# Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side inL1

Authors: Abdeluaab Lidouh; Rachid Messaoudi
Journal: International Journal of Differential Equations
 (2018)

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## Abstract

We consider the standardaffine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in L∞Ω and the right-hand side belongs to L1Ω; we extend the results where the case oflinear finite elements approximation is considered. We prove that the unique solution of the discrete problem converges in W01,qΩ for every q with 1≤q<d/d-1 (d=2 or d=3) to the unique renormalized solution of the problem. Statements and proofs remain valid in our case, which permits obtaining a weaker result when the right-hand side is a bounded Radon measure and, when the coefficients are smooth, an error estimate in W01,qΩ when the right-hand side f belongs to LrΩ verifying Tkf∈H1Ω for every k>0, for some r>1.


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## Body

## 1. Introduction

In this work we consider, in dimensiond=2 or 3, the P1 discontinuous Galerkin (dG) method approximation of the Dirichlet problem(1)-divA∇u=finΩ,u=0on∂Ω,where Ω is an open bounded set of Rd,  A is a coercive matrix with coefficients in L∞Ω, and f belongs to L1Ω.The solution of (1) does not belong to H01(Ω) for a general right-hand side in L1Ω. Actually, in order to correctly define the solution of (1), one has to consider a specific framework, the concept of renormalized (or equivalently entropy) solution (see for example [1, 2]). These definitions allow one to prove that in this new sense problem (1) is well-posed in the terminology of Hadamard.For this problem the standardP1-nonconforming finite elements approximation, related to a triangulation Th of Ω, namely,(2)Finduh∈Vh,∀vh∈Vh,ahswipuh,vh=∫Ωfvhdx,where(3)Vh=vh∈L2Ω:∀T∈Th,vh∣T∈P1T,∀F∈Fh,∫Fvh=0with the discrete bilinear formahswip yet to be designed, has a unique solution, since the right-hand side (2) ∫Ωfvhdx is correctly defined for f∈L1Ω and the bilinear form ahswip is consistent.Using the ideas which are at the root of the SWIP (Symmetric Weighted Interior Penalty) method, in the casef∈L2Ω, D. A. Di Pietro and A. Ern have proved, in [3], that the unique solution uh of (2) converges to the unique solution u of (1) in the following sense:(4)uh⟶ustrongly inL2Ω,∇huh⟶∇ustrongly inL2Ωd,uhJ⟶0,whenh⟶0,with the broken gradient∇h and the jump seminorm |·|J yet to be designed.To do that, the authors in [3] assume that the family of triangulations Th belong to an admissible mesh sequence in the sense of 17 and is compatible with the partition PΩ (see Assumption 3).The framework in this paper is the same as in [3]. The unique difference here is that f∈L1Ω is considered instead of f∈L2Ω; and we ourselves focus on the two cases d=2 and d=3. The same convergence results are proved.Notations. In the present work, Ω denotes an open bounded subset of Rd with d=2 or d=3. A particular case is the case where Ω is an open bounded polyhedron. We use the notation Avw for the scalar product of the vector Av by the vector w (which is often denoted by tw·Av). For a measurable set S⊂Ω, we denote by S the measure of S and by Sc the complement Ω∖S of S.For1<p<+∞ and m≥0, we have (5)HmTh≔v∈L2Ω/∀T∈Th,vT∈HmT,Wm,pTh≔v∈LpΩ/∀T∈Th,vT∈Wm,pT.We define also the following function spaces:(6)Hdiv,Ω≔τ∈L2Ωd/∇·τ∈L2Ω,Hdiv,Th≔τ∈L2Ωd/∀T∈Th,τT∈Hdiv,T.Fork≥0, we define the broken polynomial space (7)PdkTh≔v∈L2Ω/∀T∈Th,vT∈PdkT,with polynomial degree k ≥ 1.In that case,Pd1(Th)⊂H1(Th), which leads us to define the broken gradient ∇h:H1(Th)→[L2(Ω)]d such that, ∀v∈H1(Th), (8)∀T∈Th:∇hvT≔∇vT,and the broken divergence operator∇h:H(div,Th)→[L2(Ω)]d such that, ∀v∈H(div,Th), (9)∀T∈Th:∇h·vT≔∇·vT.Moreover, for any mesh elementT∈Th, we denote (10)∀T∈Th,FT≔F∈Fh/F∈∂T,∀F∈Fh,TF≔T∈Th/F∈∂T.And for a scalar-valued function v defined onΩ (which can admit two possible traces) the average of v is defined as(11)vFx≔12vT1x+vT2x;ifF∈FhisuchthatF⊂∂T1∩∂T2,vFx≔vTx,ifF∈FhbsuchthatF⊂∂T∩∂Ω,and the jump of v as(12)vFx≔vT1x-vT2x,ifF∈FhisuchthatF⊂∂T1∩∂T2,vFx≔vTx,ifF∈FhbsuchthatF⊂∂T∩∂Ω.For any face F∈Fh and for any integer l≥0, we define the (local) lifting operator rF,Al:L2(F)→Pdl(Th) as follows. For all φ∈L2(F),(13)∫ΩArF,Alφ·τh=∫FAτh·nFφ∀τh∈PdlThd,and for any functionv∈H1(Th), we define the (global) lifting of its interface and boundary jumps as(14)Rh,Alv=∑F∈FhrF,Alv∈PdlThd.We also introduce the normal diffusion coefficient to one face F as(15)∀F∈Fhi,F=∂T1∩∂T2:Ai≔ATinF·nF,i∈1,2,the diffusion-dependent penalty parameter (harmonic average of normal diffusion) as(16)γA,F≔2A1A2A1+A2ifF⊂∂T1∩∂T2,ATn·nifF⊂∂T∩∂Ω,the weighted average operator for allF∈Fhi such that F⊂∂T1∩∂T2 as(17)ωT1,F≔A2A1+A2,ωT2,F≔A1A1+A2,the weighted average operator for allF∈Fhi and for a.e. x∈F as(18)vω,Fx≔ωT1,FvT1x+ωT2,FvT2x,on boundary faces F∈Fhb such that F ⊂∂T∩∂Ω, we set(19)vω,Fx≔vTx,and the skew-weighted average operator for all F∈Fhi and for a.e. x∈F, as(20)vω¯,Fx≔ωT2,FvT1x+ωT1,FvT2x.The SWIP bilinear form is defined by (see Lemma 4.47 in [3])(21)ahswipv,w≔∫ΩA∇hv∇hwdx︸consistencyterm+∑F∈FhηγA,FhF∫Fv,w︸penaltyterm-∑F∈Fh∫FA∇hvω¯·nFw+A∇hwω·nFv︸symmetryterm,where the quantityη>0 denotes a user-dependent penalty parameter which is independent of the diffusion coefficient.And the SWIP norms are defined by(22)vswip≔A1/2∇hvL2Ωd2+vA,J21/2,∀q≥1:vswip,q≔A1/2∇hvLqΩdq+vJ,A,qq1/q,with the diffusion-dependent jump seminorms(23)vJ,A2≔∑F∈FhγA,FhFvL2F2,∀q≥1:vJ,A,qq≔∑F∈FhγA,FhFq-1vLqFq.The discrete Galerkin norm is defined by(24)vdG≔∇hvL2Ωd2+vJ21/2,with the jump seminorm(25)vJ2≔∑F∈Fh1hFvL2F2.For everyr with 1<r<+∞, we denote by Lr,∞Ω the Marcinkiewicz space whose norm is defined by(26)vLr,∞Ω=supλ>0λx∈Ω:vx≥λ1/r.For every real numberk>0 we define the truncation Tk:R→R by(27)Tks=max-k,min⁡k,s=sifs≤k,kssifs≥k.For everyd-simplexT in Rd, we adopt the following notations:(i)
ai,T,  i=0,…,d, denote the vertices of T.(ii)
mi,T,  i=0,…,d, denote the centers of the faces Fi∈T.(iii)
λi,T,  i=0,…,d, designate the barycentric coordinates with respect to the ai,T’s.(iv)
for everyx∈Rd we put(28)φi,Tx≔1-dλi,Txfori=0,…,d,where(φi,T)0≤i≤d are the P1 shape functions related to T; it is known that(29)φi∈L2Ωh,φiT∈P1T,∀i∈0,1,…,d,∀T∈Th,φiFi=1,φiai=1-d,φiaj=1suchthatj≠i,1Fj∫Fjφi=φimj=δi,j,∑i=0dφi,Tx=1,foreveryx∈Rd,withΩh≔∪T,T∈Th⊂Ω¯.(v)
If N designate the number of all interior centersmi of faces F in Th we define the interpolation operator Πh and the truncated interpolation operator Ihk by(30)∀v∈L2Ω¯avec∫Fvh=0,Πhv∈Vh,Πhv≔∑1≤i≤Nαivφi,Ihkv≔∑1≤i≤NTkαivφi,withαiv=Πh(v)(mi)=1/Fi∫FiΠh(v)≔1/Fi∫Fiv.(vi)
Finally, we define theN×N stiffness matrix Q=(Qij); namely,(31)Qi,j=ahswipφi,φjfori,jin1,2,…,N.As in [4], the main assumption of the present paper is that Q is a diagonally dominant matrix; namely,(32)∀i∈1,2,…,N:Qii-∑1≤j≤Nj≠iQij≥0.


