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Interior Penalty Discontinuous Galerkin BEM for the Helmholtz equation: Theoretical and numerical analysis
Nadir-Alexandre Messai, Pernet Sébastien
To cite this version:
Nadir-Alexandre Messai, Pernet Sébastien. Interior Penalty Discontinuous Galerkin BEM for the Helmholtz equation: Theoretical and numerical analysis. 2019. �hal-02425037�
Interior Penalty Discontinuous Galerkin BEM for the Helmholtz equation: Theoretical and numerical analysis
Messai Nadir-Alexandrea, Pernet Sébastiena
aONERA/DTIS, Université fédérale de Toulouse F-31000 Toulouse France
Abstract
In this study we propose the construction and the firsthp a priorierror analysis of a discontinuous Galerkin numerical scheme applied to the integral equation related to the Helmholtz problem in 3D. The main results of this article are an error bound in a broken norm suited to the problem and in a more classicalL2-norm. Those bounds are quasi optimal for thehconvergence and sub optimal for thepconvergence. Various formulation choices and penalty functions are theoretically and numerically discussed. We confirm the advantage of using a symmetric formulation in the context of DG in integral equations. We also give an extensive numerical study of the scheme, particularly thehandpconvergence. The very important cases of non conforming mesh and varying polynomial order are treated, which indicates the method’s ability to handle hp refinement strategies.
Keywords: Integral equation, Boundary element method, Helmholtz equation, Discontinuous Galerkin, ahp priori error analysis
1. Introduction
Wave propagation and scattering phenomenon appear in many fields of science, engineering and industry. It is of significant importance in geo-science, petroleum engineering, telecommu- nications, defense industry, and obviously acoustics. The simplest model of wave scattering by an object is the famous Helmholtz equation (recalled in section 1) which is the rule equation of acoustics in an homogeneous medium but also indirectly arises in more complex wave models (electromagnetism and elastodynamics). Despite its apparently simple form, it is a difficult equa- tion to numerically solve as it is strongly non-Hermitian and its solutions are very oscillatory.
These properties in turn make it hard to build a stable and efficient numerical scheme under practical mesh constraint.
An other difficulty of this equation is that the propagation domain is generally infinite. Roughly speaking, two main families of approaches have been explored to overcome this numerical dif- ficulty: solving the Helmholtz equation while bounding the calculation domain with artificial boundary conditions (see PML [EM79, Ber94] and the references therein); or using integral equations (see [McL00, Ned01, Cos88]). The last one is known to be a very powerful formalism when a large propagation domain is considered. It consists in transforming the initial Helmholtz equation into an equivalent integral equation with its unknowns being Cauchy data living on the surface of the scattering object. In our study we choose to work in the scope of boundary integral equations.
In real life applications, the scattering object is often large in comparison with the wavelength of interest. Moreover, the geometrical singularities of this object make the solution non-smooth.
As a consequence a classical Boundary Element Method (BEM) requires a fine mesh in order to compute an accurate numerical solution.
Recent works were devoted to the development of aposteriorierror estimates for wave Bound- ary integral equation [Bak]. This enabled the development of auto adaptive loop strategies with local refinement procedures [MK13]. Those architectures appear to be an elegant methodology to solve very large problems with an optimal mesh size regarding a desired accuracy.
Those auto-adaptive loops’ computational efficiency could be largely improved by the use of non conforming finite element method, or Discontinuous Galerkin (DG) scheme. Indeed, in the same fashion as Galerkin approximation of elliptic PDE equations, the absence of conforming constraint would allow a greater mesh flexibility. This would also enable to work with non con- forming meshes (hanging nodes), to locally enrich the polynomial space (variation of the local polynomial order of the approximation space). This so called non conforming hp-refinement couldn’t be accomplished with a BEM. The use of a DG would also ease the mesh generation process for complex geometries [PLL13](generation of a complex mesh per part and fusion). We see all those potential advantages as a motivation to use discontinuous approaches in the field of integral equations.
Unfortunately, as far as we know relatively little is known about the non conforming ap- proximations of those integral operators. Most of the litterature is focused on the case of the Laplace equation [HM13, CH12, Heu01]. The case of the oscillating kernel is less treated and the literature found was mostly focused on numerical study and validation ([KS18, PLL13]) As far as we know, [HS15] is the only article proposing a theoretical study of the hyper-singular operator, in a domain decomposition fashion. More precisely they consider finite element patches with non conforming interfaces.
