arXiv:1610.04482v1 [math.NA] 14 Oct 2016
JOURNAL ? (????), 1 – 29 DOI 10.1515/ JOURNAL-????-???
© de Gruyter ????
Fictitious domain method with boundary value correction using penalty-free Nitsche method Thomas Boiveau, Erik Burman, Susanne Claus and Mats G. Larson Abstract. In this paper, we consider a fictitious domain approach based on a Nitsche type method without penalty. To allow for high order approximation using piecewise affine approximation of the geometry we use a boundary value correction technique based on Taylor expansion from the approximate to the physical boundary. To ensure stability of the method a ghost penalty stabilization is considered in the boundary zone. We prove 1 optimal error estimates in the H 1 -norm and estimates suboptimal by Oph 2 q in the L2 norm. The suboptimality is due to the lack of adjoint consistency of our formulation. Numerical results are provided to corroborate the theoretical study. Keywords. Nitsche’s method, fictitious domain method, boundary value correction. 2010 Mathematics Subject Classification. 65N12, 65N30, 65N85.
1
Introduction
Mesh generation is an important challenge in computational mechanics, in fact for complex geometries this can be highly nontrivial. In some cases for time dependent problems, such as a solid body embedded in a flow, the geometry of the problem changes each time step imposing conintuous remeshing, at least locally. The main idea of the fictitious domain method [1, 7, 8, 11–14, 18] is to relax the constraint that imposes the mesh to fit with the computational domain. In fact the principle is to embed the computational domain in a mesh that is easy to generate, without matching the elements with the boundary. In the early developments of fictitious domain [11], the method was faced with the choice of either integrating the equations over the whole computational mesh including the nonphysical part This work received funding from EPSRC (award number EP/J002313/1) which is gratefully acknowledged. The Author, S. Claus, gratefully acknowledges the financial support provided by the Welsh Government and Higher Education Funding Council for Wales through the Sêr Cymru National Research Network in Advanced Engineering and Materials. The author M. G. Larson gratefully acknowledges the financial support from the Swedish Foundation for Strategic Research Grant No. AM13-0029, the Swedish Research Council Grant No. 2011-4992, and Swedish strategic research programme eSSENCE.
2
T. Boiveau, E. Burman, S, Claus and M. G. Larson
or only integrate inside the physical domain. In the first case, the method is robust but inaccurate, the second approach is accurate but can generate bad conditioning of the system matrix depending on how the boundary crosses the mesh. As a fix to solve the conditioning problem a boundary penalty term was introduced in [3] the effect of this term is that it extends the stability in the physical domain to the whole mesh domain, provided the distance from the mesh boundary to the physical boundary is Ophq. Nitsche’s method was first introduced for the weak imposition of the boundary conditions in [19] and designed to be consistent and preserve the symmetry of the original problem. The stability of the method relies on a penalty term that needs to be sufficiently large. In the context of fictitious domain methods Nitsche’s method can suffer from instability for certain mesh boundary configurations. A solution to this problem using the ghost penalty approach was suggested in [8, 18].
In [10] a non-symmetric version was proposed where the penalty parameter only needs to be strictly greater than zero for the stability to be ensured. The possibility of considering the penalty parameter equal to zero for the non-symmetric case was suggested in [15], however, coercivity cannot be proved for this nonsymmetric penalty-free method and stability was not established. In [4] the stability of the nonsymmetric Nitsche’s method without penalty for elliptic problems was proved, using an inf-sup argument, drawing on earlier work on discontinuous Galerkin methods [16]. Recently the work on penalty free methods has been extended to compressible and incompressible elasticity in [2]. The penalty-free method can be seen as a Lagrange multiplier method where the Lagrange multipliers has been replaced by the the boundary fluxes of the discrete elliptic operator. In multiphysics problems and particularly for fluid-structure interaction the loose coupling of this method appears to have some advantages that has been observed numerically in [6]. We consider a cut finite element method (CutFEM) [5] in the fictitious domain fashion, the implementation of this method often requires an approximation of the physical domain due to the boundary that can arbitrarily cut through the elements of the mesh. In this paper we propose a method to control the error introduced by this approximation of the physical domain. We follow the method that has been developed in [9] where a piecewise affine approximate geometry was used for the integration of the equation with a correction based on Taylor expansion from the approximate to the physical boundary to improve order when higher polynomial orders are used. In this work we use the penalty-free Nitsche’s method to impose the boundary conditions. This eliminates one penalty parameter at the price of loss 1 of symmetry of the algebraic system and Oph 2 q suboptimality in the L2 -norm. We believe that the method nevertheless may be of interest, in particular for problems
Boundary value correction, penalty-free Nitsche method
3
that are not symmetric, such as the advection–diffusion equation. We present and analyse the method in the two-dimensional case, but the results hold also in the three dimensional case. We end this section by introducing our model problem. Let Ω be a bounded domain in R2 with smooth boundary Γ and exterior unit normal n. The Poisson problem is given by: ´∆u “ f in Ω, u“g
on Γ, 3
where f P L2 pΩq the given body force and g P H 2 pΩq the boundary condition. The following regularity estimate holds }u}H s`2 pΩq À }f }H s pΩq ,
s ě ´1.
(1)
In this paper C will be used as a generic positive constant that may change at each occurrence, we use the notation a À b for a ď Cb. We will also use a „ b to denote a À b and b À a.
2 Preliminaries Let tTh uh be a family of quasi-uniform and shape regular triangulations. In a generic sense a node of the triangulation is designated by xi , K denotes a triangle of Th and F denotes a face of a triangle K. hK “ diampKq is the diameter of K and h “ maxKPTh hK the mesh parameter for a given triangulation Th . Pp pKq defines the space of polynomials of degree less than or equal to p on the element K, ΩT is the domain covered by the mesh Th , let us introduce the following finite element space ( Vhp “ v P H 1 pΩT q : v|K P Pp pKq @K P Th . For simplicity we will write the L2 -norm on a domain Θ, } ¨ }L2 pΘq as } ¨ }Θ . The domain Ω is embedded in a mesh Th . Figure 1 gives an example of a simple configuration. We define ρ as the signed distance function, negative on the inside and positive on the outside of Ω. The tubular neighbourhood of Γ is defined as Uδ pΓq “ tx P R2 : |ρpxq| ă δu. We consider a constant δ0 ą 0 such that the closest point mapping ppxq : Uδ0 pΓq Ñ Γ is well defined and we have the identity ppxq “ x ´ ρpxqnpppxqq. We suppose that δ0 is chosen small enough such that ppxq is a bijection. Let the polygonal domain Ωh with boundary Γh , be a domain approximating Ω. For simplicity we will assume that the discrete domain is defined by the zero levelset of the nodal interpolant of ρ on Vh1 . Then on a triangle K cut by the boundary, Γh restricted to K is a straight line. The
4
T. Boiveau, E. Burman, S, Claus and M. G. Larson
Ω
Figure 1. Domain Ω embedded in a background mesh Th .
discrete normal nh denotes the exterior unit normal to Γh . Observe that nh is constant on Γh |K . We define ph px, ςq “ x ` ςnh pxq, the function %h is defined by %h : Γh ÝÑ R such that ph px, %h pxqq P Γ for all x P Γh , for simplicity let ph pxq “ ph px, %h pxqq. Observe that %h is well defined for h small enough (see [9]). We assume that ph px, ςq P Uδ0 pΩq for all x P Γh and all ς between 0 and %h pxq. By our definition of Ωh we have δh “ }%h }L8 pΓh q “ Oph2 q.
