Stabilization of Higher Order Cut Finite Element Methods on ... - arXiv

arXiv:1710.03343v1 [math.NA] 9 Oct 2017

Stabilization of Higher Order Cut Finite Element Methods on Surfaces∗ Mats G. Larson

†

Sara Zahedi

‡

Abstract We develop and analyze a stabilization term for cut finite element approximations of an elliptic second order partial differential equation on a surface embedded in Rd . The new stabilization term combines properly scaled normal derivatives at the surface together with control of the jump in the normal derivatives across faces and provides control of the variation of the finite element solution on the active three dimensional elements that intersect the surface which may be used to show that the condition number of the stiffness matrix is O(h−2 ), where h is the mesh parameter. The stabilization term works for higher order elements and the derivation of its stabilizing properties is simple. We also present numerical studies confirming our theoretical results.

1

Introduction

CutFEM for Surface Partial Differential Equations. CutFEM (Cut Finite Element Method) provides a new high order finite element method for solution of partial differential equations on surfaces. The main approach is to embed the surface into a three dimensional mesh and to use the restriction of the finite element functions to the surface as trial and test functions. More precisely, let Γ be a smooth closed surface embedded in R3 with exterior unit normal n and consider the Laplace-Beltrami problem: find u ∈ H 1 (Γ) such that a(u, v) = l(v) ∀v ∈ H 1 (Γ) (1.1) The forms are defined by Z

Z ∇Γ w · ∇Γ v,

a(w, v) = Γ

l(v) =

fv

(1.2)

Γ

∗

This research was supported in part by the Swedish Foundation for Strategic Research Grant No. AM13-0029, the Swedish Research Council Grants Nos. 2013-4708 and 2014-4804, and the Swedish Research Programme Essence † Department of Mathematics and Mathematical Statistics, Ume˚ a University, SE-90187 Ume˚ a, Sweden ‡ Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden

1

where ∇Γ is surface gradient and f ∈ L2 (Γ) a given function with average zero. Let now Γ be embedded into a polygonal domain Ω equipped with a quasiuniform mesh Th,0 and let Γh be a family of discrete surfaces that converges to Γ in an appropriate manner. Let the active mesh Th consist of all elements that intersect Γh . Let Vh be the space of continuous piecewise polynomial functions on Th with average zero on Γh . The basic CutFEM takes the form: find uh ∈ Vh such that ah (uh , v) = lh (v) where

∀v ∈ Vh

Z

Z ∇Γ w · ∇Γ v,

ah (w, v) =

(1.3)

lh (v) =

Γh

fh v

(1.4)

Γh

and fh is a suitable representation of f on Γh . In the context of surface partial differential equations this approach, also called trace finite elements, was first introduced in [21] and has then been developed in different directions including higher order approximations in [24], discontinuous Galerkin methods [5], transport problems [23], embedded membranes [9], coupled bulk-surface problems [7] and [15], minimal surface problems [8], and time dependent problems on evolving surfaces [18], [20], [22], and [25]. We also refer to the overview article [2] and the references therein. Previous Work on Preconditioning and Stabilization. The basic CutFEM method (1.3) manufactures a potentially ill conditioned linear system of equations and stabilization or preconditioning is therefore needed. In [19] it was shown that diagonal preconditioning works for piecewise linear approximation and in [24] it was shown that also quadratic elements can be preconditioned if the full gradient is used instead of the tangential gradient in the bilinear form ah , see also [10] where this formulation was proposed. In [3] a stabilized version of (1.3) was introduced. The method takes the form ah (uh , v) + sh (uh , v) = l(v)

∀v ∈ Vh

(1.5)

where sh is a stabilization form which is added in order to ensure stability of the method. More precisely we show that there is a constant such that for all v ∈ Vh , kvk2L2 (Th ) . h(ah (v, v) + sh (v, v))

(1.6)

Using (1.6) and the properties of ah we can show that the condition number κ of the stiffness matrix satisfies the optimal bound κ . h−2

(1.7)

In [3] only piecewise linear finite elements were considered and the stabilization term was defined by sh (w, v) = cF ([DnF w], [DnF v])Fh 2

(1.8)

where [DnF v] is the jump in the normal derivative of v at the face F , Fh is the set of internal faces (i.e. faces with two neighbors) in the active mesh Th , and cF > 0 is a stabilization parameter. Optimal order bounds on the error and condition number were established. Furthermore, for linear elements the so called full gradient stabilization sh (v, w) = h(∇v, ∇w)Th

(1.9)

was proposed and analyzed in [6] as a simpler alternative, and for higher order elements the normal gradient stabilization sh (w, v) = hα (nh · ∇w, nh · ∇v)Th

α ∈ [−1, 1]

(1.10)

was proposed and analysed independently in [4] and [14]. The restrictions on α essentially comes from a lower bound which implies that optimal order a priori error estimates holds and an upper bound which implies that the desired stability estimate (1.6) holds. Note that stronger control than (1.6) is sometimes demanded, for instance in cut discontinuous Galerkin methods [5], which may lead to stronger upper restrictions on α. New Contributions. In this work we consider stabilization of higher order elements using a combination of face stabilization and normal derivative stabilization at the actual surface. More precisely the stabilization term takes the form sh (v, w) = sh,F (v, w) + sh,Γ (v, w)

(1.11)

with sh,F (v, w) =

p X

cF,j h2(j−1+γ) ([Dnj F v], [Dnj F w])Fh

(1.12)

cΓ,j h2(j−1+γ) (Dnj h v, Dnj h w)Γh

(1.13)

j=1 p

sh,Γ (v, w) =

X j=1

where γ ∈ [0, 1] is a parameter, [Dnj F v] is the jump in the normal derivative of order j across the face F , and cF,j > 0, cΓ,j > 0, are stabilization constants. This stabilization term with γ = 1 has also been used in a high order space-time CutFEM on evolving surfaces [25]. We note that also for this stabilization term there is a certain range of scaling with the mesh parameter h, and in fact the following stronger version of (1.6) holds kvk2L2 (Th ) + h2γ k∇vk2L2 (Th ) . h(ah (v, v) + sh (v, v))

(1.14)

where we note that we get stronger control of the gradient for smaller γ ∈ [0, 1], which corresponds to stronger stabilization. Using the combination of the face and surface stabilization terms the proof of (1.6) consists of two steps: 3

• Using the face terms we can estimate the L2 norm of a finite element function at an element in terms of the L2 norm at a neighboring element which share a face and the stabilization term on the shared face. Repeating this procedure we can pass from any element to an element which has a large intersection, |T ∩ Γh | & h2 , with Γh . • Using the stabilization of normal derivatives at the surface we can control a finite element function on elements which have large intersection with Γh . Using shape regularity we can show that one can pass from any element in Th to an element with a large intersection in a uniformly bounded number of steps, see [3] and [12]. Outline. In Section 2 we introduce the model problem, in Section 3 we formulate the finite element method, in Section 4 we collect some basic preliminary results, in Section 5 we formulate the properties of the stabilization term, in Section 6 we establish an optimal order condition number estimate, in Section 7 we prove optimal order a priori error estimates, and in Section 8 we present numerical results that support our theoretical findings.