In this article, we present the first theoretical and numerical analysis of an Interior Penalty Dis- continuous Galerkin (IPDG) method for the oscillating hypersingular operator used for solving the Helmholtz equation. The structure of the analysis differs from [HS15]. Indeed, we propose an hp non-conforming analysis for symmetric and anti-symmetric scheme on closed surface for any wavenumber, where [HS15] treats the case of the anti-symmetric bilinear form with low order approximation space for open surface and low wave number.
The main technical contributions of the article are:
• The ellipticity of the Laplace bilinear form in the symmetric case (the anti-symmetric case is proved in [HM13]), see proposition 3.
• A continuity and a Garding inequalities for the symmetric and the anti symmetric bilinear forms (see proposition 6).
• An a priori error bound in a DG norm and in a L2 norm. The bounds are optimal in h convergence and sub optimal for thepconvergence (see main theorems).
• The definition and the use of a lifting bilinear form for the proof, and an error estimate of the residual (see subsection 4.3).
• A discussion about the condition number and the comparison of the symmetric and anti symmetric schemes.
• Numerical test cases in 3D consideringhpnon conformity.
The remainder of the article is organized as follow. We first recall the problem model and the main integral operators in the section 2.1 before we build the IPDG scheme in 2.2. The
construction is based on a non trivial integration per part formula which is recalled and briefly justified. We present afterward the main theorems of the article: an apriori error estimate in a DG norm and in aL2norm. An originality of the problem is the great influence of the scheme parameters (the penalty function) over the theoretical error bound. Several penalty functions’
form are proposed and discussed. The section 4 contains the proof of the main results. The main ingredients are a conforming projector, the use of an original lifting operator, a Garding-type inequality and a duality approach. Finally, section 5 deals with numerical convergence rates on an example to illustrate the theorem and also addresses practical considerations: the condition number of the Galerkin matrix and the robustness of the method in case of non conforming meshes (hanging nodes and varying polynomial order).
2. Construction of the Discontinuous Galerkin formulation
In this first part the problem model and its main properties will be reminded. The construc- tion of the discontinuous Galerkin formulation will be exposed afterward.
During all the rest of the article, we note Γ a closed polyhedral and Lipschitz surface.
We note (Γi)i∈[1,N] its faces. Nevertheless the construction of the scheme introduced in the section below is also valid for an open surface.
2.1. Problem model and integral operators
We constantly use in this work surface fractional Sobolev spaces. Fors≥0 we noteHs(Γ) and H˜s(Γ) the completion ofC∞(Γ) and respectivelyCo∞(Γ) with the norms||.||Hs(Γ)and||.||H˜s(Γ). We call H−s(Γ) and ˜H−s(Γ) the dual spaces (with L2 as pivot space) of ˜Hs(Γ) and Hs(Γ), respectively. For a complete description of those spaces, see [McL00] and [Gri11].
The problem model and its discontinuous Galerkin discretization are expressed with two inte- gral operators living on the Sobolev spaces introduced above. We need to recall those operators before introducing the model problem.
Definition 1 (Single layer operator). Let s ∈ [−1,0]. We define the single layer integral operatorV :Hs(Γ)→Hs+1(Γ) such that∀u∈Hs(Γ):
∀x∈Γ, Vu(x) = Z
Γ
g(x, y)u(y)dΓ(y), (1)
withg(x, y) = eik||x−y||
4π||x−y|| being the Green kernel of the Helmholtz equation, wherek∈R+. We also need the following second integral operator.
Definition 2 (Hypersingular operator). Lets∈[0,1]. We define the hypersingular integral operatorW:Hs(Γ)→Hs−1(Γ) such that∀u∈Hs(Γ):
∀x∈Γ, Wu(x) =− ∂
∂n(x) Z
Γ
u(y)∂g(x, y)
∂n(y) dΓ(y). (2)
Remark 1. In the definition of the double layer operator the integral has to be understood as a finite part integral.
An extensive description of those operators can be found in [McL00], [Ned01] and [Cos88]. They will explicitly be present in the definition of the problem model below.
Our physical scope of interest is the scattering of an acoustic wave by a rigid object in an homogeneous medium. The initial problem corresponding to this physical set is the well know exterior Helmholtz equation:
∆uv(x) +k2uv(x) = 0∀x∈R3\Ω
∂uv(x)
∂n(x) =g∀x∈Γ lim|x|→∞|x| ∂|x|uv−ikuv
= 0
(3)
whereuv is the scattered field’s potential (either for the pressure or the velocity),ga boundary data (for example the Neumann trace of a plane wave), and k ∈ R+ the wave number of the problem. We notenthe exterior normal unit vector on Γ, and Ω the bounded domain such that
∂Ω = Γ.