(2)
Let ω be a subset of ΩT , we define ( Kh pωq “ K P Th | K X ω ‰ H ,
Nh pωq “ YKPKh pωq K,
and the norm ˛1
¨
2
ÿ }v}Nh pωq
“˝
}v}2K ‚
KPKh pωq
We will use the notations Kh “ Kh pΩ Y Ωh q,
KΓ “ Kh pΓq Y Kh pΓh q,
Nh “ Nh pΩ Y Ωh q.
.
Boundary value correction, penalty-free Nitsche method
5
We now recall the following trace inequalities for v P H 1 pNh q ´1
1
´ 21
1 2
2 }v}BK À phK 2 }v}K ` hK }∇v}K q
}v}KXΓ À phK }v}K ` hK }∇v}K q
@K P Kh ,
(3)
@K P Kh .
(4)
Inequality (4) has been shown in [13], we note that this is also true if we consider Γh instead of Γ. For vh P Vhp the following inverse estimate holds }∇vh }K À h´1 K }vh }K
@K P Kh .
(5)
The following inequality has been shown in [9] for all v P H 1 pΩh q }v}2Ωh zΩ À δh2 }∇v ¨ n}2Ωh zΩ ` δh }v}2Γh .
(6)
The ghost penalty [3] is introduced to ensure the well conditioning of the system matrix, it also provides the control of the gradient in case of small cut elements Jh puh , vh q “ γg
p ÿ ÿ F PFG l“1
h2l´1 xJDnl F uh KF , JDnl F vh KF yF ,
with FG “ tF P KΓ | F X pΩ Y Ωh q ‰ Hu , γg the ghost penalty parameter and nF the unit normal to the face F with fixed but arbitrary orientation. Dnl F is the partial derivative of order l in the direction nF and JwKF “ wF` ´ wF´ , with wF˘ “ limsÑ0` wpx ¯ snF q is the jump across a face F . The following estimate has been shown in [18] for all vh P Vhp }∇vh }2Nh À }∇vh }2Ωh ` Jh pvh , vh q À }∇vh }2Nh .
(7)
We now construct an interpolation operator πh . Let E be an H s -extension on Nh , E : H s pΩq Ñ H s pNh q such that for all w P pEwq|Ω “ w and }Ew}H s pNh q À }w}H s pΩq
@w P H s pΩq, s ě 0.
(8)
For simplicity we will write w instead of Ew. Let πh˚ : H s pNh q Ñ Vhp be the Lagrange interpolant, we construct the interpolation operator πh such that πh u “ πh˚ Eu.
(9)
We have the interpolation estimate for 0 ď r ď s ď p ` 1, }u ´ πh˚ u}H r pKq À hs´r |u|H s pKq
@K P Kh .
(10)
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T. Boiveau, E. Burman, S, Claus and M. G. Larson
Using the estimate (8) together with (10) we have }u ´ πh u}H r pNh q À hs´r |u|H s pΩq .
(11)
Let us introduce the norms ~w~2h “ }∇w}2Ωh ` h´1 }w}2Γh ` Jh pw, wq, }w}2˚ “ ~w~2h ` h}∇w ¨ nh }2Γh ` h´1 }T1,k pwq}2Γh , with the Taylor expansion defined such that Tm,k puqpxq “
k ÿ Dni h upxq i %h pxq, i! i“m
(12)
Dni h is the derivative of order i in the direction nh . Using the estimate (11) combined with the trace inequality (3) it is straightforward to show ~u ´ πh u~h ď }u ´ πh u}˚ À hp |u|H p`1 pΩq .
3
(13)
Finite element formulation
Here we use the boundary value correction approach from [9], we write the extensions of f and u respectively as f “ Ef and u “ Eu. pf, vqΩh “ pf ` ∆u, vqΩh ´ p∆u, vqΩh “ pf ` ∆u, vqΩh zΩ ` p∇u, ∇vqΩh ´ x∇u ¨ nh , vyΓh . We know that f ` ∆u “ 0 on Ω. On Ωh zΩ we have f ` ∆u “ Ef ` ∆Eu ‰ 0. Let us enforce weakly the boundary condition u “ g on Γ by adding a consistent boundary term pf, vqΩh “ pf ` ∆u, vqΩh zΩ ` p∇u, ∇vqΩh ´x∇u ¨ nh , vyΓh ` x∇v ¨ nh , u ˝ ph ´ g ˝ ph yΓh . Remark that this is equivalent to the penalty-free Nitsche’s method [2,4]. However, we cannot access u ˝ ph , so we use a Taylor approximation in the direction nh (12) u ˝ ph pxq « T0,k puqpxq.
(14)
We note that in (12) we could replace nh by n ˝ p and %h by % if these quantities were available (as mentioned in [9]). Adding and subtracting the Taylor expansion
7
Boundary value correction, penalty-free Nitsche method
in the Nitsche antisymmetric term and rearranging we obtain p∇u, ∇vqΩh ´ x∇u ¨ nh , vyΓh ` x∇v ¨ nh , T0,k puqyΓh ` pf ` ∆u, vqΩh zΩ ` x∇v ¨ nh , u ˝ ph ´ T0,k puqyΓh “ pf, vqΩh ` x∇v ¨ nh , g ˝ ph yΓh . (15) The discrete formulation is obtained by dropping the terms pf ` ∆u, vqΩh zΩ and x∇v ¨ nh , u ˝ ph ´ T0,k puqyΓh . Find uh P Vhp Ah puh , vh q ` Jh puh , vh q “ Lh pvh q
@vh P Vhp ,
(16)
with the linear forms Ah puh , vh q “ p∇uh , ∇vh qΩh ´ x∇uh ¨ nh , vh yΓh ` x∇vh ¨ nh , T0,k puh qyΓh , Lh pvh q “ pf, vh qΩh ` x∇vh ¨ nh , g ˝ ph yΓh . Using the definition of the Taylor expansion, the bilinear form Ah can be written as Ah puh , vh q “ p∇uh , ∇vh qΩh ´ x∇uh ¨ nh , vh yΓh `x∇vh ¨ nh , uh yΓh ` x∇vh ¨ nh , T1,k puh qyΓh . The terms that has been dropped in the discrete formulation are defined as Bh pu, vh q “ pf ` ∆u, vh qΩh zΩ ` x∇vh ¨ nh , u ˝ ph ´ T0,k puqyΓh
@vh P Vhp .
In Section 4 we show an inf-sup condition for the discrete formulation (16). In Section 5 the high order terms of Bh pu, vh q are treated. Section 6 presents the error estimates.