2

The Laplace-Beltrami Problem on a Surface

The Surface. Let Γ be a smooth, closed, simply connected surface in R3 with exterior unit normal n embedded in a polygonal domain Ω ⊂ R3 . Let Uδ (Γ) be the open tubular neighborhood of Γ with thickness δ > 0, Uδ (Γ) = {x ∈ R3 : |d(x)| < δ}

(2.1)

where d(x) is the signed distance function of Γ with d < 0 in the interior of Γ and d > 0 in the exterior. Note that n = ∇d is the outward-pointing unit normal on Γ. Then there is δ0 > 0 such that for each x ∈ Uδ0 (Γ) the closest point projection: p(x) = x − d(x)n(p(x))

(2.2)

maps x to a unique closest point on Γ. In particular, we require δ0 max kκi kL∞ (Γ) < 1

(2.3)

i=1,2

where for x ∈ Γ, κi are the principal curvatures. We also define the extension of a function u on Γ to Uδ0 (Γ) by ue (x) = u(p(x)), x ∈ Uδ0 (Γ) (2.4) The Problem. We consider the problem: find u ∈ H01 (Γ) = {v ∈ H 1 (Γ) | such that a(u, v) = l(v) ∀v ∈ H01 (Γ)

4

R Γ

v ds = 0} (2.5)

where

Z

Z ∇Γ u · ∇Γ v ds,

a(u, v) =

l(v) =

Γ

f v ds

(2.6)

Γ

Here ∇Γ is the tangential gradient on Γ defined by ∇Γ u = PΓ ∇ue

(2.7)

PΓ = I − n ⊗ n

(2.8)

where We used the extension of u which is constant in the normal direction to Γ to formally define ∇Γ u. However, the tangential derivative depends only on the values of u on Γ and does not depend on the particular choice of extension. Using Lax-Milgram’s Lemma we conclude that there is a unique solution to (2.5), and we also have the elliptic regularity estimate kukH s+2 (Γ) . kf kH s (Γ) ,

3

s ∈ Z, s ≥ −1

(2.9)

The Method

3.1

The Mesh and Finite Element Space

• The background mesh and space: Let Th,0 be a quasiuniform partition of Ω with mesh p parameter h ∈ (0, h0 ] into shape regular tetrahedra. On Th,0 we define Vh,0 to be the space of continuous piecewise polynomials of degree p ≥ 1. • The discrete surfaces: Let Γh be a sequence of surfaces that converges to Γ. We specify the precise assumptions on the convergence below. • The active mesh and the induced surface mesh: We denote by Th the set consisting of elements in the background mesh Th,0 that are cut by the discrete geometry Γh . These elements form the so called active mesh. See Fig. 1 for an illustration. The restriction of the active mesh to the discrete surface Γh manufactures an induced cut surface mesh that we denote by Kh = {K = T ∩ Γh : T ∈ Th }

(3.1)

• The finite element space: The R restriction of the background space to the active mesh together with the condition Γh v dsh = 0 defines our finite element space Vhp , Vhp

 Z p = vh ∈ Vh,0 |Th : Γh

5

 v dsh = 0

(3.2)

Figure 1: The discrete surface Γh , the background mesh Th,0 , and the active mesh Th .

3.2

The Finite Element Method

Method. Find uh ∈ Vhp such that ∀vh ∈ Vhp

(3.3)

lh (v) = (fh , v)Γh

(3.4)

Ah (uh , v) := ah (uh , v) + sh (uh , v) = lh (v) Forms. • The forms ah and lh are defined by ah (w, v) = (∇Γh w, ∇Γh v)Γh , where the tangential gradient ∇Γh is defined by ∇Γh uh = PΓh ∇uh

(3.5)

PΓh = I − nh ⊗ nh

(3.6)

with • The stabilization form sh is defined by sh (w, v) = sh,F (w, v) + sh,Γ (w, v)

(3.7)

where sh,F (w, v) =

p X

cF,j h2(j−1+γ) ([Dnj F w], [Dnj F v])Fh

(3.8)

cΓ,j h2(j−1+γ) (Dnj h w, Dnj h v)Kh

(3.9)

j=1

sh,Γ (w, v) =

p X j=1

6

Here (Daj v)|x is the j:th directional derivative of v along the line defined by the unit vector a(x), [w]|F denotes the jump of w over the face F , Fh is the set of internal faces (i.e. faces with two neighbors) in the active mesh Th , cΓ,j > 0, cF,j > 0 are stabilization parameters, and γ ∈ [0, 1] is a parameter. Remark 3.1 In the special case p = 1 we obtain sh (w, v) = cF h2γ ([Dnj F w], [Dnj F v])Fh + cΓ h2γ (Dnj h w, Dnj h v)Kh

(3.10)

where γ ∈ [0, 1] which can be compared with the pure face penalty term sh (w, v) = cF ([Dnj F w], [Dnj F v])Fh

(3.11)

developed in [3]. Note that the scalings with the mesh parameter h are different in the two terms and that for γ > 0 the face penalty term is weaker in the stabilization term we introduce here.

3.3

Assumptions

A1. We assume that the closed, simply connected approximation Γh and its unit normal nh are such that for all h ∈ (0, h0 ], kdkL∞ (Γh ) . hp+1 kn − nh kL∞ (Γh ) . hp

(3.12) (3.13)

that n · nh > 0 on Γh , and that we have a uniform bound on the curvature κh . A2. We assume that the mesh is sufficiently small so that there is a constant c independent of the mesh size h such that Th ⊂ Uδ (Γ) with δ = ch ≤ δ0 and that the closest point mapping p(x) : Γh → Γ is a bijection. A3. We assume that the data approximation fh is such that k|B|f e −fh kL2 (Γh ) . hp+1 kf kL2 (Γ) . Here ds = |B|dsh (see Section 4 for details).

4 4.1

Preliminary Results Norms

Given a positive semidefinite bilinear form b we let kvk2b = b(v, v) denote the associated seminorm. In particular, we will use the following norms on Vhp + H p+1 (Th ), kvk2ah = ah (v, v),

kvk2sh = kvk2sh,F + kvk2sh,Γ ,

kvk2Ah = kvk2ah + kvk2sh

(4.1)

We let H s (ω), with ω ⊂ Rd , denote the standard Sobolev spaces of order s. For ω ⊂ Γ or ω ⊂ Γh H s (ω) denotes the surface Sobolev space which is based on tangential derivatives. 7

4.2

Some Inequalities

• Under A1 and A2 there is a constant such that the following Poincar´e inequality holds for h ∈ (0, h0 ], kvkL2 (Γh ) . kvkah ∀v ∈ Vhp (4.2) see Lemma 4.1 in [3]. • For T ∈ Th we have the trace inequalities kvk2L2 (∂T ) . h−1 kvk2L2 (T ) + hk∇vk2L2 (T ) kvk2L2 (T ∩Γh ) . h−1 kvk2L2 (T ) + hk∇vk2L2 (T )

v ∈ H 1 (T ) v ∈ H 1 (T )

(4.3) (4.4)

where the first inequality is a standard trace inequality, see e.g.[1], and the second inequality is proven in [16] and the constant is independent of how Γh intersects T and of h ∈ (0, h0 ]. • For T ∈ Th we have the inverse inequality |v|2H j (T ) . h−2(j−s) kvk2H s (T )

0 ≤ s ≤ j, v ∈ Vhp

(4.5)

see [1].