The integral formalism is well-adapted for solving this kind of unbounded problem. This formalism is therefore used in this work. It consists in building an equivalent integral equa- tion using the classical integral representation formulas. For details about the construction of those equivalent problems, see for instance [Ned01] or [McL00]. In our case, the corresponding equivalent integral equation is the following problem model.
Problem 1 (Problem model). For a givenf ∈L2(Γ), findu∈H12(Γ) such that:
Wu=f (4)
Remark 2. From [Ned01], this problem admits an unique solution in H12(Γ), expect for a discrete set of wavenumbers. It is well known from the properties of W that the regularity of the right-hand sidef impliesu∈H1(Γ).
Remark 3. The right-hand side f is naturally connected to the data of the initial Helmholtz problem (3) such that f = [[uv]]. So the apparent limiting condition f ∈ L2(Γ) is finally very general and englobes almost all the physical configurations.
Proposition 1 (Relation between V and W). From [Ned01], the following relation exists between the hypersingular and the single layer potential:
∀u∈H12(Γ), Wu=curlΓ(VcurlΓu)−k2n· V(un). (5) Where curlΓ being the surface curl operator on Γ and curlΓ its adjoins operator. They are studied in details in [Heu01].
The relation above is essential for practical application of the method. Indeed it enables to decrease the order of singularity of the operator. This formula will also be instrumental in the construction of the discontinuous Galerkin formulation.
2.2. Discontinous Galerkin construction 2.2.1. Integration per part formula
The construction of the discontinuous Galerkin method generally requires a local integration by parts formula in order to introduce the trace of the function on the skeleton of the mesh.
In the case of classical PDE systems (Maxwell, Helmholtz, Poisson, etc.), one generally uses classical Stoke identities (see [MPS13, HPSS05, HTX]). The situation is a bit more difficult in the case of a field living on a surface. Indeed one has to be careful to maintain the trace of the function on the skeleton well defined.
Our need here is to obtain a integration per part formula in order to transform a term of the formhcurlΓ(VcurlΓu), viΓ.
For regular functions the result is straightforward. Indeed, applying the trivial proposition 10 (which is a consequence of the Stoke formula) on a sufficiently regular sub domainQ⊂Γ, one can get foru, v∈H1(Γ):
hcurlQ(VcurlΓu), viQ=hVcurlΓu,curlΓviQ+ht∂Q· VcurlQu, vi∂Q, (6) witht∂Qbeing the tangent unit vector of∂Qwith positive orientation. In the formula above and in all the remaining of the article,curlQ will denote the restriction of curlΓ to the sub domain Q.
Unfortunately, the term on the frontier∂Qis not well defined forv∈L2(Q) as the trace operator γo:Hs(Q)→Hs−12(∂Q) exists and is continuous only fors > 12 (see [Cos88]). But the formula (6) can be extended through the definition of the linear and bounded operatort∂Q· VcurlΓu:
u∈H12(Γ);Wu∈L2(Γ) −→ H−(∂Q)
u 7−→ t∂Q· VcurlΓu (7)
defined by the relation:
∀v∈H12+(Q), ht∂Q· VcurlΓu, vi∂Q =hcurlQ(VcurlΓu), viQ− hVcurlΓu,curlQviQ, (8) with >0. See [Heu01] or [GHH] for the details of the proof. This integration per part formula will be used for the construction of the DG formulation. The formula shows that an extra regularity to the test function space is required. It needs to be at least in H12+(Γ) in order to have a skeleton trace well defined. It also reveals the impossibility of building an equivalent
"broken" formulation of the problem model.
Remark 4. This extension to a bounded operator only serves for a problem on an open surface.
Indeed in this case there would beu∈H1−ε(Γ) and so the formula (6) can no longer be applied.
This result can serve for further investigation on open surface case.
2.2.2. Construction of the Discontinuous Galerkin formulations
Let Th be a conforming quasi-uniform mesh whose elements are shape regular trianglesK.