## 2. Statement of the Main Result

We consider a matrixA such that(33)A∈L∞Ωd×d,(34)a.ex∈Ω:∀ξ∈Rd:Axξξ≥αξ2,for some α>0, and a right-hand side f such that(35)f∈L1Ω.A functionu is the renormalized solution of the problem (1) if u satisfies(36)u∈L1Ω,(37)∀k>0,Tku∈H01Ω,(38)limk→∞1k∫Ω∇Tku2dx=0,(39)∀k>0,∀S∈Cc1RwithsuppS⊂-k,k,∀v∈H01Ω∩L∞Ω,∫ΩA∇Tku∇vSudx+∫ΩA∇Tku∇TkuS′uvdx=∫ΩfSuvdx.It is known (see [1, 5]) that when f belongs to L1Ω∩H-1Ω, the usual weak solution of (1), namely,(40)u∈H01Ω,∀v∈H01Ω,∫ΩA∇v∇udx=∫Ωfvdx,is a renormalized solution of (1) and conversely the main interest of definition of renormalized solution is the following existence, uniqueness, and continuity theorem (see [1, 4]).Theorem 1.
Assume thatA and f satisfy (33), (34), and (35). Then there exists a renormalized solution of (1). This solution is unique. Moreover this unique solution belongs to W01,qΩ for every q with 1≤q<d/d-1. It depends continuously on the right-hand side f in the following sense: if fε is a sequence which satisfies(41)fε⟶fstronglyinL1Ω,when ε tends to zero, then the sequence uε of the renormalized solutions of (1) for the right-hand sides fε satisfies for every k>0 and for every q with 1≤q<d/d-1(42)Tkuε⟶TkustronglyinH01Ω,uε⟶ustronglyinW01,qΩ,when ε tends to zero, where u is the renormalized solution of (1) for the right-hand side f. Finally, if f1 and f2 belong to L1Ω, and if u1 and u2 are the renormalized solutions of (1) for the right-hand sides f1 and f2, then, for every k>0, the function Tku1-u2 belongs to H01Ω and for every q with 1≤q<d/d-1 one has(43)αTku1-u2H01Ω2≤kf1-f2L1Ω,αu1-u2W01,qΩ≤Cd,Ω,qf1-f2L1Ω,Where the constant Cd,Ω,q only depends on d, Ω, and q.Remark 2.
Throughout all this paper, we denote byCp1,p2,p3,.. any real constant which only depends on the parameters p1, p2, and p3 …. We can use the same notation for different constants.Now we consider a family of triangulationsTh satisfying for each h>0 the following assumption:(44)thetriangulationThismadeofafinitenumberofclosedd-simplicesT(namelytriangleswhend=2andtetrahedrawhend=3)suchthat:iΩh=⋃T:T∈Th⊂Ω¯,iiforeverycompactsetKwithK⊂Ω,thereexistsh0K>0suchthatK⊂Ωhforeveryhwithh<h0K,iiiforT1,T2∈Th2withT1≠T2,onehasT1∩T2=0,iveveryfaceofeveryTofThiseitherasubsetof∂Ωh,orafaceofanotherT′ofTh.Note that because of (iv) the triangulations are conforming. A particular case is the case whereΩ is a polyhedron of Rd, and where Ωh coincides with Ω for every h.In practice, the diffusion coefficient (i.e., matrix A) has more regularity than just belonging toL∞(Ω). Henceforth, we make the following assumption (assumption 4.43 [3]):(45)thereisapartitionPΩ≔Ωi1≤i≤NΩofΩsuchthat:iEachΩi,1≤i≤NΩ,isapolyhedron;iiTherestrictionofAtoeachΩi,1≤i≤NΩ,isconstant.An important assumption on the mesh sequenceTh≔THh∈H is its compatibility with the partition PΩ in the following sense (assumption 4.45 [3]).Assumption 3 (mesh compatibility).
We suppose that the admissible mesh sequenceTH is such that, for each h∈H, each T∈Th is a subset of only one set Ωi of the partition PΩ. In this situation, the meshes are said to be compatible with the partition PΩ.For everyT∈Th, we denote by hT the diameter of T and by ρT the diameter of the ball inscribed in T. We set(46)h=supT∈ThhTand we assume that h tends to zero.We also assume that the family of triangulationsTh is regular in the sense of P. G. Ciarlet [6]; namely, there exists a constant σ such that(47)∀h,∀T∈Th,hTρT≤σ.For every triangulationTh, we consider the discrete problem:(48)Finduh∈Vh,∀vh∈Vh,ahswipuh,vh=∫Ωfvhdx.Note that the right-hand side of (48) makes sense since f∈L1Ω and Vh⊂L∞Ω. The discrete bilinear forme ahswip is consistent and coercive (see (128)) on Vh, so a straightforward consequence of the Lax-Milgram Lemma is that the discrete problem (48) is well-posed. The solution uh of (48) exists and is unique.As in [4], the main result of this paper is the following.Theorem 4.
Assume thatA, f, and Th satisfy (33), (34), (35), (44), (46), (47), and (32). Then the unique solution uh of (48) satisfies for every k>0 and for every q with 1≤q<d/d-1(49)uh⟶ustronglyinLqΩ,∇huh⟶∇ustronglyinLqΩd,uhJ,A,q⟶0,Ihkuh⟶TkustronglyinL2Ω,∇hIhkuh⟶∇TkustronglyinL2Ωd,IhkuhJ,A⟶0,when h tends to zero, where u is the unique renormalized solution of (1).This theorem will be proved in Section4, using the tools that we will prepare in Section 3. In Section 5, we will explain why the results of [4] when f is a bounded Radon measure remain valid in our case. In Section 6 we also show that if we assume in addition that Tkf∈H1Ω for every k>0, we obtain for smooth solutions an Oh41-1/r error estimate in swip,q-norm (Section 6.1), and for Low-Regularity solutions an Oh4αp1-1/r error estimate in swip,q-norm (Section 6.2). Finally, in Section 7 we show that in the case where A is the identity matrix, condition (32) remains satisfied when every inner angle of every d-simplex of Th is acute.