4 Inf-sup condition on Ωh We assume that Ωh is defined by the zero level set of the nodal interpolant Ih ρ of the distance function ρ. We also assume that h is small enough so that a band of elements in Kh pΓh q is in the tubular Uδ0 pΓq and that in every K P Kh pΓh q there holds }ρ ´ Ih ρ}L8 pKq ` h}∇pρ ´ Ih ρq}L8 pKq ď cρ h where the constant cρ only depends on the regularity of the interface and, since ρ is a distance function, for x P Kh pΓh q we have 1 ď |∇ρpxq| ď C1 ,
(17)
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T. Boiveau, E. Burman, S, Claus and M. G. Larson
with C1 ą 1 a constant of order 1. We now introduce boundary patches that will be useful for the upcoming inf-sup analysis. Let us consider the set Kh pΓh q and split it into Np smaller disjoint sets of elements Kj with j “ 1, . . . , Np , then we define Pj “ Kj Y tK P Th : K X Ωh ‰ H, DK 1 P Kj such that K X K 1 ‰ Hu. This means that Pj consists of the elements on Kj and its neighbours that intersect Ωh (we assume here that the mesh is truncated beyond Kh pΓh q so that there are no exterior neighbours, otherwise it is straightforward to handle them separately). Observe that the patches Pj overlap. For each patch Pj we define the faces Fj1 and Fj2 where BPj X Γh ‰ H. We define the interior elements of the patch by Kj˝ :“ tK P Kh pΓh q X Pj : K X pFj1 Y Fj2 q “ Hu. Let IPj be the set of vertices txi u in the patch Pj and the cardinality of IPj is NPj . We define the set of mesh vertices Ij that are in the interior of the patch Pj or on the outer boundary, ( Ij “ xi P K : K P Kj˝ , Figure 2 shows an example of a patch. Let Γj “ Γh X Kj denote the part of the boundary included in the patch Pj , for all j, the patch Pj has the following properties meas1 pΓj q „ h and meas2 pPj q „ h2 . (18) In (18) we can control the constant in both relations by choosing the patches to contain more elements (but uniformly under refinement). řNp The function vΓ is defined such that vΓ “ j“1 vj , for each patch Pj , the function vj has the form vj “ ζ ϕ˜ j with ζ P R. Let ϕ˜ j P Vh1 be defined for each node xi P Th such that # 0 for xi P IPj zIj ϕ˜ j pxi q “ ´ρpxi q for xi P Ij , with i “ 1, . . . , NPj . By the Poincaré inequality on a patch Pj the following inequality holds }vj }Pj À h}∇vj }Pj . (19) Lemma 4.1. For every patch Pj with 1 ď j ď Np ; @rj P R there exists vj P Vhp such that ż measpΓj q´1 ∇vj ¨ nh ds “ rj , (20) Γj
9
Boundary value correction, penalty-free Nitsche method
K7 K4 K3
K6 K5
K2 K1
Figure 2. Example of a patch Pj , in this case K3 Y K4 Y K5 “ Kj “ Kj˝ , ϕ˜ j is equal to zero on the nonfilled nodes.
and the following property holds 1
}∇vj }Pj À }h 2 rj }Γj .
(21)
Proof. The functions vj and ϕ˜ j are defined as previously described, let ż ∇ϕ˜ j ¨ nh ds. Ξj “ measpΓj q´1 Γj
We will first prove that for shape regular meshes, Ξj is strictly negative and bounded away from zero uniformly in h, provided the hidden constant of the upper bound in (18) is chosen large enough. Let K1 , . . . , Km be the triangles crossed by Γh within a patch Pj , the numbering of the crossed elements is done in order following a path of Γj as in figure 2. The assumption (18) says that the number of these triangles should be uniformly bounded by some m. We will show that the upper bound m only depends on the shape regularity of the mesh. The mesh size on the other hand must be small enough to resolve the boundary. The triangles K1 , K2 , Km´1 and Km will have nodes on the boundary of BPi where ϕ˜ “ 0 (on Fj1 or Fj2 ). Let us merge K1 and K2 (resp. Km´1 and Km ) into one quadrilateral ˝ ). Observe that for the triangles K , i “ 3, ..., m ´ 2 there element K1˝ (resp. Km i ˝ holds Ki P Kj and with Ih denoting the standard nodal interpolant on piecewise linear functions, pnh ¨ ∇ϕ˜ j |Ki q “ |∇Ih ρ|Ki |. ˝ the following upper bound holds trivially On K1˝ and Km
pnh ¨ ∇ϕ˜ j |Ki˝ q ă |∇ϕ˜ j |Ki˝ |,
i “ 1, m.
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T. Boiveau, E. Burman, S, Claus and M. G. Larson
Consider now ż Γh XPj
∇ϕ˜ j ¨ nh ds “
m ż ÿ i“1 Γj XKi
˝ ∇ϕ˜ j ¨ nh ds “ T1˝ ` Tm `
m´2 ÿ
Ti .
i“3
For i P t3, . . . , m ´ 2u using (17) we have Ti “ |∇πh ρ|Ki |measpΓj X Ki q ě measpΓj X Ki qp1 ´ |∇p1 ´ Ih qρ|Ki |q ě p1 ´ cρ hqmeaspΓj X Ki q. For i P t1, mu we have 1
Ti˝ ď h 2 }∇ϕ˜ j }Fj XKi˝ ď }∇ϕ˜ j }Ki˝ ď Ci h´1 }ϕ˜ j }Ki˝ ď Ci h´1{2 }Ih ρ}BKi˝ XBKj˝ ď Ci h´1{2 p}ρ}BKi˝ XBKj˝ ` }ρ ´ Ih ρ}BKi˝ XBKj˝ q ď cB h. The right hand side of these two inequalities depend only on the shape regularity of the mesh. Hence, using that measpΓ X Kj˝ q ě measpΓj q ´ 2h measpΓj q
´1
m ż ÿ i“1 Γj XKi
∇ϕ˜ j ¨ nh ds
ě p1 ´ cρ hqp1 ´ 2hmeaspΓj q´1 q ´ 2cB hmeaspΓj q´1 . Assume now that h is sufficiently small so that p1 ´ cρ hq ą measpΓj q´1
m ż ÿ i“1 Γj XKi
∇ϕ˜ j ¨ nh ds ě
1 2
then
1 ´ p2cB ` 1qhmeaspΓj q´1 . 2
If the lower constant of the left relation of (18) is larger than 4p2cB ` 1q we may conclude that m ż ÿ 1 Ξj “ measpΓj q´1 ∇ϕ˜ j ¨ nh ds ě , 4 i“1 Γj XKi which shows the uniform lower bound. Note that the constant cB only depends on the curvature of the boundary and the mesh geometry. Thanks to this lower bound we may define the normalised function ϕj by ϕj “ Ξ´1 ˜j. j ϕ
11
Boundary value correction, penalty-free Nitsche method
By definition there holds ż measpΓj q´1 Γj
∇ϕj ¨ nh ds “ 1,
(22)
and }∇ϕj }Pj “ Ξ´1 ˜ j }Pj . j }∇ϕ
(23)
The right hand side can be bounded as follows }∇ϕ˜ j }Pj ď }∇ϕ˜ j }Kj˝ ` }∇ϕ˜ j }Pj zKj˝ “ T1 ` T2 , 1
1
T1 ď }∇pπh ρ ´ ρq}Kj˝ ` }∇ρ}Kj˝ ď Ch 2 measpΓj q 2 , T2 ď Ci h´1 }ϕ˜ j }Pi zKj˝ ď Ch´1{2 }Ih ρ}BKj˝ 1
1
ď Ch´1{2 p}ρ}BKj˝ ` }Ih ρ ´ ρ}BKj˝ q ď Ch 2 measpΓj q 2 . We conclude that
1
}∇ϕj }Pj ď Ch 2 measpΓj q.