For all elementK∈ Th we notehK its diameter. We also defineh= max
K∈Th
hK. The set of edges ofThis notedEh. We also need the skeleton of the meshγh= [
e∈Eh
e. We restrict ourselves in the theoretical work toconformingandγ-shape regularmeshes [Mel05] for sake of simplicity.
Notation: In all the rest of the document we noteab(resp. ab ora'bif there exists c ∈ R+ independent of the approximation parameters (i.e. h and p) such that a ≤ cb (resp.
a≥cbora=cb).
Now let’s construct the DG formulation. The idea is to get a "broken" formulation by splitting the expression on the mesh. Thanks to the L2 regularity of f, the duality product in (4) corresponds to anL2(Γ) scalar product. So by linearity, (4) becomes:
X
K∈Th
hWu, viK=hf, viΓ, ∀v∈L2(Γ). (9)
Then we use the integration by part formula (8) and the relation (5) betweenVandW, equation (9) becomes, forv∈H12+(K) (for anyK∈ Th):
X
K∈Th
hVcurlΓu,curlKviK−k2hV(un), vniK+htK.VcurlΓu, vi∂K =hf, viΓ. (10) Following what’s classically done in DG literature, we write the skeleton term as a sum over the edges so that the jumps of the test function appear. As for each edgee∈ Ehthere isK1, K2∈ Th
such thate=K1∩K2, we obtain:
X
K∈Th
htK.VcurlΓu, vi∂K = X
e=K1∩K2⊂γh
htK1.VcurlΓu, vie+htK2.VcurlΓu, vie, (11) with tK1 =−tK2. We choose for each e ∈ Eh an arbitrary but fixed tangent vector te= tK1. We can rewrite the term so that:
X
K∈Th
htK· VcurlΓu, vi∂K = X
e∈Eh
hte· VcurlΓu,[[v]]ie, (12) with: [[v]] =v|K1−v|K2 being the jump ofv on an edgee=K1∩K2.
Remark 5. We implicitly used the regularityVcurlΓu∈H1(Γ), which implies a weak continu- ity: for anye=K1∩K2∈ Eh, VcurlΓu|K1 =VcurlΓu|K2 almost everywhere one.
If we consider a conforming approximation spaceV ⊂H12+(Γ), the term (12) would vanish as [[v]] = 0 for any v ∈V. The formulation would then become a classic boundary finite element method. But the equation (9) also accepts more general (broken) test space of the structure
Hdg(Th) ={v∈L2(Γ), v|K ∈H12+ε(K)∀K∈ Th}. (13) withε >0. For practical reasons, we define the brokencurloperatorcurlh by parts such that:
∀K∈ Th, curlh(v)|K=curlK(v|K). (14) This notation enables to write the initial weak formulation under a compact form. The equations from (9) to (12) imply:
Proposition 2 (Discontinuous weak formulation). Letube the solution of problem 1 then:
∀v∈Hdg(Th), A(u, v) =hf, vi (15)
with:
A(u, v) =hVcurlhu,curlhviΓ−k2hV(un), vniΓ+hte.Vcurlhu,[[v]]iγh. (16) Unfortunately, It is known that this kind of weak formulation doesn’t imply stable numerical schemes (it will be showed later). Following the IPDG literature ([FW08, MPS13, HPSS05, HTX]), a penalty term is added in the formulation. It has the following general form:
Pσ(u, v) =hσh[[u]],[[v]]iγh, ∀u, v∈Hdg(Th), (17) whereσh:γh→R+is the penalty function to determine. It’s role is to stabilize the formulation.
It will also force the continuity of the solution.
For numerical and practical reasons, we choose to work with a broken polynomial space Xhp={v∈L2(Γ), v|K ∈Pp(K)∀K∈ Th} ⊂Hdg(Th) (18) with Pp(K) being the polynomial space of degree p on K. We also define pK the local poly- nomial order on the elementK ∈ Th. We then derive from proposition 2 the discrete Galerkin formulation:
Problem 2 (Discontinuous Galerkin formulation). Seekuh∈Xhp such that:
∀v∈Xhp, Aθh(uh, v) =hf, vi (19) where
Aθh(u, v) =hVcurlhuh,curlhviΓ−k2hV(uhn), vniΓ
+hT uh,[[v]]iγh+Pσ(uh, v) +θhT v,[[uh]]iγh, (20) withhT u, viγh=hte· Vcurlhu, viγh simply being an abbreviate notation.