## 3. Tools

We are going to prove Theorem4 in several steps. We begin by proving the following result which is a piecewise P1 variant of a result of L. Boccardo & T. Gallouët [2, 5].Theorem 5.
Assume thatvh∈Vh satisfies(50)∀k>0,∫Ω∇hIhkvh2dx≤kMfor some M>0. Then, for every q with 1≤q<d/d-1,(51)vhswip,q≤C2d,Ω,q,σ,AL∞Ωd×dMwhere the constant C2d,Ω,q only depends on d, Ω, and q.As in [4], to prove Theorem 5, we use the following lemmas.Lemma 6.
Under assumption (47), for all T ∈Th and all F∈FT, one has(52)hFF≤2σdTProof.
Indeed, letT∈Th and F∈FT, so(53)hFF≤hTdand by (47) one has(54)hFF≤σdρTdwhich combined with the fact that(55)T≥cρTd,where c=Π/4 in 2D, and c=Π/6 in 3D, implies (52).Lemma 7.
Under assumption (47), for every q such that 1 ≤ q, the following bound holds for any vh∈Vh:(56)vJ,A,qq≤Cq,σ,d,AL∞Ωd×d∇hvLqΩ.Proof.
For everyT∈Th, we denote by (∇hv)T the (constant) gradient of the restriction of v to T. With this notation, using the continuity of v across any F in FT at the mass center xF of any internal F, the fact that v vanishes at the mass center xF of any external F, and the known inequality(57)∀q≥1,a+bq≤2q-1aq+bqand using (52) we get(58)vJ,A,qq=∑F∈FhiγA,FhFq-1∫Fvqdx+∑F∈FhbγA,FhFq-1∫Fvqdx≤∑F∈FhiF=T1∩T2γA,FhFq-1∫F∇hvT1-∇hvT2·x-xFqdx+∑F∈FhbF∈FT∫FγA,FhFq-1∇hvT·x-xFqdx≤∑F∈FhiF=T1∩T22qmin⁡λT1,F,λT2,FhFF∇hvT1q+∇hvT2q+∑F∈FhbF∈FTλT,FhFF∇hvTq≤2σdAL∞Ωd×d∑F∈FhiF=T1∩T22qT1∇hvT1q+T2∇hvT2q+2σdAL∞Ωd×d∑F∈FhbF∈FTT∇hvTq≤2q+3σdAL∞Ω∑T∈ThT∇hvTq,
which is (56) with C(q,σ,d,AL∞Ωd×d)=2q+3σdAL∞Ωd×d.Lemma 8.
Letvh∈Vh and let k>0. If for some T∈Th there exists y∈T with vhy≥k, then there exists a d-simplex S⊂T with S=cdT such that(59)∀x∈S:Ihkvhx≥k2,and the strictly positive constant c(d) only depends on d.Proof.
LetT be a d-simplex from the triangulation Th,  vh∈Vh, and k>0, such that supTvh≥k. Consider wh=Ihkvh∈P1(T).(i)
Ifmaxmi∈T|vh|<k, thus(60)∀i∈0,1,…,d,Tkαivh=Tkvhmi=vhmi=αivhsowh=vh, and ∃y∈T such that |wh(y)|≥k.(ii)
Ifmaxmi∈T|vh|≥k, thus ∃mi∈T, such that |vh(mi)|≥kso|Tk(vh(mi))|=k, and |wh(mi)|=Tk(αivh)=k.
In both cases, there exists an elementy in T such that |wh(y)|≥k.
Butwh∈P1(T), so (61)why-whx=∇why-x,In other words (62)∀i≠j≠kdans0,1,…,d,∇λiai-ak=1,∇λiaj-ak=0.and since (63)y-x=∑k=1dλky-λkxak-a0,one obtains(64)∇wh=-d∑i=0dTkαivh∇λi,∇whak-ao=-dTkαkvh-Tkα0vh,so (65)∇whak-ao≤2kd,and (66)why-whx≤2d∑k=1dλky-λkx≤k2,as soon as (67)∑k=1dλky-λkx≤14d.For this purpose we define the d-simplex S=y-a0+S0 such that (68)S0=x,λ0≥1-14d=4d-14d,so(69)whx≥why-why-whx≥k-k2=k2,and to estimate the measure of S, it is clear to verify first that |S|=|S0|. Let T^ be the reference unit d-simplex with vertices a^0=0,  a^1=e1,…,a^d=ed, where {ei,i=1,…,d} is the canonical basis of Rd. Let FT be the invertible affine mapping that maps FT(T^)=T onto T and set S^=FT-1(S).
SinceFT is affine, it is easy to check that λ^i=FT-1∘λi,T for i=0,1,…,d are the barycentric coordinates with respect to the a^i’s and that(70)S^=x^∈T^:λ0^x^≥4d-14d,S=S^T^T=CdT,where c(d)=|S^|/|T^| is a constant that depends only on d. This proves the result.Lemma 9.
