MATHEMATICSOFCOMPUTATION Volume00,Number0,Pages000–000 S0025-5718(XX)0000-0 MAXIMAL Lp ANALYSIS OF FINITE ELEMENT SOLUTIONS FOR PARABOLIC EQUATIONS WITH 6 NONSMOOTH COEFFICIENTS IN CONVEX POLYHEDRA 1 0 2 BUYANGLIANDWEIWEISUN r p A Abstract. The paper is concerned with Galerkin finite element solutions of parabolic equations in a convex polygon or polyhedron with a diffusion co- 4 efficient in W1,N+α for some α > 0, where N denotes the dimension of the 1 domain. Weprovethe analyticity ofthesemigroupgenerated bythe discrete ellipticoperator,thediscretemaximalLp regularityandtheoptimalLp error ] estimateofthefiniteelementsolutionfortheparabolicequation. A N . h 1. Introduction t a m Let Ω be a bounded domain in RN (with N = 2 or N = 3), and let Sh be a finite element subspace ofH1(Ω) consisting of continuouspiecewise polynomials of [ 0 degree r ≥ 1 subject to certain quasi-uniform triangulation of the domain Ω. We 2 consider the parabolic equation v 5 ∂tu−∇·(a∇u)=f in Ω×(0,∞), 4 (1.1)  u=0 on ∂Ω×(0,∞), 3  u(·,0)=u0 in Ω, 7 and its finite element approximation 0 1. (1.2) ∂tuh,vh +(a∇uh,∇vh)=(f,vh), ∀ vh ∈Sh, 0 (cid:26) (cid:0)uh(0)=u(cid:1)0h, 5 where f is a given function, and a = (a (x)) is an N ×N symmetric matrix 1 ij N×N : which satisfies the ellipticity condition v i N X (1.3) Λ−1|ξ|2 ≤ a (x)ξ ξ ≤Λ|ξ|2, for x∈Ω, ij i j r iX,j=1 a for some positive constant Λ. If we define the elliptic operator A : H1(Ω) → H−1(Ω) and its finite element 0 approximationA :S →S by h h h (1.4) (Aw,v):=(a∇w,∇v), ∀ w,v ∈H1(Ω), 0 (1.5) (A w ,v ):=(a∇w ,∇v ), ∀ w ,v ∈S , h h h h h h h h 2010 Mathematics Subject Classification. Primary65M12,65M60,Secondary35K20. TheworkofB.LiwassupportedinpartbyNSFC(grantno. 11301262),andtheresearchstay oftheauthoratUniversit¨atTu¨bingenwassupportedbytheAlexandervonHumboldtFoundation. The work of W. Sun was supported in part by a grant from the Research Grants Council of theHongKongSAR,China(projectno. CityU11301915). Thispaperwasaccepted forpublicationinMath. Comp. onDecember2,2015. (cid:13)cXXXXAmericanMathematicalSociety 1 2 BUYANGLIANDWEIWEISUN then the solutions of (1.1) and (1.2) can be expressed by t (1.6) u(t)=E(t)u0+ E(t−s)f(s)ds, Z 0 t (1.7) u (t)=E (t)u0 + E (t−s)f(s)ds, h h h Z h 0 where {E(t)=e−tA}t>0 and {Eh(t)=e−tAh}t>0 denote the semigroups generated by the operators −A and −A , respectively. By the theory of parabolic equa- h tions and [33], it is well known that {E(t)} is an analytic semigroup on C (Ω) t>0 0 satisfying (1.8) kE(t)vk +tk∂ E(t)vk ≤Ckvk , ∀v ∈C (Ω), ∀t>0, L∞ t L∞ L∞ 0 which is equivalent to the resolvent estimate k(λ+A)−1vk ≤Cλ−1kvk , ∀v ∈C (Ω), ∀λ∈Σ , L∞ L∞ 0 θ+π/2 where Σ := {z ∈ C : |arg(z)| < θ +π/2}. The counterparts of these two θ+π/2 inequalities above for the discrete finite element operator A