(24)
Let vj “ rj ϕj , then condition (20) is verified considering (22). The upper bound (21) is obtained using (24), (18) and 1
1
1
}∇vj }Pj “ |rj |}∇ϕj }Pj À measpΓj q 2 |rj |h 2 “ }h 2 rj }Γj .
Lemma 4.2. For uh , vh P Vhp with vh “ uh ` αvΓ , there exists a positive constant β0 such that the following inequality holds β0 ~uh ~2h ď Ah puh , vh q ` Jh puh , vh q. Proof. Using (7) it is straightforward to obtain pAh ` Jh qpuh , uh q “ p∇uh , ∇uh qΩh ` x∇uh ¨ nh , T1,k puh qyΓh ` Jh puh , uh q Á }∇uh }2Nh ` x∇uh ¨ nh , T1,k puh qyΓh .
We bound the second term using the trace inequality (4), inverse inequality (5) and (2) x∇uh ¨ nh , T1,k puh qyΓh ď }∇uh ¨ nh }Γh }T1,k puh q}Γh À h´1 }∇uh }Nh pΓh q }T1,k puh q}Nh pΓh q À γphq}∇uh }2Nh pΓh q ,
12
T. Boiveau, E. Burman, S, Claus and M. G. Larson
where γphq “ Section 2, for
řk
´
i“1 the j th
δh h
¯i
, note that Opγphqq “ h. Considering vΓ as defined in
patch Pj we get
αpAh ` Jh qpuh , vj q “ αp∇uh , ∇vj qPj XΩh ´ αx∇uh ¨ nh , vj yΓj ` αJh puh , vj q ` αx∇vj ¨ nh , uh yΓj ` αx∇vj ¨ nh , T1,k puh qyΓj , applying Cauchy-Schwarz inequality together with (7) we can write αp∇uh , ∇vj qPj XΩh ` αJh puh , vj q
1
1
ď α}∇uh }Pj XΩh }∇vj }Pj XΩh ` αJh puh , uh q 2 Jh pvj , vj q 2 , ď }∇uh }2Pj `
Cα2 }∇vj }2Pj . 4
Using the trace inequality (4), the inverse inequality (5) and the inequality (19) we can write the following αx∇uh ¨ nh , vj yΓj ď α}∇uh ¨ nh }Γj }vj }Γj À αh´1 }∇uh }Pj }vj }Pj À α}∇uh }Pj }∇vj }Pj ď }∇uh }2Pj ` Let us consider the average uj “ measpΓj q´1 choosing rj “
h´1 uj
ş Γj
Cα2 }∇vj }2Pj . 4
uh ds. Using Lemma 4.1 and
we get the inequality 1
}∇vj }Pj À }h´ 2 uj }Γj .
(25)
Our choice of rj allows us to write 1
αx∇vj ¨ nh , uh yΓj “ α}h´ 2 uj }2Γj ` αx∇vj ¨ nh , uh ´ uj yΓj . It is straightforward to observe }uh ´ uj }Γj ď Ch}∇uh }Γj ,
(26)
combining this result with the trace and inverse inequalities, we can show αx∇vj ¨ nh , uh ´ uj yΓj ď }∇uh }2Pj `
Cα2 }∇vj }2Pj . 4
Using (2) once again αx∇vj ¨ nh , T1,k puh qyΓj ď γphq2 }∇uh }2Pj `
Cα2 }∇vj }2Pj . 4
Boundary value correction, penalty-free Nitsche method
13
Each term has been bounded, we can now get back to pAh ` Jh qpuh , vh q ě pC ´ γphqq}∇uh }2Nh ` α
Np ÿ
1
}h´ 2 uj }2Γj
j“1 Np ÿ
´ 4
}∇uh }2Pj ´
j“1
Np Cα2 ÿ }∇vj }2Pj , j“1
using (25) and rearranging, we obtain pAh ` Jh qpuh , vh q ě pC ´ γphq ´
4q}∇uh }2Nh
Np ´ Cα ¯ ÿ ´ 1 j 2 `α 1´ }h 2 u }Γj . j“1
Using (26), the trace inequality and the inverse inequality we can show 1
1
}h´ 2 uj }2Γj ě }h´ 2 uh }2Γj ´ C 1 }∇uh }2Pj , using this result together with (7) we obtain ` ˘ pAh ` Jh qpuh , vh q ě pC ´ γphq ´ 4 ´ C 1 αq }∇uh }2Ωh ` Jh puh , uh q Np ´ Cα ¯ ÿ ´ 1 }h 2 uh }2Γj . `α 1´ j“1
It is easy to choose and α such that the two terms of this expression are positive, for example, by choosing “
1 16
we obtain α “ minp
C´γphq´ 14 C1
1 , 16C q.
Theorem 4.3. There exists a positive constant β such that for all function uh P Vhp the following inequality holds β~uh ~h ď sup
vh PVhp
Ah puh , vh q ` Jh puh , vh q . ~vh ~h
Proof. Considering Lemma 4.2 the only thing to show is ~vh ~h À ~uh ~h , Np ÿ
~vh ~h ď ~uh ~h ` ~vΓ ~h with
~vΓ ~h ď
~vj ~h ,
j“1 1
~vj ~2h “ }∇vj }2Pj XΩh ` }h´ 2 vj }2Γj ` Jh pvj , vj q. Using the trace inequality together with (19) and (25) we observe 1
1
1
}h´ 2 vj }2Γj À }∇vj }2Pj À }h´ 2 uj }Γj À }h´ 2 uh }Γj . We conclude using (7).