4.3

Interpolation

Let πhp : L2 (Th ) → Vhp be the Scott-Zhang interpolation operator. For all v ∈ H p+1 (Th ) and T ∈ Th the following standard estimate holds kv − πhp vkH m (T ) . hs−m kvkH s (Nh (T ))

m ≤ s ≤ p + 1,

m = 0, · · · , p + 1

(4.6)

where Nh (T ) ⊂ Th is the union of the elements in Th which share a node with T . In particular, we have the stability estimate kπhp vkH m (Th ) . kvkH m (Th )

4.4

(4.7)

Some Results for Extended and Lifted Functions

Extension. Recall that we define the extension of a function u on Γ to Uδ0 (Γ) by ue (x) = u(p(x)). For u ∈ H s (Γ), s ≥ 0 the following holds kue kH s (Uδ (Γ)) . δ 1/2 kukH s (Γ)

(4.8)

where 0 < δ ≤ δ0 and for h small enough we may take δ ∼ h. See Lemma 3.1 in [24]. Using the definition of the extension of a function u on Γ, the definition of the closest point projection (2.2), the chain rule, and the identity n = ∇d we obtain ∇ue = (PΓ − dH)∇u 8

in Uδ0 (Γ)

(4.9)

where H(x) = D2 d(x). For x ∈ Uδ0 (Γ), the eigenvalues of H(x) are κ1 (x), κ2 (x), and 0, with corresponding orthonormal eigenvectors a1 (x), a2 (x), and ne (x). We have the identities κei (x) κi (x) = , ai (x) = aei (x), i = 1, 2 (4.10) 1 + d(x)κei (x) and H(x) may be expressed in the form H(x) =

2 X i=1

κei (x) aei (x) ⊗ aei (x) e 1 + d(x)κi (x)

(4.11)

see [13, Lemma 14.7]. Using the fact that H is tangential, PΓ H = HPΓ = H, in the identity (4.9) we obtain ∇ue = (I − dH)∇Γ u

(4.12)

∇Γh ue = B T ∇Γ u

(4.13)

B = PΓ (I − dH)PΓh

(4.14)

Thus, with

Lifting. Define the lifting v l of a function v on Γh to Γ as (v l )e = v l ◦ p = v

on Γh

(4.15)

Using equation (4.13) it follows that ∇Γh v = ∇Γh (v l )e = B T ∇Γ (v l )

(4.16)

∇Γ (v l ) = B −T ∇Γh v

(4.17)

and thus we have Here, B

−1

n ⊗ nh  = I− (I − dH)−1 n · nh 

(4.18)

see [11], and using the fact that n and nh are unit normals it is easy to show that BB −1 = PΓ ,

B −1 B = PΓh

(4.19)

Estimates Related to B. It can be shown, see [3] for instance, that the following estimates hold kBkL∞ (Γh ) . 1,

kB −1 kL∞ (Γ) . 1,

9

kBB T − IkL∞ (Γh ) . hp+1

(4.20)

Surface Measures. For the surface measure on Γh , dsh (x), and the surface measure on Γ, ds(p(x)), we have the following identity ds(p(x)) = |B(x)|dsh (x)

x ∈ Γh

(4.21)

where |B(x)| = n · nh (1 − d(x)κ1 (x))(1 − d(x)κ2 (x))

(4.22)

see Proposition A.1 in [11]. Using the identity kn(x) − nh (x)k2R3 = 2(1 − n(x) · nh (x)) and our assumptions we obtain the following estimates:

−1

|B| ∞ k1 − |B|k ∞ . hp+1 , k|B|k ∞ . 1, .1 (4.23) L (Γh )

L (Γh )

L (Γh )

Norm Equivalences. Using the identities (4.13) and (4.17) and the bounds in equations (4.20) and (4.23) we obtain the following equivalences kv l kL2 (Γ) ∼ kvkL2 (Γh ) ,

kukL2 (Γ) ∼ kue kL2 (Γh )

(4.24)

k∇Γ ukL2 (Γ) ∼ k∇Γh ue kL2 (Γh )

(4.25)

and k∇Γ v l kL2 (Γ) ∼ k∇Γh vkL2 (Γh ) , We will also use the following bound kDj ue kL2 (Γh ) . kukH m (Γ)

5

u ∈ H m (Γ), j ≤ m

(4.26)

Properties of the Stabilization Term

Here we formulate properties of a general stabilization term that are sufficient to prove that the resulting linear system of equations have an optimal scaling of the condition number with the mesh parameter compared to standard finite elements and that the convergence of the method is of optimal order. Properties. Let sh be a semi positive definite bilinear form such that P1. For v ∈ H p+1 (Γ) ∩ H01 (Γ),

P2. For v ∈ H p+1 (Γ) ∩ H01 (Γ),

kv e − πhp v e ksh . hp kvkH p+1 (Γ)

(5.1)

kv e ksh . hp kvkH p+1 (Γ)

(5.2)

kπhp v e ksh . hkvkH 2 (Γ)

(5.3)

kvksh . h−3/2 kvkL2 (Th )

(5.4)

P3. For v ∈ H 2 (Γ), P4. For v ∈ Vhp ,

10

P5. For v ∈ Vhp ,

kvk2L2 (Th ) . h(kvk2ah + kvk2sh )

(5.5)

Remark 5.1 The first three properties are used to establish the a priori error estimates: P1 is needed to show optimal interpolation error estimates, P2 is used to estimate the consistency error, and P3 is used to show optimal L2 -error estimates. In P3 the assumption that v ∈ H 2 (Γ) reflects the fact that the solution to the dual problem resides in H 2 (Γ). The two last properties are used to show the condition number estimate: P4 is the inverse inequality and P5 is the Poincar´e inequality. Similar properties have also been stated in [14]. Verification of P1–P5. We now show that the stabilization term defined in (3.7) satisfies P1-P5. P1. Using a standard trace inequality on the faces (4.3) and the trace inequality (4.4) for the contributions on Γh we obtain kwk2sh =

p X X

cF,j h2(j−1+γ) k[Dnj F w]k2L2 (F )

(5.6)

j=1 F ∈Fh p

+

X X

cΓ,j h2(j−1+γ) kDnj h wk2L2 (K)

j=1 K∈Kh p

.

XX

2(j−1+γ)

h



−1

h kD

j

wk2L2 (T )

+ hkD

j+1

wk2L2 (T )



(5.7)

j=1 T ∈Th p

. =

XX

h2(j+γ)−1 kDj+1 wk2L2 (T )

j=0 T ∈Th p X 2(j+γ)−1

h

kDj+1 wk2L2 (Th )

(5.8)

(5.9)

j=0

Next, setting w = v e − πh v e , using the interpolation error estimate (4.6) and the stability (4.8) of the extension operator with δ ∼ h, we obtain e

kv −

πh v e k2sh

.

p X

h2(j+γ)−1 kDj+1 (v e − πh v e )k2L2 (Th )

(5.10)

h2(j+γ)−1 h2(p−j) kDj+1 v e k2H p+1 (Th )

(5.11)

j=0 p

.

X j=0

. h2(p+γ) kvk2H p+1 (Γ) Finally, h2(p+γ) . h2p for γ ∈ [0, 1] since h ∈ (0, h0 ], and thus P1 follows. 11

(5.12)

P2. Note first that for v ∈ H p+1 (Γ) we have kv e ksh,F = 0 and thus kv e ksh = kv e ksh,Γ . Next, subtracting Dnj v e = 0, using Assumption A1, and inequality (4.26) we obtain kv e k2sh

=

p X

cΓ,j h2(j−1+γ) kDnj h v e k2Kh

(5.13)

h2(j−1+γ) k(Dnj h − Dnj )v e k2L2 (Kh )

(5.14)

h2(j−1+γ) knh − nk2L∞ (Kh ) kDj v e k2L2 (Kh )

(5.15)

j=1

.

p X j=1 p

.