A consistent termhT v,[[uh]]iγh is added in the bilinear form, where the parameterθcan take the values−1,0,1. It enables to have formulations with distinct properties. The caseθ = 1 is the
"symmetric" formulation and therefore shall have good numerical properties. Choosingθ=−1 gives the "anti-symmetric" formulation. We call theθ= 0 case the "natural formulation".
The θ values {−1,0,1} respectively lead to the "anti-symmetric", "natural" and "symmetric"
formulation. This latter is expected to have good numerical properties ([Riv08]).
3. A priori error estimates : main results
The main result of the article is an hp a priori error analysis for the GD formulation from problem 2. The nature of the operators and the spaces involved in the formulation guided us in the definition of a suited norm for this study.
We equip the spaceHdg(Th) with the following "DG norm":
||u||dg=
||curlhu||2
H−12(Γ)+||u||2L2(Γ)+||σh12[[u]]||2L2(γh)
12
. (21)
The error analysis can be accomplished using a suited penalty function. It will be showed later that a particular behavior forσhis required to stabilize the DG formulation. For all the rest of the article, we adopt the following form for the penalty function:
Definition 3 (Penalty function). We define the penalty functionσh:γh→R+such that:
∀e∈ Eh, σh(e) =σopnep
hneh
, (22)
wherenp≥1,nh>1 andσo>0 are real constants, called the penalty function’s parameters.
The mesh’s parameters pe and he are defined using a local averaging of the functionsh andp such that:
∀e∈ Eh, pe=pK1+pK2
2 andhe= hK1+hK2
2 , (23)
whereK1, K2∈ Th such thate=K1∩K2.
3.1. Error estimate in DG norm
Let’s now introduce the main results of this paper.
Theorem 1 (Main result: a priori error estimate). Letu∈Hr(Γ)be the solution of prob- lem 1, withr≥1. Letσh be the penalty function from definition 3 withnh>3 andnp>4.
Let uh denote the DG approximation defined in problem 2. There exists ho>0 andpo≥1such that for any spaceXhp such thatXhopo ⊂Xhp, the following error estimate holds:
||u−uh||dg max
K∈Th
h
nh−3 2 +µK K
pr+
np−4 2
K
||u||Hr(Γ)+ max
K∈Th
hmin(0,
nh−4 2 ) K
pmin(0,
np−5 2 ) K
inf
v∈Xhp∩Co(Γ)||v−u||dg, (24) with : µK = min(pK+ 1, r), for anyK∈ Th.
A proof of this theorem will be given in section 4. Following the lines of [VD99], the theorem (1) also serves to prove the existence ofuh.
Corollary 1 (Existence of the numerical solution). let the conditions from the main the- orem 1 be fulfilled. Let σh be the penalty function from definition 3, withnh >3 andpK >4.
Then the problem 2 admits a unique solutionuh∈Xhp.
Proof. Using a finite dimensional argument, we just have to show thatuh= 0 when the right hand sidef = 0. Iff = 0, then u= 0 as well and then the apriori estimate from theorem (1) leads to||uh||dg≤0.
We can use standard interpolation theory results (see [Ste08, EG04]) to estimate the infimum, in order to obtain anhpexplicit form of the apriori error estimate of theorem 1.
Corollary 2 (Explicit hp version). With the same conditions as in theorem 1, the explicit hp error estimate holds:
||u−uh||dg max
K∈Th
hmin(r−
1
2,r+nh2−5) K
pmin(r−
1
2,r−6−2np) K
||u||Hr(Γ). (25)
Keeping the polynomial order p≥1 fixed on the mesh and varyinghleads to the h-version of the discontinuous Galerkin scheme. We can particularize the main theorem in this case to get theh−convergence estimate theorem below.
Corollary 3 (explicith-version error estimate). With the same conditions as in theorem 1, the explicitperror estimate holds:
||u−uh||dg max
K∈Th
hmin(r−
1
2,r+nh2−5) K
||u||Hr(Γ). (26) Remark 6 (On the theoretical importance of the penalty parameternh). The penalty parameternh plays an important role in the convergence rate of the formulation. Depending on the selected value, the DG method can be in three distinct regimes:
• r+ nh2−5 ≤0: the error isn’t controlled by the theorem. Nothing can be said about the convergence of the numerical scheme.
• 12 ≤r+nh2−4 ≤r: convergence rate ofO
hr+nh2−5
which is slower than the BEM rate of convergence.
• nh≥4: convergence rate ofO hr−12
, as fast as the conforming BEM method.