are the analyticity of h the semigroup {E (t)} on L∞∩S : h t>0 h (1.9) kE (t)v k +tk∂ E (t)v k ≤Ckv k , ∀v ∈S , ∀t>0, h h L∞ t h h L∞ h L∞ h h and the resolvent estimate k(λ+A )−1v k ≤Cλ−1kv k , ∀v ∈S , ∀λ∈Σ . h h L∞ h L∞ h h ϕ+π/2 The estimates of the discrete semigroup have attracted much attention in the past severaldecades. Withtheseestimates,onemayreachmorepreciseanalysesoffinite element solutions, such as maximum-norm analysis of FEMs [31, 45, 46, 48], error estimatesoffullydiscreteFEMs[30,34,45]andthediscretemaximalLp regularity for parabolic finite element equations [14, 15, 22, 25, 27]. The proof of (1.9) dates back to Schatz et. al. [38], who proved (1.9) with a logarithmicfactorfortheheatequationinatwo-dimensionalsmoothconvexdomain with the linear finite element method. The logarithmic factor was removed in the case r≥4 for N =1,2,3 in [32], and the analysis wasfurther extended to the case 1≤N ≤5 in [4]. Later, a unified approachwas presented in [39] by Schatz et. al., where they proved (1.9) with the Neumann boundary condition for all r ≥ 1 and N ≥1. Theresultwasextendedtothe Dirichletboundaryconditionin[47]forthe linear finite element method. Some other maximum-norm error estimates can be found in [7, 8, 11, 20, 24, 28], and the resolvent estimates can be found in [1, 2]. A related topic is the discrete maximal Lp regularity (when u0 = 0 and 1 < p,q<∞) (1.10) k∂ u k +kA u k ≤C kfk , t h Lp((0,T);Lq) h h Lp((0,T);Lq) p,q Lp((0,T);Lq) which resembles the maximal Lp regularity of the continuous parabolic problem and was proved by Geissert [14, 15]. A straightforwardapplication of (1.10) is the Lp-norm error estimate (1.11) kP u−u k ≤C (kP u0−u0k +kP u−R uk ), h h Lp((0,T);Lq) p,q h h Lq h h Lp((0,T);Lq) where R is the Ritz projection associated with the operator A and P is the L2 h h projection onto the finite element space. MAXIMAL REGULARITY OF FEMS FOR PARABOLIC EQUATIONS 3 All these estimates were established under the assumption that the coefficients a and the domain Ω are smooth enough so that the parabolic Green’s function ij satisfies (1.12) |∂tγ∂xβG(t,x,y)|≤C(t1/2+|x−y|)−(N+2γ+|β|)e−|x−Cyt|2, ∀ 0≤γ ≤2, 0≤|β|≤2. Althoughtheconditiononthecoefficientswasrelaxedtoa ∈C2+α(Ω)in[14],this ij assumption is still too strong for many physical applications. One of the examples is an incompressible miscible flow in porous media [9, 26], where the diffusion- dispersion tensor [a ]N is only a Lipschitz continuous function of the velocity ij i,j=1 field. Inarecentwork[25], the firstauthorproved(1.9)ina smoothdomainunder the assumption a ∈ W1,∞(Ω), together with the estimate (when u0 = 0 and ij 1<p,q <∞) (1.13) ku k ≤C kfk , h Lp((0,T);W1,q) p,q Lp((0,T);W−1,q) which were then applied to the incompressible miscible flow in porous media [27]. Moreover,the problemin a polygonor a polyhedron is of high interestin practical cases, while the inequality (1.12) does not hold in arbitrary convex polygons or polyhedra, and all the analyses of (1.10)-(1.13) are limited to smooth domains so far. For the problem in two-dimensional polygons with constant coefficients, the inequality (1.9) with an extra logarithmic factor was provedin [3, 35, 45] by using the following estimate of the discrete Green’s function Γ : h |Γ (t,x,x )|dx≤C|lnh|. h 0 Z Ω The corresponding results in three-dimensional polyhedra are unknown. More interested is whether these stability estimates hold with the natural regularity a ∈ W1,p(Ω) for some 1 < p < ∞, since such estimates are important for the ij extension of the analysis to a general nonlinear model. This paper focuses on (1.9)-(1.10) and (1.13) in a convex polygon or polyhe- dron with a weaker regularity of the diffusion coefficient. Instead of estimating Γ h directly, we present a more precise estimate for the error function F := Γ −Γ h (see Lemma 2.2) with which the logarithmic factor can be removed (this idea was used in [39]), where Γ is a regularizedGreen’s function. To compensate the lack of pointwise estimate of the second-order derivatives of the Green’s function, we use localW1,∞ estimateandlocalenergyestimatesofthesecond-orderderivatives(see Lemma 4.1). Our main result is the following theorem. Theorem 1.1. Assume that a ∈ W1,N+α(Ω) for some α > 0, satisfying the ij condition (1.3), and assume that Ω is either a convex polygon in R2 or a convex polyhedron in R3. Then (1) the semigroup estimate (1.9) holds, (2) the solution of (1.2) satisfies (1.10) when f ∈Lp((0,T);Lq) and u0 =0, (3) the solution of (1.2) satisfies (1.13) when f ∈Lp((0,T);W−1,q) and u0 =0. Under the assumptions in Theorem 1.1 and assuming that the solution of (1.1) satisfies u∈C(Ω×[0,T]), (1.11) follows immediately from (1.10). The rest of this paper is organized as follows. In section 2, we introduce some notationsandpresentakeylemmabasedonwhichourmaintheoremcanbeproved. In section 3, we present superapproximation results for smoothly truncated finite 4 BUYANGLIANDWEIWEISUN elementfunctions andpresentseveralestimates forthe parabolicGreen’sfunctions under the assumed regularity of the coefficients and the domain. Based on these estimates, we prove our key lemma in section 4. 