14
5
T. Boiveau, E. Burman, S, Claus and M. G. Larson
Boundary value correction
The goal of this section is bound the two high order terms that has been dropped in the finite element formulation (16). Theorem 5.1. Let Bh be the bilinear form as defined in (3) the following holds @vh P Vhp 1
Bh pu, vh q À ph´ 2 δhk`1 sup }Dk`1 u}Γt ` δhl`1 sup }Dnl pf ` ∆uq}Γt q~vh ~h . 0ďtďδ0
0ďtďδ0
Proof. Using the Cauchy-Schwarz inequality |Bh pu, vh q| ď }u ˝ ph pxq ´ T0,k puqpxq}Γh }∇vh ¨ nh }Γh ` }f ` ∆u}Ωh zΩ }vh }Ωh zΩ . By definition of the Taylor approximation we have ˇ ˇ ż %h pxq ˇ ˇ k Dnk`1 upxpsqqp% pxq ´ sq ds |u ˝ ph pxq ´ T0,k puqpxq| “ ˇ ˇ. h h 0
Using the Cauchy Schwarz inequality ż }u ˝ ph pxq ´
T0,k puqpxq}2Γh
}Dnk`1 u}2Ix }p%h pxq ´ sqk }2Ix ds h
ď Γh
ż }Dnk`1 u}2Iδ |%h pxq|2k`1 ds h
ď
h
Γh
ď δh2k`1 }Dk`1 u}2Uδ
h
pΓh q
ď δh2k`2 sup }Dk`1 u}2Γt , 0ďtďδ0
where Γt “ tx P Ω : |ρpxq| “ tu is the levelset with distance t to the boundary Γ, following the approach from [9]. Suppose that f ` ∆u P H l pUδ0 pΩqq, this property holds if f P H l pΩq by applying (1) and (8). Using Ωh zΩ P Uδ pΓq and δ „ δh it follows that l` 21
}f ` ∆u}Ωh zΩ À δhl }Dnl pf ` ∆uq}Ωh zΩ À δh
sup }Dnl pf ` ∆uq}Γt , 0ďtďδ0
Boundary value correction, penalty-free Nitsche method
15
1{2
Then we can bound the bilinear form Bh , using }v}Ωh zΩ À δh ~v~h (deduced from (6)) the trace and inverse inequalities @vh P Vhp |Bh pu, vh q| l` 12
À δhk`1 sup }Dk`1 u}Γt }∇vh ¨ nh }Γh ` δh 0ďtďδ0
À
1 ph´ 2 δhk`1
sup }D
k`1
u}Γt `
0ďtďδ0
δhl`1
sup }Dnl pf ` ∆uq}Γt }vh }Ωh zΩ
0ďtďδ0
sup }Dnl pf ` ∆uq}Γt q~vh ~h . 0ďtďδ0
(27)
6
A priori error estimate
The formulation (16) satisfies the following consistency relation (Galerkin orthogonality). Lemma 6.1. Let uh P Vhp be the solution of (16) and u P H 2 pNh q be the solution of (1), then Ah pu ´ uh , vh q ´ Jh puh , vh q ` Bh pu, vh q “ 0 , @vh P Vhp . Proof. Subtracting (15) and (16) this is straightforward. Lemma 6.2. Let w P H 2 pNh q ` Vhp and vh P Vhp there exists a positive constant M such that Ah pw, vh q ď M }w}˚ ~vh ~h . Proof. Using the Cauchy-Schwarz inequality, trace inequality and inverse inequality we have p∇w, ∇vh qΩh ´ x∇w ¨ nh , vh yΓh ` x∇vh ¨ nh , T0,k pwqyΓh À }∇w}Ωh }∇vh }Ωh ` }∇w ¨ nh }Γh }vh }Γh ` }∇vh }Ωh h´1 p}w}Γh ` }T1,k pwq}Γh q.
Using the inf-sup condition from Section 4 and the estimate obtained in Section 5 the error in the triple norm can now be estimated.
16
T. Boiveau, E. Burman, S, Claus and M. G. Larson
Proposition 6.3. Let u P H 2 pNh q be the solution of (1) and uh P Vhp the solution of (16), then 1
~u ´ uh ~h À hp }u}H p`1 pΩq ` h´ 2 δhk`1 sup }Dk`1 u}Γt 0ďtďδ0
` δhl`1 sup }Dnl pf ` ∆uq}Γt . 0ďtďδ0
1
p
hp
k
h´ 2 δhk`1
l
δhl`1
1 2 3 4
h1 h2 h3 h4
0 1 2 3
h1.5 h3.5 h5.5 h7.5
0 1 2 3
h2 h4 h6 h8
Table 1. Order of the terms in the estimation of the H 1 -error depending on p, k and l, assuming δh “ Oph2 q.
Proof. Using the Galerkin orthogonality of Lemma 6.1 we obtain Ah puh ´ πh u, vh q ` Jh puh ´ πh u, vh q “ Ah pu ´ πh u, vh q ´ Jh pπh u, vh q ` pf ` ∆u, vh qΩh zΩ ` x∇vh ¨ nh , u ˝ ph ´ T0,k puqyΓh . Using this result, Theorem 4.3, the Lemma 6.2 and Jh pu, vh q “ 0 given by the regularity of u, we can write β~uh ´πh u~h ď
Ah pu ´ πh u, vh q ´ Jh pπh u, vh q ` Bh pu, vh q ~vh ~h 1
1
Jh pπh u ´ u, πh u ´ uq 2 Jh pvh , vh q 2 ` Bh pu, vh q ď M }u ´ πh u}˚ ` ~vh ~h ď pM ` 1q}u ´ πh u}˚ `
Bh pu, vh q ~vh ~h
Applying the triangle inequality we can write ~u ´ uh ~h ď ~u ´ πh u~h ` ~uh ´ πh u~h 1´ Bh pu, vh q ¯ ď ~u ´ πh u~h ` pM ` 1q}u ´ πh u}˚ ` , β ~vh ~h Using the estimate (13) we conclude applying Theorem 5.1.
17
Boundary value correction, penalty-free Nitsche method
The next proof requires these two inequalities and the assumption δh À h2 3
}T1,k pu ´ uh q}Γh À hp`1 }u}H p`1 pΩq ` ph´ 2 δh qh~u ´ uh ~h 1 2
h }∇pu ´ uh q ¨ nh }Γh À hp }u}H 1 pΩq ` ~u ´ uh ~h ,
(28) (29)
inequality (28) is proved in [9], (29) can be shown using the trace and inverse inequalities in the following way 1
h 2 }∇pu ´ uh q ¨ nh }Γh À }∇pu ´ uh q ¨ nh }N pΓh q ` h}D2 pu ´ uh q}N pΓh q À }∇pu ´ πh uq ¨ nh }N pΓh q ` h}D2 pu ´ πh uq}N pΓh q ` }∇puh ´ πh uq ¨ nh }N pΓh q ` h}D2 puh ´ πh uq}N pΓh q À hp }u}H 1 pΩq ` }∇puh ´ πh uq}N pΓh q À hp }u}H 1 pΩq ` }∇pu ´ uh q}N pΓh q . Theorem 6.4. Let u P H 2 pNh q be the solution of (1) and uh P Vhp the solution of (16), we assume δh À h2 then 1
}u ´ uh }Ωh À hp` 2 }u}H p`1 pΩq ` δhk`1 sup }Dk`1 u}Γt 0ďtďδ0 1
` h 2 δhl`1 sup }Dnl pf ` ∆uq}Γt . 0ďtďδ0
1
1
p
hp` 2
k
δhk`1
l
h 2 δhl`1
1 2 3 4
h1.5 h2.5 h3.5 h4.5
0 1 2 3
h2 h4 h6 h8
0 1 2 3
h2.5 h4.5 h6.5 h8.5
Table 2. Order of the terms in the estimation of the L2 -error depending on p, k and l, assuming δh “ Oph2 q.