X j=1

. h2(p+γ) kvk2H p (Γ)

(5.16)

where we used the estimate knh − nkL∞ (Kh ) . hp . We conclude that P2 holds for γ ∈ [0, 1] since h ∈ (0, h0 ]. P3. For w ∈ Vhp we have the estimate kwk2sh . h2γ k[DnF w]k2L2 (Fh ) + h2γ kDnh wk2L2 (Kh ) + h2(1+γ)−1 kD2 wk2L2 (Th )

(5.17)

Verification of (5.17). We start from the definition kwk2sh = kwk2sh,F + kwk2sh,Γ

(5.18)

and estimate the two terms on the right hand side as follows. First kwk2sh,F

2γ

.h

k[DnF w]k2L2 (Fh )

+

p X

h2(j−1+γ) k[Dnj F w]k2L2 (Fh )

(5.19)

j=2

. h2γ k[DnF w]k2L2 (Fh ) + h2(1+γ)−1 k[D2 w]k2L2 (Th )

(5.20)

where for each j = 2, . . . , p and each face F ∈ Fh with neighboring elements T1 and T2 we used the inverse estimate k[Dnj F w]k2L2 (F ) . h−1 kDj wk2L2 (T1 ∪T2 ) . h−1 h2(2−j) kD2 wk2L2 (T1 ∪T2 )

(5.21)

Second, using a similar approach kwk2sh,Γ

2γ

.h

kDnh wk2L2 (Kh )

+

p X

h2(j−1+γ) kDnj h wk2L2 (Kh )

(5.22)

2(j−1+γ) −1 2(2−j) 2 2 h h |h {z } kD wkL2 (Th )

(5.23)

j=2 p

. h2γ kDnh wk2L2 (Kh ) +

X j=2

2γ

.h

kDnh wk2L2 (Kh )

2(1+γ)−1

+h

12

=h2(1+γ)−1 kD2 wk2L2 (Th )

(5.24)

where we used the inverse estimate kDnj h wk2L2 (K) ≤ kDj wk2L2 (K) . h−1 kDj wk2L2 (T ) . h−1 h2(2−j) kD2 wk2L2 (T )

(5.25)

for j = 2, . . . , p. Thus (5.17) holds. Setting w = πh v e in (5.17) we obtain kπh v e k2sh . h2γ k[DnF πh v e ]k2L2 (Fh ) + h2γ kDnh πh v e k2L2 (Kh ) | {z } | {z } I 2(1+γ)−1

+h

= I + II +

(5.26)

II

2

kD πh v e k2L2 (Th ) h2(1+γ) kvk2H 2 (Γ)

(5.27)

where we used the stability of the interpolation operator (4.7) and the stability of the extension operator (4.8). Next we have the following bounds. Term I. Using the fact that [DnF v e ] = 0 for v ∈ H 2 (Γ) we obtain k[DnF πh v e ]k2L2 (Fh ) = k[DnF (πh v e − v e )]k2L2 (Fh ) . h−1 k[D(πh v e − v e )]k2L2 (Th ) + hk[D2 (πh v e − v e )]k2L2 (Th ) . .

hkD2 v e k2L2 (Th ) h2 kvk2H 2 (Γ)

(5.28) (5.29) (5.30) (5.31)

where we used the trace inequality (4.3), the interpolation estimate (4.6), and finally the stability (4.8) of the extension operator. Thus we have I . h2(1+γ) kvk2H 2 (Γ)

(5.32)

Term II. Using the fact that Dne v e = 0 we obtain kDnh πh v e k2L2 (Kh ) . kDne πh v e k2L2 (Kh ) + kne − nh k2L∞ (Kh ) kDπh v e k2L2 (Kh ) . kDn (πh v e − v e )k2L2 (Kh ) + h2p−1 kDπh v e k2L2 (Th ) 2

.h

kvk2H 2 (Γ)

2p

+h

kvk2H 1 (Γ)

(5.33) (5.34) (5.35)

and thus we arrive at II . h2(1+γ) kvk2H 2 (Γ)

(5.36)

since p ≥ 1 and h ∈ (0, h0 ]. Collecting the bounds we obtain kπh v e ksh . h1+γ kvkH 2 (Γ) and thus P3 holds. 13

(5.37)

Simplified Proof of P3 in the Case γ = 1. Using estimate (5.9) with w ∈ Vhp and inverse inequality (4.5) we obtain kwk2sh

.

p X

h2j+1 kDj+1 wk2L2 (Th )

(5.38)

j=0

.

hkDwk2L2 (Th )

+

p X

h2j+1 kDj+1 wk2L2 (Th )

(5.39)

j=1

|

{z

.h3 kD2 wk2 2

}

L (Th )

.

hkDwk2L2 (Th )

3

+ h kD

2

wk2L2 (Th )

(5.40)

Setting w = πh v e and using the stability of the interpolation operator (4.7) and the stability of the extension operator (4.8) we obtain kπh v e k2sh . hkDπh v e k2L2 (Th ) + h3 kD2 πh v e k2L2 (Th ) 2

.h

kvk2H 1 (Γ)

4

+h

kvk2H 2 (Γ)

(5.41) (5.42)

P4. Starting from (5.9) with w ∈ Vhp and using the inverse inequality we get kwk2sh

.

p X

h2j+1 kDj+1 wk2L2 (Th ) . h−1 kwk2L2 (Th )

(5.43)

j=0

Using that h ∈ (0, h0 ] we have h−1 . h−3 and thus Property P4 holds. P5. See the proof of Lemma 6.3 below.

6

Condition Number Estimate

We shall show that the spectral condition number of the stiffness matrix resulting scales as h−2 , independent of the position of the geometry relative to the background mesh. In particular, we show that the stabilization term defined in (3.7) has Property P5 and controls the condition number for linear as well as for higher-order elements. p p b N ⊂ RN denote the vector Let {ϕi }N b∈ R i=1 be a basis in Vh and given v ∈ Vh let v containing the coefficients in the expansion: v=

N X

vbi ϕi

(6.1)

i=1

R b N is Recall that functions v in Vhp satisfy the condition Γh v dsh = 0 and therefore R Z Z N N b R = {b v ∈ R | vb · ( ϕ1 dsh , · · · , ϕN dsh ) = 0} (6.2) Γh

Γh

14

Since Th is quasiuniform we have the equivalence kvkTh ∼ hd/2 kb v kRb N

(6.3)

where d is the dimension of the embedding space Rd . Let Ah be the stiffness matrix associated with Ah , (Ah w, b vb)Rb N = Ah (w, v) ∀v, w ∈ Vhp (6.4) and recall that the condition number is defined by κ(Ah ) = kAh kRN kA−1 h kRN

(6.5)

which in terms of the eigenvalues of the symmetric positive definite matrix Ah is equal to κ(Ah ) =

λN λ1

(6.6)

where λN (λ1 ) is the largest (smallest) eigenvalue of Ah . Theorem 6.1 If there are constants, independent of the mesh size h and of how the surface cuts the background mesh, such that the following hold: • Ah is continuous: Ah (w, v) . kwkAh kvkAh

∀v, w ∈ Vhp + H p+1 (Th )

∀v ∈ Vhp

• Ah is coercive: kvk2Ah . Ah (v, v)

• The inverse inequality: kvkAh . h−3/2 kvkL2 (Th )

v ∈ Vhp

• The Poincar´e inequality: kvkL2 (Th ) . h1/2 kvkAh

v ∈ Vhp

Then, the spectral condition number κ(Ah ) satisfies κ(Ah ) . h−2

(6.7)

Proof. For v, w ∈ Vhp it follows from the continuity of the form Ah , the inverse inequality, and the equivalence between the norms, equation (6.3), that Ah (w, v) . kwkAh kvkAh . h−3 kwkL2 (Th ) kvkL2 (Th ) . h−3 hd kwk b Rb N kb v kRb N

(6.8)

From coercivity, the Poincar´e inequality, and the equivalence between the norms, equation (6.3), we obtain Ah (v, v) & kvk2Ah & h−1 kvk2L2 (Th ) & h−1 hd kb v k2Rb N (6.9) Using the definition (6.6) of the spectral condition number we get κ(Ah ) =

maxu∈Rb N ,kbuk

=1 RN

(Ah u b, u b)Rb N

minu∈Rb N ,kbuk

b, u b)Rb N =1 (Ah u RN

.

h−3 hd = h−2 h−1 hd

(6.10)

In the following Lemmas we show that all the conditions in Theorem 6.1 are satisfied. 15

Lemma 6.1 (Continuity and Inf-Sup Condition) There is a constant independent of the mesh size h and of how the surface cuts the background mesh, such that Ah is continuous: Ah (w, v) ≤ kwkAh kvkAh ,

∀v, w ∈ Vhp + H p+1 (Th )

(6.11)

and Ah satisfies the inf-sup condition kwkAh .