2. Notations, assumptions and sketch of the proof 2.1. Notations. For any nonnegative integer k and 1 ≤ p ≤ ∞, we let Wk,p(Ω) be the conventional Sobolev space of functions defined in Ω, and let W1,p(Ω) be 0 the subspace of W1,p(Ω) consisting of functions whose traces vanish on ∂Ω. As conventions,wedenotethedualspaceofW1,p(Ω)byW−1,p′(Ω)for1≤p<∞,and 0 denoteHk(Ω):=Wk,2(Ω)andLp(Ω):=W0,p(Ω)foranyintegerk and1≤p≤∞. Let Q := Ω ×(0,T). For any Banach space X and a given T > 0, we let T Lp((0,T);X) be the Bochner spaces equipped with the norm T 1 p kf(t)kp dt , 1≤p<∞, kfkLp((0,T);X) = (cid:18)esZs0sup kf(tX)k (cid:19). p=∞, X t∈(0,T)  To simplify notations, in the following sections, we write Lp, Hk and Wk,p as the abbreviationsof Lp(Ω), Hk(Ω) and Wk,p(Ω), respectively, and denote by (·,·) the inner product in L2(Ω). For any subdomain Q⊂Q , we define T Qt :={x∈Ω: (x,t)∈Q}, kfk :=ess sup kf(·,t)k , L∞,2(Q) L2(Qt) t∈(0,T) 1 p kfk := |f(x,t)|pdxdt , ∀1≤p<∞, Lp(Q) (cid:18)ZZ (cid:19) Q and denote w(t)=w(·,t) for any function w defined on Q . T We assume that Ω is partitioned into quasi-uniform triangular elements τh, l = l 1,··· ,L, with h = max {diamτh}, and let S be a finite element subspace of l l h H1(Ω) consisting of continuous piecewise polynomials of degree r ≥ 1 subject to 0 the triangulation. Let a(x)=(a (x)) be the coefficient matrix and define the ij N×N operators A:H1 →H−1, A :S →S , 0 h h h R :H1 →S , P :L2 →S , h 0 h h h by Aφ,v = a∇φ,∇v for all φ,v ∈H1, 0 (cid:0) (cid:1) (cid:0) (cid:1) A φ ,v = a∇φ ,∇v for all φ ∈S , v ∈S , h h h h h h (cid:0) (cid:1) (cid:0) (cid:1) A R w,v = Aw,v for all w ∈H1 and v ∈S , h h 0 h (cid:0) (cid:1) (cid:0) (cid:1) P φ,v = φ,v for all φ∈L2 and v ∈S . h h (cid:0) (cid:1) (cid:0) (cid:1) Clearly, R is the Ritz projection operator associated to the elliptic operator A h and P is the L2 projection operator onto the finite element space. The following h estimates are useful in this paper. MAXIMAL REGULARITY OF FEMS FOR PARABOLIC EQUATIONS 5 Lemma 2.1. If Ω is a bounded convex domain and a ∈W1,N+α(Ω), N ≥2, then ij we have (2.1) kwk ≤Ck∇·(a∇w)k , ∀w ∈H1, H2 L2 0 (2.2) k∇wk ≤C k∇·(a∇w)k , for any given p>N, ∀w ∈H1, L∞ p Lp 0 and the solution of (1.1) with u0 =0 satisfies (2.3) k∂ uk +kAuk ≤C kfk , t Lp((0,T);Lq) Lp((0,T);Lq) p,q Lp((0,T);Lq) (2.4) k∂ uk +kuk ≤C kfk , t Lp((0,T);W−1,q) Lp((0,T);W1,q) p,q Lp((0,T);W−1,q) for all 1<p,q <∞. In the Lemma above, (2.1) is the standard H2-regularity estimate in convex domains and (2.2) is a simple consequence of the Green’s function estimates given in Theorem 3.3–3.4 of [18], and (2.3)-(2.4) are consequences of the maximal Lp regularity (see Appendix for details). 2.2. Properties of the finite element space and Green’s functions. Forany subdomain D ⊂ Ω, we denote by S (D) the space of functions restricted to the h domain D, and denote by S0(D) the subspace of S (D) consisting of functions h h whichequalzerooutsideD. ForanygivensubsetD ⊂Ω,wedenoteB (D)={x∈ d Ω : dist(x,D) ≤ d} for d > 0. Then there exist positive constants K and κ such that the triangulation and the corresponding finite element space S possess the h following properties (K and κ are independent of the subset D and