Proof. We define the function ψ such that # u ´ uh in Ωh ψ“ 0 in ΩzΩh .
18
T. Boiveau, E. Burman, S, Claus and M. G. Larson
Let z satisfy the adjoint problem ´∆z “ ψ
in Ω,
z“0
on Γ,
z is extended to Uδ0 pΩq using the extension operator. In this framework, the following estimates hold }z}H 2 pΩq À }ψ}ΩXΩh ,
(30)
}z}Ωh zΩ À δh }∇z ¨ n}Uδ
h
pΓq ,
(31)
pΓq ,
(32)
1
}z}Γh À δh2 }∇z ¨ n}Uδ
h
inequalities (31) and (32) has been shown in [9]. Using integration by parts, the L2 -error on Ωh can be written as }u´uh }Ωh “ pu ´ uh , ψ ` ∆zqΩh ´ pu ´ uh , ∆zqΩh “ pu ´ uh , ψ ` ∆zqΩh zΩ ` p∇pu ´ uh q, ∇zqΩh ´ xu ´ uh , ∇z ¨ nh yΓh “ pu ´ uh , ψ ` ∆zqΩh zΩ ` Ah pu ´ uh , zq ` x∇pu ´ uh q ¨ nh , zyΓh ´ 2x∇z ¨ nh , u ´ uh yΓh ´ x∇z ¨ nh , T1,k pu ´ uh qyΓh . Using (6), the property of the extension operator (8) and (30), we can write pu ´ uh ,ψ ` ∆zqΩh zΩ ď }u ´ uh }Ωh zΩ }ψ ` ∆z}Ωh zΩ 1
ď pδh2 }∇pu ´ uh q ¨ n}2Ωh zΩ ` δh }u ´ uh }2Γh q 2 p}ψ}Ωh zΩ ` }∆z}Ωh zΩ q 1
À pδh2 ` hδh q 2 ~u ´ uh ~h p}u ´ uh }Ωh zΩ ` }z}H 2 pΩq q 1
À pδh2 ` hδh q 2 ~u ´ uh ~h }u ´ uh }Ωh . Using the interpolant defined by (9) we obtain Ah pu ´ uh , zq ď |Ah pu ´ uh , z ´ πh zq ` Ah pu ´ uh , πh zq|, by Cauchy-Schwarz inequality, (28), (29) and (30) we have Ah pu ´ uh , z ´ πh zq 1
1
À p~u ´ uh ~h ` h 2 }∇pu ´ uh q ¨ nh }Γh ` h´ 2 }T1,k pu ´ uh q}Γh q~z ´ πh z~˚ À pp1 ` h´1 δh qh~u ´ uh ~h ` hp`1 }u}H 1 pΩq q}u ´ uh }Ωh .
19
Boundary value correction, penalty-free Nitsche method
The Galerkin orthogonality of Lemma 6.1 allows us to write Ah pu ´ uh , πh zq À |Bh pu, πh zq ` Jh puh , πh zq| 1
1
À |Bh pu, πh zq ` Jh puh , uh q 2 Jh pπh z, πh zq 2 |. From [8] and the properties of z we have 1
1
Jh pπh z, πh zq 2 “ Jh pπh z ´ z, πh z ´ zq 2 À h}z}H 2 pΩq À h}u ´ uh }Ωh , we also have the upper bound 1
1
Jh puh , uh q 2 À ~uh ´ πh u~h ` Jh pπh u, πh uq 2 À ~uh ´ πh u~h ` hp }u}H p`1 pΩq . Then using the proof of Proposition 6.3 we have ´ 1 |Jh pu, πh zq| À hp`1 }u}H p`1 pΩq ` h 2 δhk`1 sup }Dk`1 u}Γt 0ďtďδ0
¯ ` hδhl`1 sup }Dnl pf ` ∆uq}Γt }u ´ uh }Ωh . 0ďtďδ0
Using equation (27) the term Bh pu, πh zq can also be bounded with |Bh pu, πh zq| À δhk`1 sup }Dk`1 u}Γt }∇πh z ¨ nh }Γh 0ďtďδ0
l` 12
` δh
sup }Dnl pf ` ∆uq}Γt }πh z}Ωh zΩ .
0ďtďδ0
Using the global trace inequality }∇z ¨ nh }Γh À }z}H 2 pΩh q , (8) and (30) we can write }∇πh z ¨ nh }Γh ď }∇pπh z ´ zq ¨ nh }Γh ` }∇z ¨ nh }Γh 1
À h´ 2 ~πh z ´ z~˚ ` }z}H 2 pΩh q 1
À h 2 }z}H 2 pΩq ` }z}H 2 pΩh q À }u ´ uh }Ωh . Using inequalities (8), (30) and (31) we obtain }πh z}Ωh zΩ ď }πh z ´ z}Ωh zΩ ` }z}Ωh zΩ ď h2 }z}H 2 pΩq ` δh }∇z}Uδ ď h2 }u ´ uh }Ωh .
h
pΓq
20
T. Boiveau, E. Burman, S, Claus and M. G. Larson
Then we obtain the upper bound ´ |Bh pu, πh zq| À δhk`1 sup }Dk`1 u}Γt }∇πh z ¨ nh }Γh 0ďtďδ0 l` 12
` h2 δh
¯ sup }Dnl pf ` ∆uq}Γt }πh z}Ωh zΩ }u ´ uh }Ωh ,
0ďtďδ0
and ´ Ah pu ´ uh , zq À hp`1 }u}H p`1 pΩq ` δhk`1 sup }Dk`1 u}Γt 0ďtďδ0
l` 12
` h2 δh
¯ sup }Dnl pf ` ∆uq}Γt }u ´ uh }Ωh . 0ďtďδ0
Using (32) and (30) and the we have 1
}z}Γh À δh2 }∇z ¨ n}Uδ
h
pΓq
À δh sup }∇z ¨ n}Γt À δh }z}H 2 pΩq À δh }u ´ uh }Ωh , 0ďtďδ0
using this result with (29) and (28) we have |x∇pu ´ uh q ¨ nh , zyΓh ´ x∇z ¨ nh , T1,k pu ´ uh qyΓh | ď }∇pu ´ uh q ¨ nh }Γh }z}Γh ` }∇z ¨ nh }Γh }T1,k pu ´ uh q}Γh 1
À php }u}H 1 pΩq ` ~u ´ uh ~h qh´ 2 }z}Γh ` }z}H 2 pΩh q }T1,k pu ´ uh q}Γh 3
À ph´ 2 δh php`1 }u}H 1 pΩq ` h~u ´ uh ~h q ` }T1,k pu ´ uh q}Γh q}u ´ uh }Ωh 1
À pph´ 2 δh q~u ´ uh ~h ` hp`1 }u}H p`1 pΩq q}u ´ uh }Ωh . Also |x∇z ¨ nh , u ´ uh yΓh | ď }∇z ¨ nh }Γh }u ´ uh }Γh 1
ď }z}H 2 pΩh q h 2 ~u ´ uh ~h 1
ď }u ´ uh }Ωh h 2 ~u ´ uh ~h . Using δh À h2 we obtain the bound x∇pu ´ uh q ¨ nh , zyΓh ´ x∇z ¨ nh , T1,k pu ´ uh qyΓh ´ 2x∇z ¨ nh , u ´ uh yΓh ´ ¯ 1 À hp`1 }u}H p`1 pΩq ` h 2 ~u ´ uh ~h }u ´ uh }Ωh ´ 1 À hp` 2 }u}H p`1 pΩq ` δhk`1 sup }Dk`1 u}Γt 0ďtďδ0 1
1
l` 12
` h 2 δh2 δh The Theorem follows.
