)

For sets such that , we may also derive the estimate

(57)

where denotes the interior faces of .

The Energy Norm.

We equip the finite element space with the energy norm

(58)

where . In order to have the normal flux well defined on the Dirichlet boundary we assume that

(59)

where we recall, see (35), that for all regularization parameters . The stabilization form is not defined on , due to the low regularity, and therefore we equip with the weaker energy norm

(60)

There is constant such that for all , , and ,

(61)

where we use the norm , which does not include the stabilization, on .

Proof.

To verify this estimate we start from the definition (40) of the regularized form and using the Cauchy Schwarz inequality we get

(62)
(63)
(64)

We estimated , with , using the inverse inequality

(65)

where we first used the inverse trace inequality (54) and then the stabilization estimate (56). ∎

We will now prove a bound on the error introduced by replacing by its regularized counterpart . There is a constant such that for all , and with ,

(66)
Proof.

Using the definitions (20) and (40) of the forms and we obtain

(67)
(68)
(69)

where we used the fact that , see (32). To estimate we proceed in the same way as in (65), we first use an inverse estimate and then the stablization (56),

(70)

Next to estimate we pass over to the norm in order to extract an factor and then we use suitable inverse bounds to pass to the energy norm.

(71)
(72)
(73)
(74)
(75)
(76)

Here is a slightly larger patch of elements such that the dimensional measure of its intersection with the Dirichlet boundary satisfies ,which allows us to utilize the control available in at the Dirichlet boundary and to employ a Poincaré inequality in (74), see the appendix in [BHL18]. The patch does not in general satisfy and therefore it is enlarged by adding a suitable number of face neighboring elements in . In the last step (76) we also used the stabilization (56). Note that due to the assumption that with it follows from shape regularity that there is a uniform bound on the number of elements in . ∎

Lemma 3.4 is instrumental for the coercivity that we prove next. For large enough and , the forms , , with small enough, are coercive

(77)
Proof.

First we note that is coercive using standard techniques together with the inverse estimate (65). Next using the bound (66) of Lemma 3.4, we obtain

where in the last step we choose with and small enough. ∎

Using Lax-Milgram we conclude that for each , there is a unique solution to the regularized problem (41) such that

(78)

3.5 Technical Lemmas

In this section we collect some technical results that will be useful in the analysis. More precisely we start with four technical lemmas before proving Lemma 3.5 which is used to estimate the problematic term