Assume thatvh∈Vh satisfies (50), then(71)Bkvh≤Cd,Ω,2∗Mk2∗/2,for every k>0, where 2∗=2d/d-2=6 if d=3 and 2∗ is any real number with 2∗≥1 if d=2; Bkvh is defined by(72)Bkvh=⋃T∈Th:maxTvh≥k,and Cd,Ω,2∗ is a constant depending only on d, 2∗, θ0, and Ω.Proof.
Discrete Sobolev’s theorem (see theorem 5.3 in [3]) asserts that(73)∀v∈Vh,vL2∗Ω≤σ2,2∗d,ΩvdG,and we will also need (56) with q=2(74)∀v∈Vh,vJ2≤Cσ,d∇hvL2Ω.
Hered=2 or d=3 so 2∗=2d/d-2=6 if d=3 and 2∗ can be any real number with 1≤2∗.
Fixk>0. If T⊂Bkvh, from Lemma 8, we know that there exists S⊂T, with S=cdT and(75)∀x∈S,Ihkvhx≥k2.
Therefore(76)∫TIhkvhx2∗dx≥∫SIhkvhx2∗dx≥k22∗S=cdTk22∗.
Hence(77)Bkvh=∑T⊂BkvhT≤∑T⊂Bkvh1cd2k2∗∫TIhkvhx2∗dx≤1cd2k2∗∫ΩIhkvhx2∗dx.
Combining with (73) and (74) one has(78)Bkvh≤2σ2,2∗d,Ω2∗1+Cσ,d2∗/2cdk2∗∫Ω∇hIhkvhx2dx2∗/2.
Finally using (50), we obtain(79)Bkvh≤2σ2,2∗d,Ω2∗1+Cσ,d2∗/2cdMk2∗/2,which is (71) with (80)Cd,Ω,2∗,σ,AL∞Ωd×d=2σ2,2∗d,Ω2∗Cd1+Cσ,d2∗/2.Proof (proof of Theorem5; see [4]).
Fixq with 1≤q<d/d-1. Take r=2×2∗/2+2∗=3/2 in the case d=3, and verify 2×2∗/2+2∗>q, in the case d=2.
The embedding inequality (LqΩ↪Lr,∞Ω) writes(81)∇hvhLqΩ≤Cq,r,Ω∇vhLr,∞Ω,
and by (56) one can have(82)∀v∈W01,qTh,vJ,A,qq≤Cq,σ,d,AL∞Ωd×d∇hvLqΩ.
So to prove Theorem5, it suffices to estimate(83)∇vhLr,∞Ω=supλ>0⋃T∈Thx∈T:∇hvh≥λ1/r.
So letλ>0. For every k>0, we can write(84)⋃T∈Thx∈T:∇hvh≥λ≤⋃T∈Thx∈T:∇hvh≥λ∩Bkvh+⋃T∈Thx∈T:∇hvh≥λ∩Bkvhc≤Bkvh+⋃T∈Thx∈T:∇hvh≥λ∩Bkvhc.But x∈Ω:∇vhx≥λ∩Bkvhc coincides, up to a set of measure zero, with(85)⋃T∈Thx∈T:∇hvh≥λ∩T∈Th:maxTvh<k.On the other hand, if maxTvh<k, then Ihkvh|T=vh|T, so(86)⋃T∈Thx∈T:∇hvh≥λ∩Bkvhc≤⋃T∈Thx∈T:∇ΠhTkvh≥λ≤1λ2∫Ω∇hΠhTkvh2dx.
By the use of (50), one has(87)⋃T∈Thx∈T:∇hvh≥λ∩Bkvhc≤kMλ2.We fix k=λ2-rMr-1 to have kM/λ2=M/λr; using (71) we obtain(88)⋃T∈Thx∈T:∇hvh≥λ≤Cd,Ω,2∗Mk2∗/2+Mλr,so, for r=2×2∗/2+2∗ to have M/k2∗/2=M/λr, in both cases d=2 or d=3, one has(89)⋃T∈Thx∈T:∇hvh≥λ≤Cd,Ω,2∗+1Mλr,and then(90)λ⋃T∈Thx∈T:∇hvh≥λ1/r≤Cd,Ω,qM,for every λ>0, which finishes the proof.Lemma 10.
Letv∈Vh. For every s, k with 0≤s<k, the set Bk,sv, defined by(91)Bk,sv=⋃T∈ThT∈Th:minTv≤s,maxTv≥k,satisfies(92)Bk,sv≤h2k-s2∫Ω∇hv2dx.Proof.
IfT∈Bk,s(v), maxTv=v(a)≥k, and minTv=v(b)≤s, so(93)k-s≤va-vb≤va-vb≤∇va-b.Therefore (94)1≤∇v2a-b2k-s2Hence(95)Bk,sv=∑T∈Bk,svT=∑T∈Bk,sv∫T1dx≤∑T∈Bk,sv∫T∇v2a-b2k-s2dx≤h2k-s2∫Ω∇hv2dx.The estimate (92) follows.Lemma 11.
Letvh∈Vh and 0≤s<k, then the set B~k,svh, defined by(96)B~k,svh=⋃T∈Th:minmi∈Tvh≤s,maxTvh≥k, satisfies(97)B~k,svh≤h2k-s2∫Ω∇hIhkvh2dx.Proof.
Indeed, ifT⊂B~k,svh then there are two possibilities: (i)
Ifmaxmi∈Tvh≤k so Ihk(vh)|T=vh|T.(ii)
Ifmaxmi∈Tvh≥k so maxTIhk(vh)≥k and obviously minTIhk(vh)≤s.
In the two casesT⊂Bk,sIhkvh. The estimate (97) follows.Remark 12.
It is then clear that, under hypothesis (50),(98)B~k,svh⟶h→00,Bk,sIhkvh⟶h→00.In addition, one has the following.Proposition 13.
Letvh∈Vh and 0≤s<k. If vh satisfies (50), then(99)Bk,svh⟶h→00.Proof.
Fixk>0 and s>0,  s<k. For h>0 such that 1/h≥k, we can write(100)Bk,svh=Bk,svh∩B1/hvh∪Bk,svh∩B1/hvhc.
On the one hand, with2∗=6 in (71), one has(101)Bk,svh∩B1/hvh≤B1/hvh≤CΩh3.