Thus an user shall choose nh≥4 in order to have an optimal convergence rate. This behavior doesn’t appear in other IPDG methods ([FW08, MPS13, HPSS05, HTX]) . It is a particularity of IPDG method for the hypersingular operator.
Nevertheless we will see in the section 5 that the method is numerically more robust that the theorem statement and can handle a wider range of penalty function parameters’ values.
Keeping a constant mesh (hconstant) and varying the local polynomial orderpK andpleads to the so calledp-version of the DG method. We also particularize the error estimate from the main theorem to this case.
Corollary 4 (Explicit p-version error estimate). With the same conditions as in theorem 1, the explicitperror estimate holds:
||u−uh||dg max
K∈Th
1 pmin(r−
1
2,r−6−2np) K
||u||Hr(Γ). (27)
Remark 7 (On the theoretical importance of the penalty parameternp). The penalty parameternp plays an important role in the convergence rate of the formulation. Depending on the selected value, the DG method can be in three distinct regimes:
• r+ np2−6 ≤0: the error isn’t controlled by the theorem. Nothing can be said about the convergence of the numerical scheme.
• 12 ≤r+np2−6 ≤r: convergence rate ofO 1 pr+np−62
!
which is slower than the BEM rate of convergence.
• np≥5: convergence rate ofO
1 pr−12
, as fast as the conforming BEM method.
Thus a user shall choose np ≥5 in order to have an optimal convergence rate. As for the h- convergence, we will see in the section 5 that the method is numerically more robust that the theorem’s statement and can handle a much wider range of penalty function parameters’ values.
3.2. Error estimate inL2(Γ)norm
The theorem above guarantees the reliability of the numerical scheme. An other estimate in theL2(Γ) norm can be obtained, and is very relevant from a more practical point of view.
Theorem 2 (L2 error estimate). Let’s consider the same hypothesis as in the main theorem 1. The following error estimate inL2 norm holds:
||u−uh||L2(Γ) max
K∈Th
hrK prK
||u||Hr(Γ)+ max
K∈Th
hmin(
1 2,nh2−3)
K pmax(−
1 2,4−2np) K
||u−uh||DG. (28) We also present apart the particular cases of thehandpconvergence.
Corollary 5 (L2 error: h-version). Let’s consider the same conditions as in theorem 1. Let σh be the penalty function from definition 3 withnp ≥0 and nh >3 a positive constant. Let the polynomial orderpbe fixed on the mesh. Consideringnh∈]3,4], the following error estimate holds:
||u−uh||L2(Γ) max
K∈Th
n hr−nh
−4 2
o||u||Hr(Γ) (29)
whereas fornh≥4 there is
||u−uh||L2(Γ) max
K∈Th
{hrK} ||u||Hr(Γ). (30) This corollary enables to easily see the convergence rate difference betweenL2norm and the DG norm. In this case the scheme converges 12 times faster.
Corollary 6 (L2 error: p-version). Let’s consider the same conditions as in theorem 1. Let’s consider the mesh sizehfixed on the mesh and let’s allow a varying polynomial orderp. If the parameter’s valuenp∈]4,5] is chosen, the following error estimate holds:
||u−uh||L2(Γ) max
K∈Th
1 pr+
np−5 2
K
||u||Hr(Γ) (31)
whereas fornp≥5 there is:
||u−uh||L2(Γ) max
K∈Th
1 prK
||u||Hr(Γ). (32)
4. Proof of the error estimates
This section is devoted to the proof of the main theorem. After having recalled the useful technical lemmas in subsection 4.1, and deriving the needed properties of the problem’s bilinear form in subsections 4.2, 4.3, we will carry out the apriori error analysis in subsection 4.4.1.
4.1. Auxiliary results
We need some intermediate results in order to prove the theorem, mostly about function approximation theory and integral and surface differential operators’ properties.
4.1.1. Function interpolation
The error estimate requires the use of interpolation and projection operators. We first recall those classical results.
We start with the standardL2 projection Π2:Hs(Γ)→Xhp, with s∈R+. There exists a local and global error estimate results.
Lemma 1 (Local L2 projector). Lets∈R+ andq≤s. Letf ∈Hs(K), with K∈ Th. The following error estimate holds:
||Π2f −f||Hq(K) hmin(pK K+1,s)−q
ps−qK ||f||Hs(K). (33) Proof. See [EG04] for the case on integer orders. The results follows then by the use of the interpolation theory [CWHM15].