¯ sup }Dnl pf ` ∆uq}Γt }u ´ uh }Ωh . 0ďtďδ0
Boundary value correction, penalty-free Nitsche method
7
21
Numerical Results and Discussion
We will consider 3 examples of increasing complexity to corroborate the theoretical findings in the previous sections. The exact boundary of the domain Ω is described using analytical expressions of level set functions whose zero level set describes the boundary. We first consider a circular domain and a domain with convex and concave boundaries with zero dirichlet boundary conditions and then a flower shape domain with non-zero Dirichlet boundary conditions. We will demonstrate the effect of the boundary value correction terms for polynomial order 2 and 3. In all examples, we set the ghost penalty parameter to γp “ 0.1.
7.1
Reference Solution in Circle with Zero Dirichlet Boundary Conditions
In our first example, we consider a circular domain described by the zero level set of φ“R´1 a where R “ x2 ` y 2 . We investigate the convergence of the numerical solution to the following analytical solution
upx, yq “ cospπ
R2 q, 2
which we prescribed using ˆ ˆ 2 ˙˙ ˆ ˆ 2 ˙˙ R R f px, yq “ π R cos π ` 2π sin π . 2 2 2
2
The solution and the linear approximation of the domain are depicted in Figure 3. Figure 4 shows the convergence rates of the numerical solution in the H 1 and L2 norm. The order of convergence is optimal when a Taylor expansion of first order is used (k “ 1). Adding terms beyond the first order term in the Taylor expansion does not yield any improvment in the rate of convergence (k ą 1).
22
T. Boiveau, E. Burman, S, Claus and M. G. Larson
1.0 0.5
u
0.0 −0.5 1.0 0.5 −1.0
0.0
−0.5
0.0
x
−0.5 1.0 −1.0
0.5
y
1.0
y
0.5
0.0
−0.5
−1.0
−1.0
−0.5
0.0
0.5
1.0
x
Figure 3. Reference solution u and the cut finite element mesh in the circle geometry.
7.2
Reference Solution in Torus with Zero Dirichlet Boundary Conditions
Next, we consider a domain with convex and concave boundaries given by the zero level set of the function φ “ pR ´ 0.75q pR ´ 0.25q .
(33)
ˆ ˙ 1 f “20 4 ´ R
(34)
We set
and obtain the analytical solution u “ 20 p0.75 ´ Rq pR ´ 0.25q .
(35)
as shown in Figure 5. The convergence rates shown in Figure 6 are optimal for p “ 2, p “ 3 when a first order Taylor expansion is used in the boundary value correction terms.
23
10−2
10−2
2.34
2.34
10
0.01
0.1
log(h)
3.44
10−4
k=0 k=1 k=2
−4
1.67
10−3
2.19
10−3
log(||uex − uh ||H 1 (Ω) )
log(||uex − uh ||H 1 (Ω) )
Boundary value correction, penalty-free Nitsche method
10−5 10−6
3.44
0.01
10−3
10−3
2.16
10−4 10 10
−6
0.01
log(h)
−4
10−5 4.2
10−6
k=0 k=1 k=2
2.96
2.03
10
2.96
−5
0.1
log(h) (b) p “ 3
log(||uex − uh ||L2 (Ω) )
log(||uex − uh ||L2 (Ω) )
(a) p “ 2
k=0 k=1 k=2
0.1
10−7
4.2
10−8 0.01
(c) p “ 2
k=0 k=1 k=2
0.1
log(h) (d) p “ 3
Figure 4. Convergence rates for the first reference solution in the circle in the H1 norm for p “ 2, (b), p “ 3 for Taylor expansions of order k “ 0, 1, 2 and in the L2 norm for (c) p “ 2,(d) p “ 3.
7.3
Reference Solution in Flower Shape with Non-Zero Dirichlet Boundary Conditions
In our final example, we consider a flower like shaped domain [17] defined by φ “ pR2 ´ rθ qpR2 ´ p1.0{6.0q2 q
(36)
with rθ “ r0 ` 0.1 sinpωθq, r0 “ 1{2, ω “ 8 and θ “ arctanpx{yq. We investigate the convergence rates of our numerical solution with respect to ´ y¯ ´ x¯ cos π (37) u “ cos π 2 2
24
T. Boiveau, E. Burman, S, Claus and M. G. Larson
0.8 0.6
1.0 0.5
0.4 0.2
−0.2
0.5
x0.0
−0.4
0.0
y
0.5
0.0
y
1.5 1.0 0.5 u 0.0 0.5 1.0
−0.6
0.5
−0.8
−0.8 −0.6 −0.4 −0.2 0.0
1.0 1.0
0.2
0.4
0.6
0.8
x
(a)
(b)
Figure 5. Reference solution u in a torus-like shaped geometry and the cut finite element mesh of the domain.
f“
´ x¯ ´ y¯ π2 cos π cos π 2 2 2
(38)
The reference solution and the cut mesh are shown in Figure 7. Figure 8 shows the convergence rate for p “ 2 and p “ 3. 7.4
3D Solution in an Ellipsoid
We compute the reference solution u “ cospπ
rˆ2 q 2
(39)
with d rˆ “
y2 z2 x2 ` ` p3.0{4.0q2 p1.0{2.0q2 p1.0{2.0q2
(40)
in a 3D ellipsoid given by the function φ “ rˆ ´ 1.
(41)
Figure 9 shows the solution in the ellipsoid and Figure 10 shows the convergence for the solution for p “ 2 and k “ 1 demonstrating the optimal convergence rate of the numerical solution as predicted by the estimates of the previous section.