On the other hand,(102)Bk,svh∩B1/hvhc⊂Bk,sIh1/hvh.Indeed, if x∈Bk,svh∩B1/hvhc and T∈Th such that x∈T, then maxTvh≥k, minTvh≤s, and, for every y in T, vhy≤1/h, which means Ih1/hvh|T=vh|T, and T∈Bk,sIh1/hvh.
Therefore, with Lemma10, and (50), one has(103)Bk,svh∩B1/hcvh≤Bk,sIh1/hvh≤h2k-s2∫Ω∇hIh1/hvh2dx≤hk-s2M.
The convergence (99) is then a consequence of (101) and (103).Lemma 14.
Letvh∈Vh. For every s and every k with 0<s<k, one has(104)x∈Ω:TsIhkvh≠Tsvhx⊂Bk,svh∪Bk,sIhkvh.Proof.
Letx∈Ω such that TsIhkvh(x)≠Tsvh(x), and T∈Th with x∈T. It is easily checked that Ihkvh|T≠vh|T,  maxT~vh≥k, and maxT~Ihkvh≥k. So there are three possibilities.(i)
Ihkvhx≥s and vhx<s, and then T⊂Bk,svh,(ii)
Ihkvhx<s and vhx≥s, and then T⊂Bk,sIhkvh,(iii)
Ihkvhx<s and vhx<s, and then(105)T⊂Bk,sIhkvh∩Bk,svh.
In all casesx∈T⊂Bk,sIhkvh∪Bk,svh, and (104) follows.Proposition 15.
Assume thatvh∈Vh satisfies (50). Then, for every k>0, one has(106)Ihkvh-Tkvh⟶0inmeasure,when h tends to zero.Proof.
Fixk>0 and ε>0 such that ε<k and consider(107)Iε=x∈Ω:Ihkvhx-Tkvhx≥ε.Let x∈Iε and T∈Th with x∈T. It is easily checked that(108)IhkvhT≠TkvhT,which implies that maxTvh>k. So there are four possibilities.(i)
vh changes sign in T; then, by continuity,(109)T⊂Bk,svhforeverys∈0,k.(ii)
Ihkvh changes sign in T; then(110)T⊂Bk,sIhkvhforeverys∈0,k.(iii)
maxmi∈Tvh≤k and vh|T≥0 (or vh|T≤0) so Ihkvh|T=vh|T.
(a)
Ifvh|T≥0, then vhx≥k+ε and T⊂Bk+ε,kIhkvh.(b)
Ifvh|T≤0, then vhx≤-k-ε and T⊂Bk+ε,kIhkvh.(iv)
maxmi∈Tvh>k and vh|T≥0 (or vh|T≤0). So(111)Ihkvhx-Tkvhx=Ihkvhx-Tkvhx.
(a)
IfIhkvhx-Tkvhx≥ε. There are three possibilities:
case 1:
vhx≥k, so Ihkvhx≥k+ε and(112)T⊂Bk+ε,kIhkvh;case 2:
vhx<k-ε/2, so T⊂Bk,k-ε/2vh;case 3:
k-ε/2≤vhx<k, so Ihkvhx≥k+ε/2 and(113)T⊂Bk+ε/2,kIhkvh.(b)
IfTkvhx-Ihkvhx≥ε, then(114)Ihkvhx≤Tkvhx-ε,
so(115)T⊂Bk,k-εIhkvh.
We can then conclude that(116)Iε⊂Bk+ε/2,kΠhTkvh∪Bk,k-εΠhTkvh∪Bk,k-ε/2vh.Convergence (106) is then consequence of (98) and (99).The result and the proof of the Proposition (2.7) in [4] can be conserved without changes.Proposition 16.
Under assumption (32), one has for every vh∈Vh and every k>0(117)ahswipvh-Ihkvh,Ihkvh≥0.Proof (proof of Proposition 2.7 in [4]).
Since(118)vh=∑i=0dαivhφi,Ihkvh=∑i=0dTkαivhφi,one has(119)ahswipvh-Ihkvh,Ihkvh=∫ΩA∇hvh-Ihkvh∇hIhkvhdx-∑F∈Fhi∫FA∇hvh-Ihkvhω¯·nFIhkvhdx-∑F∈Fhi∫FA∇hIhkvhω·nFvh-Ihkvhdx+∑F∈Fhi∫FγA,FhFηvh-IhkvhIhkvhdx=∫Ω∑i=0dαivh+TkαivhA∇hφi∑j=0dTkαjvhA∇hφjdx-∑F∈Fhi∫F∑i=0dαivh+TkαivhA∇hφiω¯·nF∑j=0dTkαjvhφjdx-∑F∈Fhi∫F∑i=0dTkαivhφiω·nF∑j=0dαjvh+TkαjvhA∇hφjdx+∑F∈Fhi∫F∑i=0dαivh+TkαivhA∇hφi∑j=0dTkαjvhφjdx=∑i=0dαivh-Tkαivh∑j=0dTkαjvhahswipφi,φj=∑i=0dαivh+Tkαivh∑j=0dTkαjvhQi,j=∑i=0dSi,
where(120)Si=αivh-TkαivhTkαivhQi,i+αivh-Tkαivh∑0≤j≤dj≠iαjTkvhQi,j.Fix ∈0,1,…,d; (i)
if|Tkvh(mi)|<k, then Tk(vh(mi))=vh(mi) et αivh-Tk(αivh)=0 and Si=0,(ii)
if|Tkvh(mi)|=k,then(121)αivh-TkαivhTkαivh=vhmi-TkvhmiTkvhmi=kvhmi-Tkvhmi,and therefore(122)Si≥kvhmi-TkvhmiQi,i-∑0≤j≤dj≠iQi,jin the light of the assumption (32). This proves that Si≥0 for every i in 1,2,…,N; and so (117).