h). (P0) Quasi-uniformity: For all triangles (or tetrahedron) τh in the partition, the diameter h of τh and l l l the radius ρ of its inscribed ball satisfy l K−1h≤ρ ≤h ≤Kh. l l (P1) Inverse inequality: If D is a union of elements in the partition, then kχ k ≤Kh−(l−k)−(N/q−N/p)kχ k , ∀ χ ∈S , h Wl,p(D) h Wk,q(D) h h for 0≤k ≤l ≤1 and 1≤q ≤p≤∞. (P2) Local approximation and superapproximation: (1) There exists a linear operator I :H1(Ω)→S such that if d≥κh, then h 0 h k kv−I vk ≤K hkd−lkvk , ∀ v ∈Hk∩H1, 1≤k ≤2. h L2(D) Hk−l(Bd(D)) 0 Xl=0 Moreover, if supp(v) ⊂ D, then I v ∈ S0(B (D)). For example, the Cl´ement h h d interpolationoperatordefinedin[5]hasthese properties. Also,the Lagrangeinter- polation operator Π satisfies h kv−Π vk +hk∇(v−Π v)k ≤Kh2k∇2vk , ∀ v ∈H2∩H1. h L2(D) h L2(D) L2(Bd(D)) 0 (2) If d ≥ κh, ω = 0 outside B (D) and |∂βω| ≤ Cd−|β| for all multi-index β, 2d then for any ψ ∈S (B (D)) there exists η ∈S0(B (D)) such that h h 3d h h 3d kωψ −η k ≤Kh1−kd−1kψ k , k =0,1. h h Hk(B3d(D)) h L2(B3d(D)) Furthermore, if ω ≡1 on B (D), then η =ψ on D and d h h kωψ −η k ≤Kh1−kd−1kψ k , k =0,1. h h Hk(B3d(D)) h L2(B3d(D)\D) 6 BUYANGLIANDWEIWEISUN For example, η =Π (ωψ ) has these properties. h h h (P3) Regularized Delta function: For any x ∈ τh, there exists a function δ ∈ C3(Ω) with support in τh such 0 j x0 j that e χ (x )= χ δ dx, ∀χ ∈S , h 0 Zτh h x0 h h j e kδ k ≤Kh−l−N(1−1/p) for 1≤p≤∞, l=0,1,2,3. x0 Wl,p (P4) Discerete Delta function Let δ denote the Dirac Delta function centered at x , i.e. δ (y)ϕ(y)dy = x0 0 Ω x0 ϕ(x ) for any ϕ∈C(Ω). The discrete Delta function P δ satisRfies that 0 h x0 Phδx0(x)≤Kh−Ne−|xK−xh0|, ∀x,x0e∈Ω. The properties (P0)-e(P4) hold for any quasi-uniform partition with those stan- dard finite element spaces and also, have been used in many previous works such as [25, 39, 41, 47]. The proof can be found in the appendix of [41]. Foranelementτh andapointx ∈τh,weletG(t,x,x )betheGreen’sfunction l 0 l 0 of the parabolic equation, defined by (2.5) ∂ G(t,·,x )+AG(t,·,x )=0 for t>0 with G(0,x,x )=δ (x), t 0 0 0 x0 The regularized Green’s function Γ(t,x,x ) is defined by 0 (2.6) ∂ Γ(·,·,x )+AΓ(·,·,x )=0 for t>0 with Γ(0,·,x )=δ , t 0 0 0 x0 where δ is given in (P2), and the discrete Green’s function Γ (·,·,ex ) is defined x0 h 0 by e (2.7) ∂ Γ (·,·,x )+A Γ (·,·,x )=0 for t>0 with Γ (0,·,x )=P δ . t h 0 h h 0 h 0 h x0 The functions G(t,x,x0) and Γh(t,x,x0) are symmetric with respect to xeand x0. By the fundamental estimates of parabolic equations, there exists a positive constant C such that ([12], Theorem 1.6; note that the Green’s function in the domain Ω is less than the Green’s function in RN) (2.8) |G(t,x,y)|≤C(t1/2+|x−y|)−Ne−|x−Cyt|2. By estimating Γ(t,x,x ) = G(t,x,y)δ (y)dy, it is easy to see that (2.8) also 0 Ω x0 