Ah (w, v) , v∈Vhp \{0} kvkAh sup

∀w ∈ Vhp

(6.12)

Proof. The continuity follows directly from the Cauchy-Schwarz inequality. The bilinear form Ah is coercive Ah (v, v) = kvk2Ah by definition and the inf-sup condition (6.12) follows from coercivity. Lemma 6.2 (Inverse Inequality) There are constants, independent of the mesh size h and of how the surface cuts the background mesh, such that the following inverse inequality holds: kvkAh . h−3/2 kvkL2 (Th ) v ∈ Vhp (6.13) Proof. Using the element wise trace inequality (4.4), the inverse inequality (4.5), and Property P4 of the stabilization term we obtain kvk2ah . h−1 kvk2H 1 (Th ) + hkvk2H 2 (Th ) . h−3 kvk2L2 (Th ) kvk2sh . h−3 kvk2L2 (Th )

(6.14)

Using the above estimates and recalling the definition of kvkAh the result follows. Lemma 6.3 (Poincar´e Inequality) There are constants, independent of the mesh size h and of how the surface cuts the background mesh, such that the following Poincar´e inequality holds kvkL2 (Th ) . h1/2 kvkAh v ∈ Vhp (6.15) Proof. To prove the Poincar´e inequality we will procced in two main steps: 1. We use the face penalty to reach an element which has a large intersection with Γh . Here we employ a covering of Th , where each covering set contains elements that have a so called large intersection property, see [3]. 2. For elements with a large intersection the normal derivative control on Γh is used to control the L2 norm on the element. To establish the Poincar´e inequality (6.15) we shall prove that kvk2L2 (Th )

.

p X

2j+1

h

kDnj h vk2L2 (Γh )

+

p X j=1

j=0

16

h2j+1 k[Dnj F v]k2L2 (Fh )

(6.16)

Here we note that (6.15) follows from (6.16) since the right hand side of (6.16) satisfies the estimate p X

2j+1

h

kDnj h vk2L2 (Γh )

+

p X

h2j+1 k[Dnj F v]k2L2 (Fh )

j=1

j=0

.

h(kvk2Γh

2(1−γ)

+h

kvk2sh ) . h(kvk2ah + h2(1−γ) kvk2sh ) . hkvk2Ah

(6.17)

where we used the definition (3.7) of sh , the Poincar´e inequality (4.2), and the fact that γ ∈ [0, 1]. Large Intersection Coverings of Th . There is a covering {Th,x : x ∈ Xh } of Th , where Xh is an index set, such that: (1) Each set Th,x contains a uniformly bounded number of elements. (2) Each element in Th,x share at least one face with another element in Th,x . (3) In each set Th,x there is one element Tx ∈ Th,x which has a large intersection with Γh , hd−1 . |Tx ∩ Γh | = |Kx |

(6.18)

where d is the dimension. (4) The number of sets Th,x to which an element T belongs is uniformly bounded for all T ∈ Th . See [3] for the construction of the covering. We prove (6.16) by considering a set Th,x in the covering and the following steps: Step 1. We shall show that kvk2L2 (Th,x )

.

kvk2L2 (Tx )

+

p X

h2j+1 ([Dnj F v], [Dnj F w])Fh,x

(6.19)

j=1

where Fh,x is the set of interior faces in Th,x . To prove (6.19) we recall that for two elements T1 and T2 that share a face F it holds kvk2L2 (T1 )

.

kvk2L2 (T2 )

+

p X

h2j+1 k[Dnj F v]k2L2 (F )

(6.20)

j=1 j 0 see [17] for instance. Now let Th,x = {Tx } and for j = 1, 2, . . . let Th,x be the set of all j−1 elements that share a face with an element in Th,x . Then there is a uniform constant J j j−1 such that Th,x = Th,x for j ≥ J and we also have the estimate

kvkT j . kvkT j−1 + h,x

X

p X

h,x

j j−1 F ⊂Fh,x \Fh,x

h2j+1 k[Dnj F v]k2L2 (F )

j=1

j j where Fh,x is the set of interior faces in Th,x . Iterating (6.21) we obtain (6.19).

17

(6.21)

Step 2. We shall show that there is a constant such that for all Tx ∈ Th which have the large intersection property (6.18), kvk2L2 (Tx )

.

p X

h2j+1 kDnj h vk2L2 (Kx )

(6.22)

j=0

where Kx = Γh ∩ Tx . To verify (6.22) we define the cylinder Cylδ (Kx ) = {x ∈ Rd : x = y + tnh , y ∈ Kx , |t| ≤ δ}

(6.23)

and using Taylor’s formula for a polynomial v ∈ Pp (Cylδ (Kx )), we obtain the bound kvk2L2 (Cylδ (Kx )) .

p X

δ 2j+1 kDnj h vk2L2 (Kx )

(6.24)

j=0

Using the following inverse bound: there is a constant such that for all v ∈ Pp (Tx ) kvkL2 (Tx ) . kvkL2 (Cylδ (Kx ))

(6.25)

the desired estimate (6.22) follows for δ ∼ h. Verification of (6.25). To employ a scaling argument we will construct two regular cylinders with circular cross section and the same center line that may be mapped to a reference configuration, one containing Tx and one contained in Cylδ (Kx ). See Figure 6. Let F x be a plane with unit normal nx , which is tangent to Kx at an interior point of Kx . Then we have the bound knh − nh kL∞ (Kx ) . h (6.26) and with K x the closest point projection of Kx onto F x we conclude using shape regularity and a uniform bound on the curvature κh (Assumption A1) that there is a ball B r1 ,x ⊂ K x ⊂ F x with radius r1 ∼ h and a δ 1 ∼ h such that Cylδ1 (K x ) = {x ∈ Rd : x = y + tnh , y ∈ K x , |t| ≤ δ 1 } ⊂ Cylδ (Kx )

(6.27)

Next using shape regularity there is a larger ball B r2 ,x ⊂ F x with the same center as B r1 ,x such that T x ⊂ B r2 ,x ⊂ F x (6.28) where T x is the closest point projection of Tx onto F x , and a δ 2 ∼ h such that Tx ⊂ Cylδ2 (B r2 ,x )

(6.29)

Clearly, Cylδ1 (B r1 ,x ) ⊂ Cylδ2 (B r2 ,x ) and using a mapping to a reference configuration we conclude that there is a constant such that for all polynomials v ∈ Pp (Cylδ2 )(B r2 ,x ), kvkCyl

δ 2 (B r 2 ,x )

. kvkCyl

δ 1 (B r 1 ,x )

which in view of the inclusions (6.27) and (6.29) concludes the proof of (6.25). 18

(6.30)

Figure 2: A set of elements Th,x in the cover of Th , an element Tx with large intersection Kx = Tx ∩ Γh . Step 3. Combining equation (6.19) and (6.22) and using that {Th,x : x ∈ Xh } is a cover of Th completes the proof of (6.16) and the lemma. Remark 6.1 Using the same technique as in the proof of Lemma 6.3 we may show the stronger estimate kvk2Th + h2γ k∇vk2Th . hkvk2Ah (6.31) where γ ∈ [0, 1] is the scaling parameter in sh , see (3.8) and (3.9). The modifications are as follows. In Step 1 we show that k∇vk2L2 (Th,x )

.

k∇vk2L2 (Tx )

p X

+

h2j−1 k[Dnj F v]k2L2 (F )

(6.32)

j=1

which after multiplication by h2γ corresponds to 2γ

h

k∇vk2L2 (Th,x )

2γ

.h

k∇vk2L2 (Tx )

+h

p X

! 2(j−1+γ)

h

k[Dnj F v]k2L2 (F )

(6.33)

j=1

To show (6.32) we use the estimate k∇vk2L2 (T1 )

.

k∇vk2L2 (T2 )

+

p X

h2j−1 k[Dnj F v]k2L2 (F )

(6.34)

h2j−1 kDnj h vk2L2 (Kx )

(6.35)

j=1

In Step 2 we show that k∇vk2L2 (Tx )

.

k∇Γh vk2Kx

+

p X j=1

19

Figure 3: Schematic figure illustrating a typical configuration of an element T , the curved intersection with Γh , the plane F x which is tangent to Kx , the cylinders Cylδ (Kx ), Cylδ1 (B r1 ), and Cylδ2 (B r2 ). Note that we have the inclusions B r1 ,x ⊂ K x ⊂ T x ⊂ B r2 ,x , Tx ⊂ Cylδ2 (B r2 ), and Cylδ1 (B r1 ) ⊂ Cylδ (Kx ).