## 4. Proof of the Main Theorem

We first show an a priori estimate (compared with (50)) on the solution uh of (48).Lemma 17.
Letvh∈Vh,  T∈Th, and k>0 such that maxmi∈Tvh≤k. Then(123)maxTvh≤k2d2-1.Proof.
Letvh∈Vh, T∈Th, and k>0 such that maxmi∈Tvh≤k. For every x∈T,(124)vhx=∑i=0dαivhφix≤∑i=0dαivhφix≤∑i=0dvhmiφix≤-k∑i=0λi<1/ddφix+k∑i=0λi≥1/ddφix︸=1-∑i=0λi<1/ddφix≤k1+2∑i=0dd-1≤k2d2-1,
the inequality (123) is then proved.Proposition 18.
Under the assumption of Theorem4, the solution uh of (48) satisfies for every k>0 and every h>0(125)ahswipIhkuh,Ihkuh≤∫ΩfIhkuhdx, and, in particular, uh satisfies(126)∫Ω∇hIhkuh2dx≤k2d2-1αCηfL1Ω,where the constant Cη only depends on η,σ, and d.The proof makes use of results appearing in [3] that we reproduce as follows.Lemma 19 (Lemma 1.46 in [3], discrete trace inequality).
Under assumption (47), one has for all vh∈Pd1(Th), all T ∈Th, and all F ∈FT(127)hT1/2vhL2F≤CtrvhL2Twhere Ctr only depends on σ and d.Lemma 20 (Lemma 4.51 in [3], discrete coercivity).
For allη>η_≔(d+1)Ctr2, the SWIP bilinear form ahswip is coercive on Vh with respect to the ·swip-norm; i.e.,(128)∀vh∈Vh,ahswipvh,vh≥Cηvhswip2where Cη=η-(d+1)Ctr2/1+η.Proof (of Proposition18).
UsingIhkuh as a test function in (48) one has(129)ahswipuh,Ihkuh=∫ΩfIhkuhdx,
and from (117) we obtain (125).
On the other hand, by combining (125) with coercivities (34) and (128) one obtains successively(130)αCη∫Ω∇hIhkuh2dx≤Cη∫ΩA∇hIhkuh∇hIhkuhdx≤CηIhkuhswip2≤ahswipIhkuh,Ihkuh≤∫ΩfIhkuhdx≤maxΩIhkuh∫Ωfdx≤k2d2-1fL1Ω,
and by (123) the estimation (126) is then proved.Theorem 21.
Under the assumptions of Theorem4, the solution uh of (48) satisfies for every q with 1≤q<d/d-1(131)uh⟶ustronglyinLqΩ,∇huh⟶∇ustronglyinLqΩd,uhJ,A,q⟶0,when h tends to zero, where u is the unique renormalized solution of (1).Proof (proof of Theorem 3.2 in [4]).
Consider a sequencefε in L2Ω, converging strongly in L1Ω to f (for example, fε=T1/εf). Let uhε to be the unique solution of (48) for the right-hand side fε. Then uh-uhε satisfies(132)uh-uhε∈Vh,∀vh∈Vh,ahswipuh-uhϵ,vh=∫Ωf-fϵvhdx.
Applying estimate (126) to this problem, we obtain for every k>0, every h>0, and every ε>0(133)∫Ω∇hIhkuh-uhϵ2dx≤k2d2-1αCηf-fϵL1Ω,
Which implies by Theorem5 that for every q with 1≤q<d/d-1, every h>0, and every ε>0(134)uh-uhϵswip,q≤C2d,Ω,q,σ,AL∞Ωd×dαCηf-fϵL1Ω.
On the other hand, sincefε∈L2Ω and Th satisfies (44), (46), and (47), it is known (see [3]) that for every fixed ε(135)uhε⟶uεstronglyinL2Ω,(136)∇huhε⟶∇uεstronglyinL2Ωd,(137)uhεJ,A⟶0,
whenh tends to zero, where uε is the unique solution of(138)uε∈H01Ω,-divA∇uε=fεinD′Ω.
Finally, the functionuε is also the unique renormalized solution of the problem(139)-divA∇uε=fεinΩ,uε=0on∂Ω.
The estimate (43) combined with the inequality (140)uϵ-uswip,q≤AL∞Ωd×duϵ-uW01,qΩ
allows one to have(141)uϵ-uswip,q≤AL∞Ωd×dC1d,Ω,qαf-fϵL1Ω,for every q with 1≤q<d/d-1, where u is the unique renormalized solution of (1).
Writing now(142)uh-uswip,q≤uh-uhϵswip,q+uϵ-uswip,q+uhϵ-uϵswip,q,Using (134), (136), (137), and (141), one has for every ε>0 and every q with 1≤q<d/d-1(143)limsuph→0uh-uswip,q≤C3d,Ω,q,σ,AL∞Ωd×dαCηf-fϵL1Ω,and passing to the limit when ε tends to zero proves Theorem 21.To complete the proof of Theorem4, it remains to prove the following proposition.Proposition 22.
Under the assumptions of Theorem4, the solution uh of (48) satisfies(144)Ihkuh⟶h→0TkustronglyinL2Ω,(145)∇hIhkuh⟶h→0∇TkustronglyinL2Ωd,(146)IhkuhJ,A⟶h→00,
for everyk>0.Proof (proof of Proposition 3.3 in [4]).
First by result of Proposition15 and the estimate (126) one can have (see theorem 5.7 in [3]) the two following convergences:(147)Ihkuh⟶h→0TkustronglyinL2Ω,(148)GhlIhkuh⇁h→0∇TkuweaklyinL2Ωd,for every k>0.
On the other hand, using (123) one has(149)fIhkuh≤k2d2-1f∈L1Ω,
so, by Lebesgue’s dominated convergence theorem combined with discrete Rellich-Kondrachov’s compactness theorem (see theorem 5.6 in [3]) one has(150)∫ΩfIhkuhdx⟶h→0∫ΩfTkudx.
Therefore passing to the limit with respect toh in (125) yields (151)limsuph→0ahswipIhkuh,Ihkuh≤∫ΩfTkudx.
Consequently(152)limsuph→0∫ΩAGhlIhkuhGhlIhkuh≤∫ΩfTkudx.
By the fact that(153)∫ΩAGhlIhkuhGhlIhkuh≤ahswipIhkuh,Ihkuh,
and sinceu is the renormalized solution of (1), it is known that (see [4])(154)∫ΩA∇Tku∇Tku=∫ΩfTkudx.
Finally, from (152) and (154), we deduce that(155)limsuph→0∫ΩAGhlIhkuhGhlIhkuh≤∫ΩA∇Tku∇Tku,
which combined with the weak convergence (148) implies(156)GhlIhkuh⟶h→0∇TkustronglyinL2Ωd.
Owing to Proposition 4.36 in [3], for all vh∈Vh and all η>(d+1)Ctr2, one can have(157)IhkuhJ,A≤ahswipIhkuh,Ihkuh-A1/2GhlIhkuhL2Ωdη-d+1Ctr2
and, since the right-hand side tends to zero, the result (146) holds.
Finally, using the result of Proposition 4.34 in [3](158)RhlIhkuhL2Ωd≤d+1CtrIhkuhJ,A
with the triangle inequality that yields(159)∇hIhkuh-∇TkuL2Ωd≤GhlIhkuh-∇TkuL2Ωd+RhlIhkuhL2Ωdconcluding the proof of (145).