holds when G is replaced byRΓ and when max(t1/2,|x−y|)≥2h. e 2.3. Decomposition of the domain Ω×(0,T). Here we present some further notationsonadyadicdecompositionofthedomainΩ×(0,T),whichwereintroduced in [39] and also used in many other articles [14, 24, 25, 47]. Let R be the smallest 0 distance between a corner and a closed face which does not contained this corner. For the given polygon/polyhedron Ω, there exists a positive constant K ≥ 0 max(1,R ) (which depends on the interior angle of the edges/corners of Ω) such 0 that (1) if z is a point in the interior of Ω and B (z ) intersects a face of Ω, then 0 ρ 0 B (z )⊂B (z ) for some z which is on a face of Ω; ρ 0 2ρ 1 1 (2) if z is on a face of Ω and B (z ) intersects another face, then B (z ) ⊂ 1 ρ 1 ρ 1 B (z ) for some z which is on an edge of Ω; ρK0 2 2 MAXIMAL REGULARITY OF FEMS FOR PARABOLIC EQUATIONS 7 (3) if z is on an edge of Ω and B (z ) intersects another face which does not 2 ρ 2 contain this edge, then B (z )⊂B (z ) for some z which is a corner of Ω. ρ 2 ρK0 3 3 Foranyintegerj ≥1,wedefined =2−j−3R K−2. Foragivenx ∈Ω,weletJ j 0 0 0 ∗ beanintegersatisfyingdJ∗ =2−J∗−3R0K0−2 =C∗hwithC∗ ≥max(10,10κ,R0K0−2/8) tobedeterminedlater. Thus,J =log [R K−2/(8C h)]≤log (2+1/h)andJ >1 ∗ 2 0 0 ∗ 2 ∗ when h<R K−2/(16C ). Let 0 0 ∗ Q (x )={(x,t)∈Ω :max(|x−x |,t1/2)≤d }, ∗ 0 T 0 J∗ Ω (x )={x∈Ω:|x−x |≤d }, ∗ 0 0 J∗ Q (x )={(x,t)∈Ω :d ≤max(|x−x |,t1/2)≤2d }, j 0 T j 0 j Ω (x )={x∈Ω:d ≤|x−x |≤2d }, j 0 j 0 j D (x )={x∈Ω:|x−x |≤2d } j 0 0 j for j ≥1; see Figure 1. Figure 1. Illustration of the subdomains Q , Ω and D . j j j Forj =0wedefineQ (x )=Q \Q (x )andΩ (x )=Ω\Ω (x ),andforj <0 0 0 T 1 0 0 0 1 0 we simplify define Q (x )=Ω (x )=∅. For all j ≥1 we define j 0 j 0 Ω′(x )=Ω (x )∪Ω (x )∪Ω (x ), j 0 j−1 0 j 0 j+1 0 Ω′′(x )=Ω (x )∪Ω′(x )∪Ω (x ), j 0 j−2 0 j 0 j+2 0 Ω′′′(x )=Ω (x )∪Ω′′(x )∪Ω (x ), j 0 j−2 0 j 0 j+2 0 Q′(x )=Q (x )∪Q (x )∪Q (x ), j 0 j−1 0 j 0 j+1 0 Q′′(x )=Q (x )∪Q′(x )∪Q (x ), j 0 j−2 0 j 0 j+2 0 Q′′′(x )=Q (x )∪Q′′(x )∪Q (x ), j 0 j−2 0 j 0 j+2 0 D′(x )=D (x )∪D (x ), j 0 j−1 0 j 0 D′′(x )=D (x )∪D′(x ), j 0 j−2 0 j 0 D′′′(x )=D (x )∪D′′(x ). j 0 j−3 0 j 0 Then we have J∗ J∗ Ω = Q (x ) ∪Q (x ) and Ω= Ω (x ) ∪Ω (x ), T j 0 ∗ 0 j 0 ∗ 0 j[=0 j[=0 8 BUYANGLIANDWEIWEISUN WerefertoQ (x )asthe“innermost”set. Weshallwrite whentheinnermost ∗ 0 ∗,j setisincludedand j whenitisnot. Whenx0 isfixed,ifPthereisnoambiguity,we simply write Qj =PQj(x0), Q′j =Q′j(x0), Q′j′ =Q′j′(x0), Ωj =Ωj(x0), Ω′j =Ω′j(x0) and Ω′′ =Ω′′(x ). j j 0 In the rest of this paper, we denote by C a generic positive constant, which will be independent of h, x , and the undetermined constant C until it is determined 0 ∗ at the end of section 4.2. 