25
Boundary value correction, penalty-free Nitsche method
10−1
1.73
10
2.28
10−3 0.01
−2
10−3
k=0 k=1 k=2
0.1
log(h)
1.59
10
2.28
−2
log(||uex − uh ||H 1 (Ω) )
log(||uex − uh ||H 1 (Ω) )
10−1
3.42
10−4 10−5
0.01
(a) p “ 2
10−2
log(||uex − uh ||L2 (Ω) )
2.02
10−4
10−6 0.01
log(h) (c) p “ 2
−3
10−4
3.58
3.57
2.03
10
10−3
10−5
0.1
log(h) (b) p “ 3
log(||uex − uh ||L2 (Ω) )
10−2
k=0 k=1 k=2
3.42
4.36
10−5
k=0 k=1 k=2
0.1
10−6
4.37
10−7 10−8
0.01
k=0 k=1 k=2
log(h)
0.1
(d) p “ 3
Figure 6. Convergence rates for the reference solution in the domain including convex and concave boundaries in the H1 norm for (a) p “ 2, (b), p “ 3 for Taylor expansions of order k “ 0, 1, 2 and in the L2 norm for (c) p “ 2,(d) p “ 3.
Bibliography [1] P. Angot, H. Lomenède and I. Ramière, A general fictitious domain method with nonconforming structured meshes, Finite volumes for complex applications IV, ISTE, London, 2005, pp. 261–272. [2] T. Boiveau and E. Burman, A penalty-free Nitsche method for the weak imposition of boundary conditions in compressible and incompressible elasticity, IMA Journal of Numerical Analysis (2015). [3] E. Burman, Ghost penalty, C. R. Math. Acad. Sci. Paris 348 (2010), 1217–1220. [4] E. Burman, A penalty free nonsymmetric Nitsche-type method for the weak imposition of boundary conditions, SIAM J. Numer. Anal. 50 (2012), 1959–1981.
26
T. Boiveau, E. Burman, S, Claus and M. G. Larson
1.0 0.8 0.6
u
0.4 0.2 1.0 0.5 −1.0
−0.5
0.0 0.0
x
−0.5 1.0 −1.0
0.5
y
0.8 0.6 0.4
y
0.2 0.0 −0.2 −0.4 −0.6 −0.8
−0.8 −0.6 −0.4 −0.2 0.0
0.2
0.4
0.6
0.8
x
Figure 7. Reference solution u in flower geometry and the cut finite element mesh of the domain. [5] E. Burman, S. Claus, P. Hansbo, M. G. Larson and A. Massing, CutFEM: Discretizing geometry and partial differential equations, International Journal for Numerical Methods in Engineering (2014), n/a–n/a. [6] E. Burman and M. A. Fernández, Explicit strategies for incompressible fluidstructure interaction problems: Nitsche type mortaring versus Robin-Robin coupling, Internat. J. Numer. Methods Engrg. 97 (2014), 739–758. [7] E. Burman and P. Hansbo, Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method, Comput. Methods Appl. Mech. Engrg. 199 (2010), 2680–2686. [8] E. Burman and P. Hansbo, Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method, Appl. Numer. Math. 62 (2012), 328–341. [9] E. Burman, P. Hansbo and M. G. Larson, A Cut Finite Element Method with Boundary Value Correction, (2015). [10] J. Freund and R. Stenberg, On weakly imposed boundary conditions for second order problems, in: Proceedings of the Ninth International Conference on Finite Elements in Fluids, pp. 327–336, Università di Padova, 1995.
27
Boundary value correction, penalty-free Nitsche method
10−2
10−3
10
10−4
−3 2.4
10
−4
10−5
1.74
10−2
1.79
log(||uex − uh ||H 1 (Ω) )
log(||uex − uh ||H 1 (Ω) )
10−1
k=0 k=1 k=2
2.4
0.01
10−5 10
0.01
(a) p “ 2
log(||uex − uh ||L2 (Ω) )
2.07
10−4 10−5
3.24
10−6
3.25
10−7 0.01
log(h) (c) p “ 2
k=0 k=1 k=2
0.1
log(||uex − uh ||L2 (Ω) )
10−2
10
0.1
log(h) (b) p “ 3
10−2 −3
k=0 k=1 k=2
3.55
−7
0.1
log(h)
3.5
10−6
2.06
10
−4
10−6 4.44
10−8
10−10
3.98
0.01
log(h)
k=0 k=1 k=2
0.1
(d) p “ 3
Figure 8. Convergence rates for the reference solution in the flower shaped domain in the H1 norm for (a) p “ 2, (b), p “ 3 for Taylor expansions of order k “ 0, 1, 2 and in the L2 norm for (c) p “ 2,(d) p “ 3.
[11] V. Girault and R. Glowinski, Error analysis of a fictitious domain method applied to a Dirichlet problem, Japan Journal of Industrial and Applied Mathematics 12 (1995), 487–514 (English). [12] V. Girault, R. Glowinski and T. W. Pan, A fictitious-domain method with distributed multiplier for the Stokes problem, Applied nonlinear analysis, Kluwer/Plenum, New York, 1999, pp. 159–174. [13] A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg. 191 (2002), 5537–5552.
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T. Boiveau, E. Burman, S, Claus and M. G. Larson
Figure 9. Reference solution u in a 3D ellipsoid. 2.51
log(||uex − uh ||Ωh )
10−1 10−2
2.87
10−3 10−4
L2 (u) H1 (u) 0.04 0.05 0.06 0.08 log(h)
0.1
0.13
Figure 10. Error for p “ 2 in ellipsoid for k “ 1. [14] J. Haslinger and Y. Renard, A New Fictitious Domain Approach Inspired by the Extended Finite Element Method, SIAM Journal on Numerical Analysis 47 (2009), 1474–1499. [15] T. J. R. Hughes, G. Engel, L. Mazzei and M. G. Larson, A comparison of discontinuous and continuous Galerkin methods based on error estimates, conservation, robustness and efficiency, Discontinuous Galerkin methods (Newport, RI, 1999), Lect. Notes Comput. Sci. Eng. 11, Springer, Berlin, 2000, pp. 135–146. [16] M. G. Larson and A. J. Niklasson, Analysis of a nonsymmetric discontinuous Galerkin method for elliptic problems: stability and energy error estimates, SIAM J. Numer. Anal. 42 (2004), 252–264 (electronic).
Boundary value correction, penalty-free Nitsche method
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[17] C. Lehrenfeld, High order unfitted finite element methods on level set domains using isoparametric mappings, Computer Methods in Applied Mechanics and Engineering 300 (2016), 716–733. [18] A. Massing, M. G. Larson, A. Logg and M. E. Rognes, A Stabilized Nitsche Fictitious Domain Method for the Stokes Problem, Journal of Scientific Computing 61 (2014), 604–628 (English). [19] J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 36 (1971), 9–15 (German).
Received ???. Author information Thomas Boiveau, Department of Mathematics, University College London, Gower Street, London, UK-WC1E 6BT, United Kingdom. E-mail:
[email protected] Erik Burman, Department of Mathematics, University College London, Gower Street, London, UK-WC1E 6BT, United Kingdom. Susanne Claus, Cardiff School of Engineering, Cardiff University, Queen’s Buildings, The Parade, Cardiff CF24 3AA, Wales, United Kingdom. Mats G. Larson, Department of Mathematics and Mathematical Statistics, Umeå University, SE-90187 Umeå, Sweden.