## 5. The Case Where f Is a Bounded Radon Measure

The materials used in [4], to handle the case where f belongs to Mb(Ω), are not specific to the case of P1 finite elements approximation; only the weak convergence (148) requires clarification; in our approach it is based on the result of Proposition 15 whose proof involves only properties of vh not f. So we can also state the following convergence result.Theorem 23 (Theorem 4.1 in [4]).
Assume that f belongs toMb(Ω) and A and Th satisfy (46), (33), (35), (44), and (47) and (32). Then there exist a subsequence, still denoted by h, and a function u such that for every k>0 and for every q with 1 ≤q<d/d-1 one has(160)GhlIhkuh⇁∇TkuweaklyinL2Ωd,∇huh⇁∇uweaklyinLqΩd,
whenh tends to zero along this subsequence, where u satisfies (161)∀k>0,Tku∈H01Ω,∀qwith1≤q<dd-1,u∈W01,qΩ,∀v∈Cc∞Ω,∫ΩA∇u∇vdx=∫Ωvdf.


## 6. Convergence Rate Estimation

### 6.1. Error Estimates for Smooth Solutions

Assumption 24 (regularity of exact solution and spaceV∗).
As in [3], we assume that Th is compatible with the partition PΩ in the sense of Assumption 3, and the unique solution v is such that(162)v∈V∗≔H01Ω∩H2Th.And we set(163)V∗h≔V∗+Vh.The convergence analysis is performed in the spirit of (Theorem 1.35 [3]) by establishing discrete coercivity, consistency, and boundedness for ahswip. The discrete bilinear form ahswip is extended to V∗h×Vh.Without further knowledge on the exact solution v apart from the domainΩ and the datum g∈L2(Ω), Assumption 24 can be asserted for instance if the domain Ω is convex; see Grisvard [7].A straightforward consequence of the Lax-Milgram Lemma is that the discrete problem (48) is well-posed.Theorem 25.
Under the assumptions of Theorem4 (d=2 or d=3), if f belongs to Lr,∞(Ω) for some r such that 1<r<2,  Ω a convex polyhedron Ωh=Ω, and A∈[W1,∞(Ω)]d×d, then ∀q with 1≤q<d/d-1(164)uh-uswip,q≤Cd,Ω,q,r,α,AW1,∞Ωd×d,σh21-1/rfLr,∞Ω.Proof (proof of Theorem 5.1 in [4]).
From [3] the unique solution uϵ of (48) with right-hand side fϵ=T1/ϵ(f)∈L∞Ω⊂L2Ω verifies(165)uhϵ-uϵswip≤CAW1,∞Ωd×dhfϵL2Ωso(166)uhϵ-uϵswip,q≤Cq,d,η,Ω,AW1,∞Ωd×dhfϵL2ΩCombined with (142), (134), and (141) allows one to have(167)uh-uswip,q≤Cq,d,η,Ω,AW1,∞Ωd×df-fϵL1Ω+hfϵL2Ω,
and by proceeding as in [4], we obtain (173).Remark 26.
Iff∈Lr,∞(Ω) for some r with 1<r<2 and(168)∀k>0:Tkf∈H1Ω,
then a small adaptation of the proof given in [4] provides an Oh41-1/r error estimate in swip,q-norm, with 1≤q<d/d-1, since, with (168), it is known that(169)uhϵ-uϵswip,q≤Cd,Ω,q,r,α,AW1,∞Ωd×d,σh2fϵL2Ω.


### 6.2. Error Estimates for Low-Regularity Solutions

Assumption 27 (regularity of exact solution and spaceV∗).
As in [3], we assume that the mesh Th is compatible with the partition PΩ in the sense of Assumption 3, d≥2, and that there is p such that 2d/d+2<p≤2; the unique solution v is such that(170)v∈V∗≔H01Ω∩W2,pTh,where W2,p(PΩ)=W2,p(T) designate that the mesh T is compatible with the partition PΩ, and we set(171)V∗h≔V∗+Vh.We also assume p<2 since, in the case p=2, Assumption 27 amounts to Assumption 24.Assumption27 requires p > 1 for d = 2 and p > 6/5 for d = 3. In particular, we observe that, in two space dimensions, v∈W2,p(PΩ) with p > 1 holds true in polygonal domains; see, e.g., Dauge [8]. Moreover, using Sobolev embeddings (see [Evans [9], Sect. 5.6] or [Brézis [10], Sect. IX.3]), Assumption 27 implies(172)v∈H1+αpΩ,withαp=d+22-dp>0.Theorem 28.
Under the assumptions of Theorem4 (d=2 or d=3), if f belongs to Lr,∞(Ω) for some r such that 1<r<2, Ω a convex polyhedron Ωh=Ω, and A ∈[W1,∞(Ω)]d×d, then ∀q with 1≤q<p(173)uh-uswip,q≤Cd,Ω,q,r,α,AW1,∞Ωd×d,σh2αp1-1/rfLr,∞Ω.Remark 29.
Moreover, under the assumption in Remark26 one can have an Oh4αp1-1/r error estimate in swip,q-norm, with 1≤q<p.