2.4. Proof of Theorem 1.1. The keys to the proof of Theorem 1.1 are several more precise estimates of the Green’s functions. Let F(t)=Γ (t)−Γ(t). Then for h any x ∈Ω, we have 0 (E (t)v )(x )=(F(t),v )+(Γ(t),v ) h h 0 h h t (2.9) = (∂ F(s),v )ds+(F(0),v )+(Γ(t),v ), t h h h Z 0 (t∂ E (t)v )(x )=(t∂ F(t),v )+(t∂ Γ(t),v ) t h h 0 t h t h t (2.10) = (s∂ F(s)+∂ F(s),v )ds+(t∂ Γ(t),v ), ss s h t h Z 0 withkF(0)k =kδ −P δ k ≤C (accordingto (P3)and (P4)). Moreover,by L1 x0 h x0 L1 the analyticity of the continuous parabolic semigroup on L1(Ω), we have e e kΓ(t)k +tk∂ Γ(t)k ≤CkΓ(0)k =Ckδ k ≤C. L1 t L1 L1 x0 L1 We present some estimates of these Green’s functionsein the following lemma. The proof of the lemma is the major work of this paper and will be given in the next two sections. Lemma 2.2. Under the assumptions of Theorem 1.1, we have ∞ (2.11) |∂ F(t,x,x ) + t∂ F(t,x,x )| dxdt≤C, t 0 tt 0 Z Z 0 Ω(cid:0) (cid:12) (cid:12) (cid:1) (2.12) |∇∂ G(t,x,x )|≤Cm(cid:12)ax((cid:12)t1/2,|x−x |)−3−N for (x,t)∈Ω×(0,1). t 0 0 The estimates in Lemma 2.2 were proved in [39] for parabolic equations with the Neumannboundaryconditionandin[47]forthe Dirichletboundarycondition. However,theirproofsareonlyvalidforsmoothcoefficientsandsmoothdomains(as clearly mentioned in their papers). Later, these estimates were proved in [25] for parabolicequationsinsmoothdomainsofarbitrarydimensionsundertheNeumann boundary condition with Lipschitz continuous coefficients. Here we are concerned with the problemin a convexpolyhedronin two orthree dimensionalspaces under the Dirichlet boundary condition with a ∈W1,N+α. ij Proof of Theorem 1.1: Firstly,from(2.9)-(2.10)we seethat(1.9)is aconsequence of (2.11). Secondly, we can view E (t) as an analytic semigroup on Lq(Ω), defined by h (E (t)v)(x ):= Γ (t,x,x )v(x)dx, h 0 h 0 Z Ω whose generator is A P . From [49, Theorem 4.2] and [50, Lemma 4.c] (with a h h duality argument for the case q ≥ 2) we know that the maximal Lp regularity MAXIMAL REGULARITY OF FEMS FOR PARABOLIC EQUATIONS 9 (1.10) holds if the following maximal ergodic estimate holds: 1 t (2.13) sup |E (s)|vds ≤Ckvk , ∀v ∈Lq(Ω), (cid:13)(cid:13)t>0 t Z0 h (cid:13)(cid:13)Lq Lq (cid:13) (cid:13) where (cid:13) (cid:13) (|E (s)|v)(x ):= |Γ (t,x,x )|v(x)dx. h 0 h 0 Z Ω Let G (t,x,x ) be a truncated Green’s function which is symmetric with respect tr 0 to x and x and satisfies G (t,x,x ) = G(t,x,x ) when (x,t) is outside Q (x ) 0 tr 0 0 ∗ 0 (see [25, Section 4.2] on its construction). Then we have (assuming that τh is the 0 triangle/tetrahedronwhich contains x ) 0 |∂ Γ(t,x,x )−∂ G (t,x,x )|dxdt t 0 t tr 0 ZZ [Ω×(0,∞)]\Q∗(x0) = ∂ G(t,x,y)δ (y)dy−∂ G(t,x,x ) dxdt ZZ (cid:12)Z t x0 t 0 (cid:12) [Ω×(0,1)]\Q∗(x0)(cid:12) Ω (cid:12) (cid:12) e (cid:12) (cid:12) (cid:12) + |∂ Γ(t,x,x )−∂ G(t,x,x )|dxdt t 0 t 0 ZZ Ω×(1,∞) ≤Ch sup ∇ ∂ G(t,x,y) dxdt y t ZZ [Ω×(0,1)]\Q∗(x0)y∈τ0h(cid:12) (cid:12) (cid:12) (cid:12) ∞ +C t−1(kΓ(t/2,·,x )k +kG(t/2,·,x )k )dt [by semigroup estimate] 0 L1 0 L1 Z 1 ∞ =Ch sup ∇ ∂ G(t,x,y) dxdt+C t−1−N/2dt [see (2.8)] y t Xj ZZQj(x0)(y,t)∈Q′j(x)(cid:12) (cid:12) Z1 (cid:12) (cid:12) h ≤C +C [see (2.12)] d Xj j ≤C. By using energy estimates, it is easy to see (|∂ Γ(t,x,x )|+|∂ G (t,x,x )|)dxdt t 0 t tr 0 ZZ Q∗(x0) ≤dN/2+1(k∂ Γ(·,·,x )k +k∂ G (·,·,x )k )≤CN/2+1, J∗ t 0 L2(Ω×(0,1)) t tr 0 L2(Ω×(0,1)) ∗ wherethe constantC willbe determined atthe endofSection4. Then(2.11) and ∗ the last two inequalities imply ∞ |∂ Γ (t,x,x )−∂ G (t,x,x )|dxdt≤C. t h 0 t tr 0 Z Z 0 Ω Inotherwords,the symmetrickernelK(x,y):= ∞|∂ Γ (t,x,y)−∂ G∗(t,x,y)|dt 0 t h t tr satisfies R sup K(x,y)dx+sup K(x,y)dy ≤C, Z Z y∈Ω Ω x∈Ω Ω andtherefore,Schur’slemma implies thatthe correspondingoperatorM , defined K by M v(x) = K(x,y)v(y)dy, is bounded on Lq(Ω) for all 1 ≤ q ≤ ∞. Let K Ω R 10 BUYANGLIANDWEIWEISUN E∗(t)v(x) = G∗(t,x,y)v(y)dy and note that E∗(t)v(x) ≤ E(t)|v|(x) (because tr Ω tr tr G∗(t,x,y)≤GR(t,x,y)). We have tr sup(|E (t)P |v)(x) h h t>0 ≤sup(|E (t)P |v−E∗(t)v)(x)+sup(E∗(t)v)(x) h h tr tr t>0 t>0 ≤sup |E (t)P −E∗(t)||v| (x)+sup(E∗(t)v)(x) h h tr tr t>0(cid:0) (cid:1) t>0 t =sup (|P δ |,|v|)+ |∂ Γ (s,x,y)−∂ G∗(s,x,y)||v(y)|dyds t>0(cid:12)(cid:12) h x Z0 ZΩ t h t tr (cid:12)(cid:12) +su(cid:12)p(E∗e(t)|v|)(x) (cid:12) (cid:12) tr (cid:12) t>0 ≤(|P δ |,|v|)+(M |v|)(x)+sup(E(t)|v|)(x) h x K t>0 e where ksupE(t)|v|k ≤C kvk , ∀ 1<q <∞, Lq q Lq t>0 is asimple consequenceofthe Gaussianestimate (2.8) (Corollary2.1.12andTheo- rem2.1.6of [16]). This provesa strongerestimate than (2.13). The proof of(1.10) is completed. Finally, (1.1)-(1.2) imply that the error e = P u−u satisfies the equation h h h (when u0 =u0 =0) h (2.14) ∂ (A−1e )+A (A−1e )=P u−R u. t h h h h h h h By applying (1.10) to the equation above, we obtain (2.15) ke k ≤C kP u−R uk ≤C hkuk h Lp((0,T);Lq) p,q h h Lp((0,T);Lq) p,q Lp((0,T);W1,q) for1<p<∞and2≤q <∞,wherewehaveusedtheinequalitykP u−R uk ≤ h h Lq C hkuk ,whichonlyholdsfor2≤q <∞inconvexpolygons/polyhedra. Then, q W1,q by using an inverse inequality and (2.4), we have ke k ≤Ch−1ke k h Lp((0,T);W1,q) h Lp((0,T);Lq) ≤C kuk p,q Lp((0,T);W1,q) ≤C kfk , p,q Lp((0,T);W−1,q) which implies (1.13) for the case 1<p<∞ and 2≤q <∞. In the case 1<p<∞ and 1<q ≤2, we define ~g =∇∆−1P f and express the h solution of (1.2) by (when u0 =0) h t ∇u =L ~g := ∇A−1/2A E (t−s)A−1/2∇·~g(s)ds. h h Z h h h h 0 InordertoprovetheboundednessoftheoperatorL onLp((0,T);(Lq)N),weonly h need to prove the boundedness of its dual operator L′ on Lp′((0,T);(Lq′)N). It is h easy to see that T T T (L ~g,~η)dt= ~g, ∇A−1/2A E (t−s)A−1/2∇·~η(t)dt ds, Z h Z (cid:18) Z h h h h (cid:19) 0 0 s which gives T L′~η := ∇A−1/2A E (t−s)A−1/2∇·~η(s)ds. h Z h h h h s