20

which after multiplication by h2γ corresponds to h2γ k∇vk2L2 (Tx ) . h2γ k∇Γh vk2Kx +h | {z } .k∇Γh vk2Kx

p X

! h2(j−1+γ) kDnj h vk2L2 (Kx )

(6.36)

j=1

Combining (6.33) and (6.36), summing over the cover, and using Property (4) of the cover we obtain (6.31). Remark 6.1 Note the following: • From (5.43) we see that the scaling of the stabilization term corresponds to the mass matrix. This scaling is the weakest which guarantees the Poincar´e inequality (6.15) (Property P4 of the stabilization term). • For the operator L = α∆Γ + β

(6.37)

where α and β are constants, we would have the inverse inequality kvkAh . (αh−3/2 + βh−1/2 )kvkTh

v ∈ Vhp

(6.38)

instead of (6.13), and the resulting condition number estimate is then κ(Ah ) . αh−2 + β

(6.39)

kvkah . (αh−3/2 + βh−1/2 )kvkTh

(6.40)

• In the case when we note that the desired inverse inequality (6.38) holds for seh = (αh−τ + β)sh

(6.41)

with τ in the interval 0 ≤ τ ≤ 2, since kvk2seh = (αh−τ + β)kvk2sh . (αh−τ + β)h−1 kvk2Th . (αh−(1+τ ) + βh−1 )kvk2Th (6.42)

7

A Priori Error Estimates

In this section we prove optimal error estimates in the energy norm and in the L2 norm.

21

7.1

Strang Lemma

Using that Ah is continuous and satisfies the inf-sup condition we first show a Strang Lemma which connects the error in the energy norm to the interpolation error and the consistency error. Lemma 7.1 Let u ∈ H01 (Γ) be the unique solution of (2.5) and uh ∈ Vhp the finite element approximation defined by (3.3). Then the following discretization error bound holds kue − uh kAh . kue − πhp ue kAh +

|Ah (ue , v) − lh (v)| kvkAh v∈Vhp \{0} sup

(7.1)

Proof. Adding and subtracting an interpolant and using the triangle inequality we get kue − uh kAh ≤ kue − πhp ue kAh + kπhp ue − uh kAh

(7.2)

Using the inf-sup condition (6.12) for Ah we have kπhp ue − uh kAh .

Ah (πhp ue − uh , v) kvkAh v∈Vhp \{0} sup

(7.3)

Adding and subtracting Ah (ue , v) and using the weak formulation (3.3) yields Ah (πhp ue − uh , v) = Ah (πhp ue − ue , v) + Ah (ue − uh , v) = Ah (πhp ue − ue , v) + Ah (ue , v) − lh (v)

(7.4) (7.5)

Finally, using the continuity of Ah we obtain Ah (πhp ue − ue , v) ≤ kπhp ue − ue kAh kvkAh

(7.6)

Collecting the estimates we end up with the desired bound kue − uh kAh . kue − πhp ue kAh +

7.2

|Ah (ue , v) − lh (v)| kvkAh v∈Vhp \{0} sup

(7.7)

The Interpolation Error

Next we prove optimal interpolation error estimates using Property P1 of the stabilization term. Lemma 7.2 For all u ∈ H p+1 (Γ) we have the following estimate kue − πhp ue kAh . hp kukH p+1 (Γ)

(7.8)

Proof. From the definition of the energy norm k · kAh , equation (4.1), we have kue − πhp ue k2Ah = kue − πhp ue k2ah + kue − πhp ue k2sh | {z } | {z } I

22

II

(7.9)

Term I. Using the element wise trace inequality (4.4), standard interpolation estimates (4.6) on elements T ∈ Th , and the stability estimate (4.8) for the extension operator with δ ∼ h, we obtain I = kue − πhp ue k2ah  X h−1 kue − πhp ue k2H 1 (T ) + hkue − πhp ue k2H 2 (T ) . T ∈Th 2p−1

.h

2p

.h

kue k2H p+1 (Th )

(7.10) (7.11) (7.12)

kuk2H p+1 (Γ)

(7.13)

Term II. From Property P1 of the stabilization term we have that II . h2p kuk2H p+1 (Γ)

(7.14)

Combining the two estimates (7.10) and (7.14) the result follows.

7.3

The Consistency Error

The approximation of the geometry Γ by Γh and the approximation of the data f by fh leads to a consistency error that we estimate in the next lemma. Lemma 7.3 Let u ∈ H p+1 (Γ) ∩ H01 (Γ) be the solution of (2.5) then, the following bound holds |Ah (ue , v) − lh (v)| sup . hp kukH p+1 (Γ) + hp+1 kf kL2 (Γ) (7.15) p kvk A v∈Vh \{0} h Proof. We have the identity |Ah (ue , v) − lh (v)| ≤ |ah (ue , v) − lh (v)| + |sh (ue , v)| | {z } | {z } I

(7.16)

II

Term I. Adding −a(u, v l ) + l(v l ) = 0, and using the triangle inequality I ≤ |ah (ue , v) − a(u, v l )| + |l(v l ) − lh (v)| {z } | {z } | II

(7.17)

III

R Term I I . Adding and subtracting Γh ∇Γh ue · ∇Γh v|B| dsh , using that B T B −T = PΓh , changing the domain of integration from Γ to Γh , using the triangle inequality, norm

23

equivalences in (4.25), estimates (4.20), and (4.23) we obtain Z e l |ah (u , v) − a(u, v )| = (1 − |B|)∇Γh ue · ∇Γh v dsh Γh Z B T (B −T ∇Γh ue ) · B T (B −T ∇Γh v)|B| dsh + ZΓh (B −T ∇Γh ue ) · (B −T ∇Γh v)|B| dsh − Γh   . k1 − |B|kL∞ (Γh ) + kBB T − IkL∞ (Γh ) k∇Γ ukL2 (Γ) k∇Γh vkL2 (Γh ) . hp+1 k∇Γ ukL2 (Γ) kvkah

(7.18)

Term I II . Changing the domain of integration we get Z Z l l fh v dsh |l(v ) − lh (v)| = f v ds − Γ Γh Z Z e = f v|B| dsh − fh v dsh

(7.19)

Γh

(7.20)

Γh

. k|B|f e − fh kL2 (Γh ) kvkL2 (Γh ) . hp+1 kf kL2 (Γ) kvkL2 (Γ)

(7.21) (7.22)

where we at last used Assumption A3 on the data approximation fh . Together, the bounds of II and III , and the Poincar´e inequality (4.2), imply I . hp+1 k∇Γ ukL2 (Γ) kvkah + hp+1 kf kL2 (Γ) kvkL2 (Γh ) . (k∇Γ ukL2 (Γ) + kf kL2 (Γ) )hp+1 kvkah

(7.23) (7.24)

Finally, using that u is the solution of (2.5) we have the stability estimate k∇Γ ukL2 (Γ) . kf kL2 (Γ) , and we obtain I . hp+1 kf kL2 (Γ) kvkah (7.25) Term II. Using the Cauchy-Schwarz inequality and Property P2 of the stabilization term, see Section 5, we obtain |sh (ue , v)| . kue ksh kvksh . hp kukH p+1 (Γ) kvksh

(7.26)

Combining the estimates of I and II we obtain the desired estimate |Ah (ue , v) − lh (v)| . hp kukH p+1 (Γ) kvkAh + hp+1 kf kL2 (Γ) kvkah

24

(7.27)

7.4

Error Estimates

We first prove optimal error estimates in the energy norm and then apply a duality argument to obtain L2 -error estimates Theorem 7.1 Let u ∈ H p+1 (Γ) ∩ H01 (Γ) be the solution of (2.5) and uh ∈ Vhp the finite element approximation defined by (3.3). If assumptions A1-A3 hold and the stabilization term has properties P1-P5 then, there is a constant independent of the mesh size h such that the following error bound holds kue − uh kAh . hp kukH p+1 (Γ) + hp+1 kf kL2 (Γ)

(7.28)

Proof. Using the Strang Lemma followed by the bounds on the interpolation error (7.8) and the consistency error (7.15) we get the desired bound. Theorem 7.2 Let u ∈ H p+1 (Γ) ∩ H01 (Γ) be the solution of (2.5) and uh ∈ Vhp the finite element approximation defined by (3.3). If assumptions A1-A3 hold and the stabilization term has properties P1-P5 then, there is a constant independent of the mesh size h such that the following error bound holds kue − uh kL2 (Γh ) . hp+1 kukH p+1 (Γ) + hp+1 kf kL2 (Γ)

(7.29)

Proof. The proof is similiar to the proof of the L2 -error estimate in [3] for R R p = 1. Let eh = ue − uh |Γh and its lift on Γ be elh = u − ulh . Recall that Γ u ds = 0 and Γh uh dsh = 0. We now define u˜h ∈ Vhp as Z u˜h = uh − |Γ|−1

ulh ds

(7.30)

Γ

so that we have we obtain

R Γ

u˜lh = 0. By adding and subtracting u˜lh and using the triangle inequality ulh − ulh kL2 (Γ) kelh kL2 (Γ) ≤ ku − u˜lh kL2 (Γ) + k˜ {z } | {z } | I

(7.31)

II

Term I. Consider the dual problem: a(v, φ) = (ψ, v)Γ ,

ψ ∈ L2 (Γ) \ R

(7.32)

It follows from the Lax-Milgram lemma that there exists a unique solution in H 1 (Γ) \ R and we also have the elliptic regularity estimate kφkH 2 (Γ) . kψkL2 (Γ)

25

(7.33)

Setting v = u − u˜lh in (7.32), adding and subtracting an interpolant, and using the weak formulations in (2.5) and (3.3) we obtain (u − u˜lh , ψ)Γ = a(u − u˜lh , φ) = a(u − = a(u − | +

u˜lh , φ u˜lh , φ

−

− {z

(7.34) (πhp φe )l ) + a(u − u˜lh , (πhp φe )l ) (πhp φe )l ) + l((πhp φe )l ) − lh (πhp φe ) }

II

ah (˜ uh , πhp φe ) |

|

{z

(7.35) (7.36)

}

III p e l l a(˜ uh , (πh φ ) ) + sh (˜ uh , πhp φe )

− {z

IIII

}

|

{z

IIV

(7.37)

}

Term I I . Using the Cauchy-Schwarz inequality, the norm equivalences (see Section 4), and the energy norm estimate (Theorem 7.1) we obtain II . k∇Γ (u − u˜lh )kL2 (Γ) k∇Γ (φ − (πhp φe )l )kL2 (Γ) . kue − uh kah kφe − πhp φe kah . (hp kukH p+1 (Γ) + hp+1 kf kL2 (Γ) )kφe − πhp φe kah

(7.38) (7.39) (7.40)

The interpolation error estimate (7.10) with s = 1 yields kφe − πhp φe kah . hkφkH 2 (Γ)

(7.41)

and finally using the elliptic regularity estimate (7.33) we get II . (hp+1 kukH p+1 (Γ) + hp+2 kf kL2 (Γ) )kψkL2 (Γ)

(7.42)

Term I II − I III . We use the estimate of Term I in the proof of Lemma 7.3, together with the following stability estimate k∇Γ u˜lh kL2 (Γ) = k∇Γ ulh kL2 (Γ) . kuh kah . kfh kL2 (Γh ) . kf kL2 (Γ)

(7.43)

to get |III + IIII | . (hp+1 k∇Γ u˜lh kL2 (Γ) + hp+1 kf kL2 (Γ) )kπhp φe kah . hp+1 kf kL2 (Γ) kπhp φe kah (7.44) Adding and subtracting an interpolant, using the interpolation error estimate (7.41) followed by the elliptic regularity estimate (7.33) yield kπhp φe kah . kπhp φe − φe kah + kφe kah . kφkH 2 (Γ) . kψkL2 (Γ)

(7.45)

|III + IIII | . hp+1 kf kL2 (Γ) kψkL2 (Γ)

(7.46)

Thus,

26

Term I IV . We have |IIV | = |sh (uh , πhp φe )| . kuh ksh kπhp φe ksh

(7.47)

Using the energy norm error estimate, Theorem 7.1, and Property P2 of the stabilization we obtain kuh ksh . kuh − ue ksh + kue ksh . hp kukH p+1 (Γ) + hp+1 kf kL2 (Γ) + hp kukH p+1 (Γ)

(7.48) (7.49)

and using Property P3 and the elliptic regularity estimate (7.33) we obtain kπhp φe ksh . hkφkH 2 (Γ) . hkψkL2 (Γ)

(7.50)

|IIV | . (hp+1 kukH p+1 (Γh ) + hp+2 kf kL2 (Γ) )kψkL2 (Γ)

(7.51)

We thus have

Final Estimate of Term I. Collecting the estimates of Terms II − IIV and taking ψ = u − u˜lh /ku − u˜lh kL2 (Γ) we obtain I = ku − u˜lh kL2 (Γ) . hp+1 kukH p+1 (Γh ) + hp+1 kf kL2 (Γ)

(7.52)

R Term II. Using the definition (7.30) of u˜h , adding Γh uh dsh = 0, changing the domain of integration and using related bounds, employing the Poincar´e inequality (4.2), and the stability estimate (7.43), we obtain Z Z l l l (7.53) uh dsh k˜ uh − uh kL2 (Γ) = uh ds − Γh Γ Z . uh (|B| − 1)dsh (7.54) Γh p+1

. h kuh kL2 (Γh ) . hp+1 kuh kah . hp+1 kf kL2 (Γ)

(7.55) (7.56) (7.57)

Conclusion of the Proof. Using the estimates of Terms I and II in equation (7.31) and the equivalence kelh kL2 (Γ) ∼ keh kL2 (Γh ) conclude the proof.