## 6.1. Error Estimates for Smooth Solutions

Assumption 24 (regularity of exact solution and spaceV∗).
As in [3], we assume that Th is compatible with the partition PΩ in the sense of Assumption 3, and the unique solution v is such that(162)v∈V∗≔H01Ω∩H2Th.And we set(163)V∗h≔V∗+Vh.The convergence analysis is performed in the spirit of (Theorem 1.35 [3]) by establishing discrete coercivity, consistency, and boundedness for ahswip. The discrete bilinear form ahswip is extended to V∗h×Vh.Without further knowledge on the exact solution v apart from the domainΩ and the datum g∈L2(Ω), Assumption 24 can be asserted for instance if the domain Ω is convex; see Grisvard [7].A straightforward consequence of the Lax-Milgram Lemma is that the discrete problem (48) is well-posed.Theorem 25.
Under the assumptions of Theorem4 (d=2 or d=3), if f belongs to Lr,∞(Ω) for some r such that 1<r<2,  Ω a convex polyhedron Ωh=Ω, and A∈[W1,∞(Ω)]d×d, then ∀q with 1≤q<d/d-1(164)uh-uswip,q≤Cd,Ω,q,r,α,AW1,∞Ωd×d,σh21-1/rfLr,∞Ω.Proof (proof of Theorem 5.1 in [4]).
From [3] the unique solution uϵ of (48) with right-hand side fϵ=T1/ϵ(f)∈L∞Ω⊂L2Ω verifies(165)uhϵ-uϵswip≤CAW1,∞Ωd×dhfϵL2Ωso(166)uhϵ-uϵswip,q≤Cq,d,η,Ω,AW1,∞Ωd×dhfϵL2ΩCombined with (142), (134), and (141) allows one to have(167)uh-uswip,q≤Cq,d,η,Ω,AW1,∞Ωd×df-fϵL1Ω+hfϵL2Ω,
and by proceeding as in [4], we obtain (173).Remark 26.
Iff∈Lr,∞(Ω) for some r with 1<r<2 and(168)∀k>0:Tkf∈H1Ω,
then a small adaptation of the proof given in [4] provides an Oh41-1/r error estimate in swip,q-norm, with 1≤q<d/d-1, since, with (168), it is known that(169)uhϵ-uϵswip,q≤Cd,Ω,q,r,α,AW1,∞Ωd×d,σh2fϵL2Ω.


## 6.2. Error Estimates for Low-Regularity Solutions

Assumption 27 (regularity of exact solution and spaceV∗).
As in [3], we assume that the mesh Th is compatible with the partition PΩ in the sense of Assumption 3, d≥2, and that there is p such that 2d/d+2<p≤2; the unique solution v is such that(170)v∈V∗≔H01Ω∩W2,pTh,where W2,p(PΩ)=W2,p(T) designate that the mesh T is compatible with the partition PΩ, and we set(171)V∗h≔V∗+Vh.We also assume p<2 since, in the case p=2, Assumption 27 amounts to Assumption 24.Assumption27 requires p > 1 for d = 2 and p > 6/5 for d = 3. In particular, we observe that, in two space dimensions, v∈W2,p(PΩ) with p > 1 holds true in polygonal domains; see, e.g., Dauge [8]. Moreover, using Sobolev embeddings (see [Evans [9], Sect. 5.6] or [Brézis [10], Sect. IX.3]), Assumption 27 implies(172)v∈H1+αpΩ,withαp=d+22-dp>0.Theorem 28.
Under the assumptions of Theorem4 (d=2 or d=3), if f belongs to Lr,∞(Ω) for some r such that 1<r<2, Ω a convex polyhedron Ωh=Ω, and A ∈[W1,∞(Ω)]d×d, then ∀q with 1≤q<p(173)uh-uswip,q≤Cd,Ω,q,r,α,AW1,∞Ωd×d,σh2αp1-1/rfLr,∞Ω.Remark 29.
Moreover, under the assumption in Remark26 one can have an Oh4αp1-1/r error estimate in swip,q-norm, with 1≤q<p.


## 7. The Case Where A Is the Identity Matrix

Returning to the definition (31) of mass matrix Q where A=I, one hasfori,j in {1,2,…,N}(174)Qi,j=ahswipφi,φj=∫Ω∇hφi∇hφjdx︸=I1+∑F∈Fhi1hFηφiφjdx︸=I2-∑F∈Fhi∫F∇hφiω·nFφjdx+∑F∈Fhi∫F∇hφjω·nFφidx︸=I3.Ifi=j, so(175)I1=∫Ω∇hφi2dx≥0,I2=∑F∈Fhi∫FηhFφi2dx≥0,I3=2∫Fi∇hφiω·nFφidx+∑F∈FhiF≠Fi∇hφiω·nFφidx=0,since[φi]Fi=0,∇hφiω·nF=-d/hFini·nF, and ∫F≠Fiφidx=0.Ifi≠j, so(176)I1=∫Ω∇hφi∇hφjdx=1d2FiFjni·nj≤0 (see Proposition 6.1 in [4]),(177)I2=∫F∈Fi∪FjηhFφiφjdx+∑F∈FhiF≠Fi≠FjηhFφiφjdx=-ηd+1∑F∈FhiF≠Fi≠FjFhF≤0,I3=-d∑F∈FhiF≠Fi≠Fj1hFini·nFφj+1hFjnj·nFφidx=0.It is therefore concluded that, under the conditionnl·nk≤0(l≠k), one can have Qlk≤0. Thus, the matrix Q verifies (32).



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*Source: 4650512-2018-07-02.xml*