8

Numerical Examples

Let the interface Γ be a circle of radius 1 centered at the origin. We generate a uniform triangular mesh with h = hx1 = hx2 on the computational domain: [−1.5, 1.5]×[−1.5, 1.5] where the interface is embedded. 27

Conditioning and Convergence Study For the Laplace-Beltrami Problem. A right-hand side f to equation (2.5) is calculated so that u(x, t) = e−4t x1 x2 + x31 x22

(8.1)

is the exact solution. We discretize (2.5) using the proposed CutFEM with polynomials of degree p=1, 2, and 3. We choose the stabilization constants in (3.7)-(3.9) to be cF,j = −1 cΓ,j = 10j! , j = 1, · · · , p and γ = 1. In Fig. 4 we show the error and the condition number of the linear systems as a function of the mesh size h. The error measured in the L2 norm behaves as O(hp+1 ) and in the H 1 norm as O(hp ), as expected. The condition number behaves as O(h−2 ) independent of the polynomial degree and how the interface cuts the background mesh. However, the constant in the O(h−2 ) estimate increases as the polynomial degree increases. We always show the condition number using the L1 norm which is an estimate of the spectral condition number. The stabilization term developed in [3] can be written in the form (3.7)-(3.9) with cΓ,j = 0 for all j and γ = 0. Therefore, we refer to it as the pure face stabilization. We −1 choose cF,j = 10j! , j = 1, · · · , p. In Fig. 5 and 6 we show the effect of the stabilization on the error and the condition number using linear (p=1) and cubic elements (p=3). We show the convergence only with respect to the L2 norm since the results are similiar when the error is measured in the H 1 norm. The proposed stabilization, represented by circles, is compared to using no stabilization, i.e., cF,j = cΓ,j = 0, represented by diamonds, and the pure face stabilization, represented by stars. For linear elements the L2 norm of the errors using the different stabilizations and no stabilization are very similiar but we clearly see that when no stabilization is used the condition number can become extremly large depending on the position of the interface relative the background mesh. For higher order elements the linear systems are severely ill-conditioned without stabilization and therefore the behaiour of the error is oscillatory and depends on how the geometry cuts the background mesh. When the approximation of the interface is slightly changed the magnitude of the error and the convergence shown in Fig. 6, may change a lot for the unstabilized method. Using the pure face stabilization we see that the L2 norm of the error initially decrease as the mesh size decreases but when the condition number becomes to large the convergence stops and rounding errors start to dominate. Note that the condition numbers are very large also for course meshes. With the proposed stabilization we obtain both optimal convergence orders and the condition number scales as O(h−2 ) independent of how the interface cuts the background mesh. Decreasing the constant in the stabilization decreases the magnitude of the error but the condition number increases. Our choice of constants in the stabilization term are not optimized and other choices may give better results. Conditioning Study for the Mass Matrix. We now consider the problem of finding uh ∈ Vhp such that Z Z uh vh dsh + sh (uh , vh ) = fh vh dsh (8.2) Γh

Γh

28

10 -2

L2-Error

10 -4 10 -6 10 -8 10 -10 10 -12 10 -3

10 -2

10 -1

10 0

10 -1

10 0

10 -1

10 0

h

10 0

H1-Error

10 -2

10 -4

10 -6

10 -8 10 -3

10 -2

h

Condition number

10 10

10 8

10 6

10 4

10 2 10 -3

10 -2

h

Figure 4: The Laplace-Beltrami equation: The error and condition number versus mesh size h for different degrees of polynomials, p, in the discretization. Squares: p=1. Stars: −1 p=2. Circles: p=3. The stabilization constants are cF,j = cΓ,j = 10j! , j = 1, · · · , p. Top: The error measured in the L2 norm versus mesh size h. The dashed lines are indicating the expected rate of convergence and are proportional to hp+1 . Middle: The error measured in the H 1 norm versus mesh size h. The dashed lines are indicating the expected rate of convergence and are proportional to hp . Bottom: The condition number versus mesh size h. The dashed lines are all proportional to h−2 . 29

10 20

Condition number

L2-Error

10 -3

10 -4

10 -5

10 15

10 10

10 5

10 -6 10 -2

10 -1

10 -2

h

10 -1

h

Figure 5: The Laplace-Beltrami equation: The error and condition number versus mesh size h using different stabilization parameters. Linear elements are used, i.e. p=1. Circles: the proposed stabilization. Stars: the pure face stabilization developed in [3]. Diamonds: no stabilization is added. Left: The error measured in the L2 norm versus mesh size h. The dashed line is indicating the expected rate of convergence and is proportional to h2 . Right: The condition number versus mesh size h. The dashed line is proportional to h−2 .

10 40

10 -2

Condition number

L2-Error

10 -4 10 -6 10

-8

10 -10 10 -12 10 -3

10 -2

10 -1

10 30

10 20

10 10

10 0 10 -3

10 0

h

10 -2

10 -1

10 0

h

Figure 6: The Laplace-Beltrami equation: The error and condition number versus mesh size h using different stabilization parameters. Cubic elements are used, i.e. p=3. Circles: the proposed stabilization. Stars: the pure face stabilization. Diamonds: no stabilization is added. Left: The error measured in the L2 norm versus mesh size h. The dashed line is indicating the expected rate of convergence and is proportional to h4 . Right: The condition number versus mesh size h. The dashed line is proportional to h−2 .

30

10 7

10 0

10

Condition number

L2-Error

10 -2 -4

10 -6

10 6

10 5

10 -8 10 -10 10 -3

10 -2

10 -1

10 4 10 -3

10 0

10 -2

10 -1

10 0

h

h

Figure 7: The mass matrix: The error and condition number versus mesh size h for different degrees of polynomials, p, in the space discretization. Squares: p=1. Stars: p=2. Circles: p=3. Left: The error measured in the L2 norm versus mesh size h. The dashed lines are indicating the expected rate of convergence and are proportional to hp+1 . Right: The condition number versus mesh size h. where sh is the stabilization term defined in (3.7). We let fh be: fh =

−(2x5 − 17x3 y 2 + 6xy 4 ) x2 + y 2

(8.3)

We study the condition number of the mass matrix using different degrees of polynomials, p, and different stabilization terms. Fig. 7 shows that the stabilization term we propose results in optimal convergence rates in the L2 norm and optimal scaling of the condition number (O(1)) of the mass matrix as it did for the Laplace-Beltrami operator. The constant in the condition number estimate however depends on the polynomial degree. −4 We have chosen cF,j = cΓ,j = 10j! , j = 1, · · · , p and γ = 1. Increasing these constants decreases the condition number but the magnitude of the error increases. Our choice of constants in the stabilization term are not optimized and other choices could give better results. In Fig. 8 and 9 we compare different stabilization terms for linear and cubic elements, −4 respectively. We use the proposed stabilization with cF,j = cΓ,j = 10j! , j = 1, · · · , p, the pure face stabilization, i.e. cΓ,j = 0 for all j, cF,j = 10−4 , j!

10−4 , j!

j = 1, · · · , p, the pure interface

stabilization i.e. cF,j = 0 for all j, cΓ,j = j = 1, · · · , p, with γ = 1 in all cases. Our observations are similiar to what we observed for the Laplace-Beltrami operator. For higher order elements the linear systems are severely ill-conditioned without stabilization and the behaviour of the error is oscillatory and the convergence rate will depend on how the geometry cuts the background mesh. We also see that neither the pure face stabilization nor a pure interface stabilization give enough control of the condition number and that we need, as we propose, the combination of them. 31

10 -1 10 30

Condition number

L2-Error

10 -2

10 -3

10 -4

10 20

10 10

10 -5 10 -2

10 -1

10 -2

h

10 -1

h

Figure 8: The mass matrix: The error and condition number versus mesh size h using different stabilization parameters. Linear elements are used, i.e. p=1. Circles: the proposed stabilization. Stars: the pure face stabilization. Diamonds: no stabilization is added. Left: The error measured in the L2 norm versus mesh size h. The dashed line is indicating the expected rate of convergence and is proportional to h2 . Right: The condition number versus mesh size h. The dashed lines are proportional to h−2 and h0 .

10 0

10 40

10

Condition number

L2-Error

10 -2 -4

10 -6 10 -8 10 -10 10 -3

10 -2

10 -1

10 30

10 20

10 10

10 0 10 -3

10 0

10 -2

10 -1

10 0

h

h

Figure 9: The mass matrix: The error and condition number versus mesh size h using different stabilization parameters. Cubic elements are used, i.e. p=3. Circles: the proposed stabilization. Stars: pure face stabilization. Diamonds: no stabilization is added. Squares: pure interface stabilization. Left: The error measured in the L2 norm versus mesh size h. The dashed line is indicating the expected rate of convergence and is proportional to h4 . Right: The condition number versus mesh size h. The dashed line is proportional to h0 .

32

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